متن درس اول زبان تخصصی برای دانشجویان کارشناسی ادبیات فارسی

متن درس اول زبان تخصصی برای دانشجویان کارشناسی ادبیات فارسی

 

CHAPTER 1

 

A.      The World of Literature

 

     The world of literature is a world where there is no reality except that of the human imagination. We see a great deal that reminds us vividly of the life we know. But in that very vividness there's something unreal. We can understand more clearly with pictures, perhaps. There are trick picturcs-trompel'oeid, the French call them-where the resemblance to life is very strong. An American painter of this school played a joke on his bitchy wife by painting one of her best napkins so expertly that she grabbed at the canvas trying to pull it off. But a painting as realistic as that isn't a reality but an illusion: it has the glittering unnatural clarity of a hallucination. The real realities, so to speak, are things that don't remind us directly of our own experience, but are such things as the wrath of Achilles or the jealousy of Othello, which are bigger and more intense experiences than anything we can reach-except in our imagination, which is what we're reaching with.

Sometimes, as in the happy endings of comedies, or in the ideal world of romances we seem to be looking at a pleasanter world than we ordinarily know. Sometimes, as in tragedy and satire, we seem to be looking at a world more devoted to suffering or absurdity than we ordinarily know. In literature we always seem to be looking either up or down. It's the vertical perspective that's important, not the horizontal one that looks out to life. Of course. In the greatest work of literature we get both the up and down views, often at the same time as different aspects of one.

 

B.   Translate the following passage into fluent Persian. Write your translation in the space provided.

 

     There are many signs of inexperienced reader. He makes fixed demands of every story he reads, and he feels frustrated and disappointed unless these demands are satisfied. Often he sticks to one type of subject matter. Instead of being receptive to any story that puts human beings in human situations. He reads only sports stories, western stories, love stories, or crime stories. If he is willing to accept a wider range of experience, he still wishes every story to conform at bottom to several strict though perhaps unconsciously formulated expectations.

 

 

 

زبان فنی برق صنعتی

 

AC جزوه زبان فنی برق – بخش

از فصل 1 تا آخر فصل 7

Chapter 1

BASIC AC THEORY

What is alternating current (AC)?

• DC stands for “Direct Current,” meaning voltage or current that maintains constant polarity or direction, respectively, over time.
• AC stands for “Alternating Current,” meaning voltage or current that changes polarity or direction, respectively, over time.
• AC electromechanical generators, known as alternators, are of simpler construction than DC electromechanical generators.
• AC and DC motor design follows respective generator design principles very closely.
• A transformer is a pair of mutually-inductive coils used to convey AC power from one coil to the other. Often, the number of turns in each coil is set to create a voltage increase or decrease from the powered (primary) coil to the unpowered (secondary) coil.
• Secondary voltage = Primary voltage (secondary turns / primary turns)
• Secondary current = Primary current (primary turns / secondary turns)

AC waveforms

• AC produced by an electromechanical alternator follows the graphical shape of a sine wave.
• One cycle of a wave is one complete evolution of its shape until the point that it is ready to repeat itself.
• The period of a wave is the amount of time it takes to complete one cycle.
• Frequency is the number of complete cycles that a wave completes in a given amount of time. Usually measured in Hertz (Hz), 1 Hz being equal to one complete wave cycle per second.
• Frequency = 1/(period in seconds)

Measurements of AC magnitude

• The amplitude of an AC waveform is its height as depicted on a graph over time. An amplitude measurement can take the form of peak, peak-to-peak, average, or RMS quantity.
• Peak amplitude is the height of an AC waveform as measured from the zero mark to the highest positive or lowest negative point on a graph. Also known as the crest amplitude of a wave.
• Peak-to-peak amplitude is the total height of an AC waveform as measured from maximum positive to maximum negative peaks on a graph. Often abbreviated as “P-P”.
• Average amplitude is the mathematical “mean” of all a waveform's points over the period of one cycle. Technically, the average amplitude of any waveform with equal-area portions above and below the “zero” line on a graph is zero. However, as a practical measure of amplitude, a waveform's average value is often calculated as the mathematical mean of all the points' absolute values (taking all the negative values and considering them as positive). For a sine wave, the average value so calculated is approximately 0.637 of its peak value.
• “RMS” stands for Root Mean Square, and is a way of expressing an AC quantity of voltage or current in terms functionally equivalent to DC. For example, 10 volts AC RMS is the amount of voltage that would produce the same amount of heat dissipation across a resistor of given value as a 10 volt DC power supply. Also known as the “equivalent” or “DC equivalent” value of an AC voltage or current. For a sine wave, the RMS value is approximately 0.707 of its peak value.
• The crest factor of an AC waveform is the ratio of its peak (crest) to its RMS value.
• The form factor of an AC waveform is the ratio of its RMS value to its average value.
• Analog, electromechanical meter movements respond proportionally to the average value of an AC voltage or current. When RMS indication is desired, the meter's calibration must be “skewed” accordingly. This means that the accuracy of an electromechanical meter's RMS indication is dependent on the purity of the waveform: whether it is the exact same waveshape as the waveform used in calibrating.

Simple AC circuit calculations

• All the old rules and laws of DC (Kirchhoff's Voltage and Current Laws, Ohm's Law) still hold true for AC. However, with more complex circuits, we may need to represent the AC quantities in more complex form. More on this later, I promise!
• The “table” method of organizing circuit values is still a valid analysis tool for AC circuits.

AC phase

• Phase shift is where two or more waveforms are out of step with each other.
• The amount of phase shift between two waves can be expressed in terms of degrees, as defined by the degree units on the horizontal axis of the waveform graph used in plotting the trigonometric sine function.
• A leading waveform is defined as one waveform that is ahead of another in its evolution. A lagging waveform is one that is behind another. Example:
• Calculations for AC circuit analysis must take into consideration both amplitude and phase shift of voltage and current waveforms to be completely accurate. This requires the use of a mathematical system called complex numbers.

Principles of radio

• James Maxwell discovered that changing electric fields produce perpendicular magnetic fields, and vice versa, even in empty space.
• A twin set of electric and magnetic fields, oscillating at right angles to each other and traveling at the speed of light, constitutes an electromagnetic wave.
• An antenna is a device made of wire, designed to radiate a changing electric field or changing magnetic field when powered by a high-frequency AC source, or intercept an electromagnetic field and convert it to an AC voltage or current.
• The dipole antenna consists of two pieces of wire (not touching), primarily generating an electric field when energized, and secondarily producing a magnetic field in space.
• The loop antenna consists of a loop of wire, primarily generating a magnetic field when energized, and secondarily producing an electric field in space.

Chapter 2

COMPLEX NUMBERS

Introduction

• A scalar number is the type of mathematical object that people are used to using in everyday life: a one-dimensional quantity like temperature, length, weight, etc.
• A complex number is a mathematical quantity representing two dimensions of magnitude and direction.
• A vector is a graphical representation of a complex number. It looks like an arrow, with a starting point, a tip, a definite length, and a definite direction. Sometimes the word phasor is used in electrical applications where the angle of the vector represents phase shift between waveforms.

Vectors and AC waveforms

• When used to describe an AC quantity, the length of a vector represents the amplitude of the wave while the angle of a vector represents the phase angle of the wave relative to some other (reference) waveform.

Simple vector addition

• DC voltages can only either directly aid or directly oppose each other when connected in series. AC voltages may aid or oppose to any degree depending on the phase shift between them.

Polar and rectangular notation

• Polar notation denotes a complex number in terms of its vector's length and angular direction from the starting point. Example: fly 45 miles ∠ 203^o (West by Southwest).
• Rectangular notation denotes a complex number in terms of its horizontal and vertical dimensions. Example: drive 41 miles West, then turn and drive 18 miles South.
• In rectangular notation, the first quantity is the “real” component (horizontal dimension of vector) and the second quantity is the “imaginary” component (vertical dimension of vector). The imaginary component is preceded by a lower-case “j,” sometimes called the j operator.
• Both polar and rectangular forms of notation for a complex number can be related graphically in the form of a right triangle, with the hypotenuse representing the vector itself (polar form: hypotenuse length = magnitude; angle with respect to horizontal side = angle), the horizontal side representing the rectangular “real” component, and the vertical side representing the rectangular “imaginary” component.

Complex number arithmetic

• To add complex numbers in rectangular form, add the real components and add the imaginary components. Subtraction is similar.
• To multiply complex numbers in polar form, multiply the magnitudes and add the angles. To divide, divide the magnitudes and subtract one angle from the other.

More on AC "polarity"

• Polarity markings are sometimes given to AC voltages in circuit schematics in order to provide a frame of reference for their phase angles.

Some examples with AC circuits

• All the laws and rules of DC circuits apply to AC circuits, with the exception of power calculations (Joule's Law), so long as all values are expressed and manipulated in complex form, and all voltages and currents are at the same frequency.
• When reversing the direction of a vector (equivalent to reversing the polarity of an AC voltage source in relation to other voltage sources), it can be expressed in either of two different ways: adding 180^o to the angle, or reversing the sign of the magnitude.
• Meter measurements in an AC circuit correspond to the polar magnitudes of calculated values. Rectangular expressions of complex quantities in an AC circuit have no direct, empirical equivalent, although they are convenient for performing addition and subtraction, as Kirchhoff's Voltage and Current Laws require.

Chapter 3

REACTANCE AND IMPEDANCE -- INDUCTIVE

 

AC resistor circuits

• Inductive reactance is the opposition that an inductor offers to alternating current due to its phase-shifted storage and release of energy in its magnetic field. Reactance is symbolized by the capital letter “X” and is measured in ohms just like resistance (R).
• Inductive reactance can be calculated using this formula: X_L = 2πfL
• The angular velocity of an AC circuit is another way of expressing its frequency, in units of electrical radians per second instead of cycles per second. It is symbolized by the lower-case Greek letter “omega,” or ω.
• Inductive reactance increases with increasing frequency. In other words, the higher the frequency, the more it opposes the AC flow of electrons.

Series resistor-inductor circuits

• Impedance is the total measure of opposition to electric current and is the complex (vector) sum of (“real”) resistance and (“imaginary”) reactance. It is symbolized by the letter “Z” and measured in ohms, just like resistance (R) and reactance (X).
• Impedances (Z) are managed just like resistances (R) in series circuit analysis: series impedances add to form the total impedance. Just be sure to perform all calculations in complex (not scalar) form! Z_Total = Z₁ + Z₂ + . . . Z_n
• A purely resistive impedance will always have a phase angle of exactly 0^o (Z_R = R Ω ∠ 0^o).
• A purely inductive impedance will always have a phase angle of exactly +90^o (Z_L = X_L Ω ∠ 90^o).
• Ohm's Law for AC circuits: E = IZ ; I = E/Z ; Z = E/I
• When resistors and inductors are mixed together in circuits, the total impedance will have a phase angle somewhere between 0^o and +90^o. The circuit current will have a phase angle somewhere between 0^o and -90^o.
• Series AC circuits exhibit the same fundamental properties as series DC circuits: current is uniform throughout the circuit, voltage drops add to form the total voltage, and impedances add to form the total impedance.

Parallel resistor-inductor circuits

• Impedances (Z) are managed just like resistances (R) in parallel circuit analysis: parallel impedances diminish to form the total impedance, using the reciprocal formula. Just be sure to perform all calculations in complex (not scalar) form! Z_Total = 1/(1/Z₁ + 1/Z₂ + . . . 1/Z_n)
• Ohm's Law for AC circuits: E = IZ ; I = E/Z ; Z = E/I
• When resistors and inductors are mixed together in parallel circuits (just as in series circuits), the total impedance will have a phase angle somewhere between 0^o and +90^o. The circuit current will have a phase angle somewhere between 0^o and -90^o.
• Parallel AC circuits exhibit the same fundamental properties as parallel DC circuits: voltage is uniform throughout the circuit, branch currents add to form the total current, and impedances diminish (through the reciprocal formula) to form the total impedance.

Inductor quirks

In an ideal case, an inductor acts as a purely reactive device. That is, its opposition to AC current is strictly based on inductive reaction to changes in current, and not electron friction as is the case with resistive components. However, inductors are not quite so pure in their reactive behavior. To begin with, they're made of wire, and we know that all wire possesses some measurable amount of resistance (unless its superconducting wire). This built-in resistance acts as though it were connected in series with the perfect inductance of the coil.

Chapter 4

REACTANCE AND IMPEDANCE -- CAPACITIVE

AC resistor circuits

• Capacitive reactance is the opposition that a capacitor offers to alternating current due to its phase-shifted storage and release of energy in its electric field. Reactance is symbolized by the capital letter “X” and is measured in ohms just like resistance (R).
• Capacitive reactance can be calculated using this formula: X_C = 1/(2πfC)
• Capacitive reactance decreases with increasing frequency. In other words, the higher the frequency, the less it opposes (the more it “conducts”) the AC flow of electrons.

Series resistor-capacitor circuits

• Impedance is the total measure of opposition to electric current and is the complex (vector) sum of (“real”) resistance and (“imaginary”) reactance.
• Impedances (Z) are managed just like resistances (R) in series circuit analysis: series impedances add to form the total impedance. Just be sure to perform all calculations in complex (not scalar) form! Z_Total = Z₁ + Z₂ + . . . Z_n
• Please note that impedances always add in series, regardless of what type of components comprise the impedances. That is, resistive impedance, inductive impedance, and capacitive impedance are to be treated the same way mathematically.
• A purely resistive impedance will always have a phase angle of exactly 0^o (Z_R = R Ω ∠ 0^o).
• A purely capacitive impedance will always have a phase angle of exactly -90^o (Z_C = X_C Ω ∠ -90^o).
• Ohm's Law for AC circuits: E = IZ ; I = E/Z ; Z = E/I
• When resistors and capacitors are mixed together in circuits, the total impedance will have a phase angle somewhere between 0^o and -90^o.
• Series AC circuits exhibit the same fundamental properties as series DC circuits: current is uniform throughout the circuit, voltage drops add to form the total voltage, and impedances add to form the total impedance.

Parallel resistor-capacitor circuits

• Impedances (Z) are managed just like resistances (R) in parallel circuit analysis: parallel impedances diminish to form the total impedance, using the reciprocal formula. Just be sure to perform all calculations in complex (not scalar) form! Z_Total = 1/(1/Z₁ + 1/Z₂ + . . . 1/Z_n)
• Ohm's Law for AC circuits: E = IZ ; I = E/Z ; Z = E/I
• When resistors and capacitors are mixed together in parallel circuits (just as in series circuits), the total impedance will have a phase angle somewhere between 0^o and -90^o. The circuit current will have a phase angle somewhere between 0^o and +90^o.
• Parallel AC circuits exhibit the same fundamental properties as parallel DC circuits: voltage is uniform throughout the circuit, branch currents add to form the total current, and impedances diminish (through the reciprocal formula) to form the total impedance.

Capacitor quirks

As with inductors, the ideal capacitor is a purely reactive device, containing absolutely zero resistive (power dissipative) effects. In the real world, of course, nothing is so perfect. However, capacitors have the virtue of generally being purer reactive components than inductors. It is a lot easier to design and construct a capacitor with low internal series resistance than it is to do the same with an inductor. The practical result of this is that real capacitors typically have impedance phase angles more closely approaching 90^o (actually, -90^o) than inductors. Consequently, they will tend to dissipate less power than an equivalent inductor.

Capacitors also tend to be smaller and lighter weight than their equivalent inductor counterparts, and since their electric fields are almost totally contained between their plates (unlike inductors, whose magnetic fields naturally tend to extend beyond the dimensions of the core), they are less prone to transmitting or receiving electromagnetic “noise” to/from other components. For these reasons, circuit designers tend to favor capacitors over inductors wherever a design permits either alternative.

Capacitors with significant resistive effects are said to be lossy, in reference to their tendency to dissipate (“lose”) power like a resistor. The source of capacitor loss is usually the dielectric material rather than any wire resistance, as wire length in a capacitor is very minimal.

Dielectric materials tend to react to changing electric fields by producing heat. This heating effect represents a loss in power, and is equivalent to resistance in the circuit. The effect is more pronounced at higher frequencies and in fact can be so extreme that it is sometimes exploited in manufacturing processes to heat insulating materials like plastic! The plastic object to be heated is placed between two metal plates, connected to a source of high-frequency AC voltage. Temperature is controlled by varying the voltage or frequency of the source, and the plates never have to contact the object being heated.

This effect is undesirable for capacitors where we expect the component to behave as a purely reactive circuit element. One of the ways to mitigate the effect of dielectric “loss” is to choose a dielectric material less susceptible to the effect. Not all dielectric materials are equally “lossy.”

Chapter 5

REACTANCE AND IMPEDANCE -- R, L, AND C

 

Review of R, X, and Z

Before we begin to explore the effects of resistors, inductors, and capacitors connected together in the same AC circuits, let's briefly review some basic terms and facts.

Resistance is essentially friction against the motion of electrons. It is present in all conductors to some extent (except superconductors!), most notably in resistors. When alternating current goes through a resistance, a voltage drop is produced that is in-phase with the current. Resistance is mathematically symbolized by the letter “R” and is measured in the unit of ohms (Ω).

Reactance is essentially inertia against the motion of electrons. It is present anywhere electric or magnetic fields are developed in proportion to applied voltage or current, respectively; but most notably in capacitors and inductors. When alternating current goes through a pure reactance, a voltage drop is produced that is 90^o out of phase with the current. Reactance is mathematically symbolized by the letter “X” and is measured in the unit of ohms (Ω).

Impedance is a comprehensive expression of any and all forms of opposition to electron flow, including both resistance and reactance. It is present in all circuits, and in all components. When alternating current goes through an impedance, a voltage drop is produced that is somewhere between 0^o and 90^o out of phase with the current. Impedance is mathematically symbolized by the letter “Z” and is measured in the unit of ohms (Ω), in complex form.

• Impedances of any kind add in series: Z_Total = Z₁ + Z₂ + . . . Z_n
• Although impedances add in series, the total impedance for a circuit containing both inductance and capacitance may be less than one or more of the individual impedances, because series inductive and capacitive impedances tend to cancel each other out. This may lead to voltage drops across components exceeding the supply voltage!
• All rules and laws of DC circuits apply to AC circuits, so long as values are expressed in complex form rather than scalar. The only exception to this principle is the calculation of power, which is very different for AC.

Parallel R, L, and C

• Analysis of series-parallel AC circuits is much the same as series-parallel DC circuits. The only substantive difference is that all figures and calculations are in complex (not scalar) form.
• It is important to remember that before series-parallel reduction (simplification) can begin, you must determine the impedance (Z) of every resistor, inductor, and capacitor. That way, all component values will be expressed in common terms (Z) instead of an incompatible mix of resistance (R), inductance (L), and capacitance (C).

Susceptance and Admittance

In the study of DC circuits, the student of electricity comes across a term meaning the opposite of resistance: conductance. It is a useful term when exploring the mathematical formula for parallel resistances: R_parallel = 1 / (1/R₁ + 1/R₂ + . . . 1/R_n). Unlike resistance, which diminishes as more parallel components are included in the circuit, conductance simply adds. Mathematically, conductance is the reciprocal of resistance, and each 1/R term in the “parallel resistance formula” is actually a conductance.

Whereas the term “resistance” denotes the amount of opposition to flowing electrons in a circuit, “conductance” represents the ease of which electrons may flow. Resistance is the measure of how much a circuit resists current, while conductance is the measure of how much a circuit conducts current. Conductance used to be measured in the unit of mhos, or “ohms” spelled backward. Now, the proper unit of measurement is Siemens. When symbolized in a mathematical formula, the proper letter to use for conductance is “G”.

Reactive components such as inductors and capacitors oppose the flow of electrons with respect to time, rather than with a constant, unchanging friction as resistors do. We call this time-based opposition, reactance, and like resistance we also measure it in the unit of ohms.

As conductance is the complement of resistance, there is also a complementary expression of reactance, called susceptance. Mathematically, it is equal to 1/X, the reciprocal of reactance. Like conductance, it used to be measured in the unit of mhos, but now is measured in Siemens. Its mathematical symbol is “B”, unfortunately the same symbol used to represent magnetic flux density.

The terms “reactance” and “susceptance” have a certain linguistic logic to them, just like resistance and conductance. While reactance is the measure of how much a circuit reacts against change in current over time, susceptance is the measure of how much a circuit is susceptible to conducting a changing current.

If one were tasked with determining the total effect of several parallel-connected, pure reactances, one could convert each reactance (X) to a susceptance (B), then add susceptances rather than diminish reactances: X_parallel = 1/(1/X₁ + 1/X₂ + . . . 1/X_n). Like conductances (G), susceptances (B) add in parallel and diminish in series. Also like conductance, susceptance is a scalar quantity.

When resistive and reactive components are interconnected, their combined effects can no longer be analyzed with scalar quantities of resistance (R) and reactance (X). Likewise, figures of conductance (G) and susceptance (B) are most useful in circuits where the two types of opposition are not mixed, i.e. either a purely resistive (conductive) circuit, or a purely reactive (susceptive) circuit. In order to express and quantify the effects of mixed resistive and reactive components, we had to have a new term: impedance, measured in ohms and symbolized by the letter “Z”.

To be consistent, we need a complementary measure representing the reciprocal of impedance. The name for this measure is admittance. Admittance is measured in (guess what?) the unit of Siemens, and its symbol is “Y”. Like impedance, admittance is a complex quantity rather than scalar. Again, we see a certain logic to the naming of this new term: while impedance is a measure of how much alternating current is impeded in a circuit, admittance is a measure of how much current is admitted.

Given a scientific calculator capable of handling complex number arithmetic in both polar and rectangular forms, you may never have to work with figures of susceptance (B) or admittance (Y). Be aware, though, of their existence and their meanings.

Summary

With the notable exception of calculations for power (P), all AC circuit calculations are based on the same general principles as calculations for DC circuits. The only significant difference is that fact that AC calculations use complex quantities while DC calculations use scalar quantities. Ohm's Law, Kirchhoff's Laws, and even the network theorems learned in DC still hold true for AC when voltage, current, and impedance are all expressed with complex numbers. The same troubleshooting strategies applied toward DC circuits also hold for AC, although AC can certainly be more difficult to work with due to phase angles which aren't registered by a handheld multimeter.

Power is another subject altogether, and will be covered in its own chapter in this book. Because power in a reactive circuit is both absorbed and released -- not just dissipated as it is with resistors -- its mathematical handling requires a more direct application of trigonometry to solve.

When faced with analyzing an AC circuit, the first step in analysis is to convert all resistor, inductor, and capacitor component values into impedances (Z), based on the frequency of the power source. After that, proceed with the same steps and strategies learned for analyzing DC circuits, using the “new” form of Ohm's Law: E=IZ ; I=E/Z ; and Z=E/I

Remember that only the calculated figures expressed in polar form apply directly to empirical measurements of voltage and current. Rectangular notation is merely a useful tool for us to add and subtract complex quantities together. Polar notation, where the magnitude (length of vector) directly relates to the magnitude of the voltage or current measured, and the angle directly relates to the phase shift in degrees, is the most practical way to express complex quantities for circuit analysis.

Chapter 6

RESONANCE

 

An electric pendulum

• A capacitor and inductor directly connected together form something called a tank circuit, which oscillates (or resonates) at one particular frequency. At that frequency, energy is alternately shuffled between the capacitor and the inductor in the form of alternating voltage and current 90 degrees out of phase with each other.
• When the power supply frequency for an AC circuit exactly matches that circuit's natural oscillation frequency as set by the L and C components, a condition of resonance will have been reached.

Simple parallel (tank circuit) resonance

• Resonance occurs when capacitive and inductive reactances are equal to each other.
• For a tank circuit with no resistance (R), resonant frequency can be calculated with the following formula:
• The total impedance of a parallel LC circuit approaches infinity as the power supply frequency approaches resonance.
• A Bode plot is a graph plotting waveform amplitude or phase on one axis and frequency on the other.

Simple series resonance

• The total impedance of a series LC circuit approaches zero as the power supply frequency approaches resonance.
• The same formula for determining resonant frequency in a simple tank circuit applies to simple series circuits as well.
• Extremely high voltages can be formed across the individual components of series LC circuits at resonance, due to high current flows and substantial individual component impedances.

Applications of resonance

• Resonance can be employed to maintain AC circuit oscillations at a constant frequency, just as a pendulum can be used to maintain constant oscillation speed in a timekeeping mechanism.
• Resonance can be exploited for its impedance properties: either dramatically increasing or decreasing impedance for certain frequencies. Circuits designed to screen certain frequencies out of a mix of different frequencies are called filters.

Resonance in series-parallel circuits

• Added resistance to an LC circuit can cause a condition known as antiresonance, where the peak impedance effects happen at frequencies other than that which gives equal capacitive and inductive reactances.
• Resistance inherent in real-world inductors can contribute greatly to conditions of antiresonance. One source of such resistance is the skin effect, caused by the exclusion of AC current from the center of conductors. Another source is that of core losses in iron-core inductors.
• In a simple series LC circuit containing resistance (an “RLC” circuit), resistance does not produce antiresonance. Resonance still occurs when capacitive and inductive reactances are equal.

Q and bandwidth of a resonant circuit

The Q, quality factor, of a resonant circuit is a measure of the “goodness” or quality of a resonant circuit. A higher value for this figure of merit correspondes to a more narrow bandwith, which is desirable in many applications. More formally, Q is the ration of power stored to power dissipated in the circuit reactance and resistance, respectively:

       Q = P_stored/P_dissipated = I²X/I²R

       Q = X/R

       where:      X = Capacitive or Inductive reactance at resonance

                   R = Series resistance.

This formula is applicable to series resonant circuits, and also parallel resonant ciruits if the resistance is in series with the inductor. This is the case in practical applications, as we are mostly concerned with the resistance of the inductor limiting the Q. Note: Some text may show X and R interchanged in the “Q” formula for a parallel resonant circuit. This is correct for a large value of R in parallel with C and L. Our formula is correct for a small R in series with L.

A practical application of “Q” is that voltage across L or C in a series resonant circuit is Q times total applied voltage. In a parallel resonant circuit, current through L or C is Q times the total applied current.

Chapter 7

MIXED-FREQUENCY AC SIGNALS

Introduction

• A sinusoidal waveform is one shaped exactly like a sine wave.
• A non-sinusoidal waveform can be anything from a distorted sine-wave shape to something completely different like a square wave.
• Mixed-frequency waveforms can be accidently created, purposely created, or simply exist out of necessity. Most musical tones, for instance, are not composed of a single frequency sine-wave, but are rich blends of different frequencies.
• When multiple sine waveforms are mixed together (as is often the case in music), the lowest frequency sine-wave is called the fundamental, and the other sine-waves whose frequencies are whole-number multiples of the fundamental wave are called harmonics.
• An overtone is a harmonic produced by a particular device. The “first” overtone is the first frequency greater than the fundamental, while the “second” overtone is the next greater frequency produced. Successive overtones may or may not correspond to incremental harmonics, depending on the device producing the mixed frequencies. Some devices and systems do not permit the establishment of certain harmonics, and so their overtones would only include some (not all) harmonic frequencies.

Square wave signals

• Square waves are equivalent to a sine wave at the same (fundamental) frequency added to an infinite series of odd-multiple sine-wave harmonics at decreasing amplitudes.
• Computer algorithms exist which are able to sample waveshapes and determine their constituent sinusoidal components. The Fourier Transform algorithm (particularly the Fast Fourier Transform, or FFT) is commonly used in computer circuit simulation programs such as SPICE and in electronic metering equipment for determining power quality.

Other waveshapes

• Any waveform at all, so long as it is repetitive, can be reduced to a series of sinusoidal waveforms added together. Different waveshapes consist of different blends of sine-wave harmonics.
• Rectification of AC to DC is a very common source of harmonics within industrial power systems.

More on spectrum analysis

• Waveforms that are symmetrical above and below their horizontal centerlines contain no even-numbered harmonics.
• The amount of DC “bias” voltage present (a waveform's “DC component”) has no impact on that wave's harmonic frequency content.

Circuit effects

• Any regular (repeating), non-sinusoidal waveform is equivalent to a particular series of sine/cosine waves of different frequencies, phases, and amplitudes, plus a DC offset voltage if necessary. The mathematical process for determining the sinusoidal waveform equivalent for any waveform is called Fourier analysis.
• Multiple-frequency voltage sources can be simulated for analysis by connecting several single-frequency voltage sources in series. Analysis of voltages and currents is accomplished by using the superposition theorem. NOTE: superimposed voltages and currents of different frequencies cannot be added together in complex number form, since complex numbers only account for amplitude and phase shift, not frequency!
• Harmonics can cause problems by impressing unwanted (“noise”) voltage signals upon nearby circuits. These unwanted signals may come by way of capacitive coupling, inductive coupling, electromagnetic radiation, or a combination thereof.

 

 از فصل ۸تا آخر فصل ۱۳

Chapter 8

FILTERS

What is a filter?

• A filter is an AC circuit that separates some frequencies from others within mixed-frequency signals.
• Audio equalizers and crossover networks are two well-known applications of filter circuits.
• A Bode plot is a graph plotting waveform amplitude or phase on one axis and frequency on the other.

Low-pass filters

• A low-pass filter allows for easy passage of low-frequency signals from source to load, and difficult passage of high-frequency signals.
• Inductive low-pass filters insert an inductor in series with the load; capacitive low-pass filters insert a resistor in series and a capacitor in parallel with the load. The former filter design tries to “block” the unwanted frequency signal while the latter tries to short it out.
• The cutoff frequency for a low-pass filter is that frequency at which the output (load) voltage equals 70.7% of the input (source) voltage. Above the cutoff frequency, the output voltage is lower than 70.7% of the input, and vice versa.

High-pass filters

• A high-pass filter allows for easy passage of high-frequency signals from source to load, and difficult passage of low-frequency signals.
• Capacitive high-pass filters insert a capacitor in series with the load; inductive high-pass filters insert a resistor in series and an inductor in parallel with the load. The former filter design tries to “block” the unwanted frequency signal while the latter tries to short it out.
• The cutoff frequency for a high-pass filter is that frequency at which the output (load) voltage equals 70.7% of the input (source) voltage. Above the cutoff frequency, the output voltage is greater than 70.7% of the input, and vice versa.

Band-pass filters

• A band-pass filter works to screen out frequencies that are too low or too high, giving easy passage only to frequencies within a certain range.
• Band-pass filters can be made by stacking a low-pass filter on the end of a high-pass filter, or vice versa.
• “Attenuate” means to reduce or diminish in amplitude. When you turn down the volume control on your stereo, you are “attenuating” the signal being sent to the speakers.

Band-stop filters

• A band-stop filter works to screen out frequencies that are within a certain range, giving easy passage only to frequencies outside of that range. Also known as band-elimination, band-reject, or notch filters.
• Band-stop filters can be made by placing a low-pass filter in parallel with a high-pass filter. Commonly, both the low-pass and high-pass filter sections are of the “T” configuration, giving the name “Twin-T” to the band-stop combination.
• The frequency of maximum attenuation is called the notch frequency.

Resonant filters

• Resonant combinations of capacitance and inductance can be employed to create very effective band-pass and band-stop filters without the need for added resistance in a circuit that would diminish the passage of desired frequencies.

 

Chapter 9

TRANSFORMERS

Mutual inductance and basic operation

• Mutual inductance is where the magnetic flux of two or more inductors are “linked” so that voltage is induced in one coil proportional to the rate-of-change of current in another.
• A transformer is a device made of two or more inductors, one of which is powered by AC, inducing an AC voltage across the second inductor. If the second inductor is connected to a load, power will be electromagnetically coupled from the first inductor's power source to that load.
• The powered inductor in a transformer is called the primary winding. The unpowered inductor in a transformer is called the secondary winding.
• Magnetic flux in the core (Φ) lags 90^o behind the source voltage waveform. The current drawn by the primary coil from the source to produce this flux is called the magnetizing current, and it also lags the supply voltage by 90^o.
• Total primary current in an unloaded transformer is called the exciting current, and is comprised of magnetizing current plus any additional current necessary to overcome core losses. It is never perfectly sinusoidal in a real transformer, but may be made more so if the transformer is designed and operated so that magnetic flux density is kept to a minimum.
• Core flux induces a voltage in any coil wrapped around the core. The induces voltage(s) are ideally in- phase with the primary winding source voltage and share the same waveshape.
• Any current drawn through the secondary winding by a load will be “reflected” to the primary winding and drawn from the voltage source, as if the source were directly powering a similar load.

Step-up and step-down transformers

• Transformers “step up” or “step down” voltage according to the ratios of primary to secondary wire turns.
• A transformer designed to increase voltage from primary to secondary is called a step-up transformer. A transformer designed to reduce voltage from primary to secondary is called a step-down transformer.
• The transformation ratio of a transformer will be equal to the square root of its primary to secondary inductance (L) ratio.

Electrical isolation

• By being able to transfer power from one circuit to another without the use of interconnecting conductors between the two circuits, transformers provide the useful feature of electrical isolation.
• Transformers designed to provide electrical isolation without stepping voltage and current either up or down are called isolation transformers.

Phasing

• The phase relationships for voltage and current between primary and secondary circuits of a transformer are direct: ideally, zero phase shift.
• The dot convention is a type of polarity marking for transformer windings showing which end of the winding is which, relative to the other windings.

Winding configurations

• Transformers can be equipped with more than just a single primary and single secondary winding pair. This allows for multiple step-up and/or step-down ratios in the same device.
• Transformer windings can also be “tapped:” that is, intersected at many points to segment a single winding into sections.
• Variable transformers can be made by providing a movable arm that sweeps across the length of a winding, making contact with the winding at any point along its length. The winding, of course, has to be bare (no insulation) in the area where the arm sweeps.
• An autotransformer is a single, tapped inductor coil used to step up or step down voltage like a transformer, except without providing electrical isolation.
• A Variac is a variable autotransformer.

Voltage regulation

• Voltage regulation is the measure of how well a power transformer can maintain constant secondary voltage given a constant primary voltage and wide variance in load current. The lower the percentage (closer to zero), the more stable the secondary voltage and the better the regulation it will provide.
• A ferroresonant transformer is a special transformer designed to regulate voltage at a stable level despite wide variation in input voltage.

Special transformers and applications

• Transformers can be used to transform impedance as well as voltage and current. When this is done to improve power transfer to a load, it is called impedance matching.
• A Potential Transformer (PT) is a special instrument transformer designed to provide a precise voltage step-down ratio for voltmeters measuring high power system voltages.
• A Current Transformer (CT) is another special instrument transformer designed to step down the current through a power line to a safe level for an ammeter to measure.
• An air-core transformer is one lacking a ferromagnetic core.
• A Tesla Coil is a resonant, air-core, step-up transformer designed to produce very high AC voltages at high frequency.
• A saturable reactor is a special type of inductor, the inductance of which can be controlled by the DC current through a second winding around the same core. With enough DC current, the magnetic core can be saturated, decreasing the inductance of the power winding in a controlled fashion.
• A Scott-T transformer converts 3-φ power to 2-φ power and vice versa.
• A linear variable differential transformer, also known as an LVDT, is a distance measuring device. It has a movable ferromagnetic core to vary the coupling between the excited primary and a pair of secondaries.

Practical considerations

• Power transformers are limited in the amount of power they can transfer from primary to secondary winding(s). Large units are typically rated in VA (volt-amps) or kVA (kilo volt-amps).
• Resistance in transformer windings contributes to inefficiency, as current will dissipate heat, wasting energy.
• Magnetic effects in a transformer's iron core also contribute to inefficiency. Among the effects are eddy currents (circulating induction currents in the iron core) and hysteresis (power lost due to overcoming the tendency of iron to magnetize in a particular direction).
• Increased frequency results in increased power losses within a power transformer. The presence of harmonics in a power system is a source of frequencies significantly higher than normal, which may cause overheating in large transformers.
• Both transformers and inductors harbor certain unavoidable amounts of capacitance due to wire insulation (dielectric) separating winding turns from the iron core and from each other. This capacitance can be significant enough to give the transformer a natural resonant frequency, which can be problematic in signal applications.
• Leakage inductance is caused by magnetic flux not being 100% coupled between windings in a transformer. Any flux not involved with transferring energy from one winding to another will store and release energy, which is how (self-) inductance works. Leakage inductance tends to worsen a transformer's voltage regulation (secondary voltage “sags” more for a given amount of load current).
• Magnetic saturation of a transformer core may be caused by excessive primary voltage, operation at too low of a frequency, and/or by the presence of a DC current in any of the windings. Saturation may be minimized or avoided by conservative design, which provides an adequate margin of safety between peak magnetic flux density values and the saturation limits of the core.
• Transformers often experience significant inrush currents when initially connected to an AC voltage source. Inrush current is most severe when connection to the AC source is made at the moment instantaneous source voltage is zero.
• Noise is a common phenomenon exhibited by transformers -- especially power transformers -- and is primarily caused by magnetostriction of the core. Physical forces causing winding vibration may also generate noise under conditions of heavy (high current) secondary winding load.

 

Chapter 10

POLYPHASE AC CIRCUITS

Single-phase power systems

• Single phase power systems are defined by having an AC source with only one voltage waveform.
• A split-phase power system is one with multiple (in-phase) AC voltage sources connected in series, delivering power to loads at more than one voltage, with more than two wires. They are used primarily to achieve balance between system efficiency (low conductor currents) and safety (low load voltages).
• Split-phase AC sources can be easily created by center-tapping the coil windings of transformers or alternators.

Three-phase power systems

• A single-phase power system is one where there is only one AC voltage source (one source voltage waveform).
• A split-phase power system is one where there are two voltage sources, 180^o phase-shifted from each other, powering a two series-connected loads. The advantage of this is the ability to have lower conductor currents while maintaining low load voltages for safety reasons.
• A polyphase power system uses multiple voltage sources at different phase angles from each other (many “phases” of voltage waveforms at work). A polyphase power system can deliver more power at less voltage with smaller-gage conductors than single- or split-phase systems.
• The phase-shifted voltage sources necessary for a polyphase power system are created in alternators with multiple sets of wire windings. These winding sets are spaced around the circumference of the rotor's rotation at the desired angle(s).

Phase rotation

• Phase rotation, or phase sequence, is the order in which the voltage waveforms of a polyphase AC source reach their respective peaks. For a three-phase system, there are only two possible phase sequences: 1-2-3 and 3-2-1, corresponding to the two possible directions of alternator rotation.
• Phase rotation has no impact on resistive loads, but it will have impact on unbalanced reactive loads, as shown in the operation of a phase rotation detector circuit.
• Phase rotation can be reversed by swapping any two of the three “hot” leads supplying three-phase power to a three-phase load.

Polyphase motor design

• AC “induction” and “synchronous” motors work by having a rotating magnet follow the alternating magnetic fields produced by stationary wire windings.
• Single-phase AC motors of this type need help to get started spinning in a particular direction.
• By introducing a phase shift of less than 180^o to the magnetic fields in such a motor, a definite direction of shaft rotation can be established.
• Single-phase induction motors often use an auxiliary winding connected in series with a capacitor to create the necessary phase shift.
• Polyphase motors don't need such measures; their direction of rotation is fixed by the phase sequence of the voltage they're powered by.
• Swapping any two “hot” wires on a polyphase AC motor will reverse its phase sequence, thus reversing its shaft rotation.

Three-phase Y and Δ configurations

• The conductors connected to the three points of a three-phase source or load are called lines.
• The three components comprising a three-phase source or load are called phases.
• Line voltage is the voltage measured between any two lines in a three-phase circuit.
• Phase voltage is the voltage measured across a single component in a three-phase source or load.
• Line current is the current through any one line between a three-phase source and load.
• Phase current is the current through any one component comprising a three-phase source or load.
• In balanced “Y” circuits, line voltage is equal to phase voltage times the square root of 3, while line current is equal to phase current.
• In balanced Δ circuits, line voltage is equal to phase voltage, while line current is equal to phase current times the square root of 3.
• Δ-connected three-phase voltage sources give greater reliability in the event of winding failure than Y-connected sources. However, Y-connected sources can deliver the same amount of power with less line current than Δ-connected sources.

Three-phase transformer circuits

• Nonlinear components are those that draw a non-sinusoidal (non-sine-wave) current waveform when energized by a sinusoidal (sine-wave) voltage. Since any distortion of an originally pure sine-wave constitutes harmonic frequencies, we can say that nonlinear components generate harmonic currents.
• When the sine-wave distortion is symmetrical above and below the average centerline of the waveform, the only harmonics present will be odd-numbered, not even-numbered.
• The 3rd harmonic, and integer multiples of it (6th, 9th, 12th, 15th) are known as triplen harmonics. They are in phase with each other, despite the fact that their respective fundamental waveforms are 120^o out of phase with each other.
• In a 4-wire Y-Y system, triplen harmonic currents add within the neutral conductor.
• Triplen harmonic currents in a Δ-connected set of components circulate within the loop formed by the Δ.

Harmonic phase sequences

Chapter 11

POWER FACTOR

Power in resistive and reactive AC circuits

• In a purely resistive circuit, all circuit power is dissipated by the resistor(s). Voltage and current are in phase with each other.
• In a purely reactive circuit, no circuit power is dissipated by the load(s). Rather, power is alternately absorbed from and returned to the AC source. Voltage and current are 90^o out of phase with each other.
• In a circuit consisting of resistance and reactance mixed, there will be more power dissipated by the load(s) than returned, but some power will definitely be dissipated and some will merely be absorbed and returned. Voltage and current in such a circuit will be out of phase by a value somewhere between 0^o and 90^o.

True, Reactive, and Apparent power

• Power dissipated by a load is referred to as true power. True power is symbolized by the letter P and is measured in the unit of Watts (W).
• Power merely absorbed and returned in load due to its reactive properties is referred to as reactive power. Reactive power is symbolized by the letter Q and is measured in the unit of Volt-Amps-Reactive (VAR).
• Total power in an AC circuit, both dissipated and absorbed/returned is referred to as apparent power. Apparent power is symbolized by the letter S and is measured in the unit of Volt-Amps (VA).
• These three types of power are trigonometrically related to one another. In a right triangle, P = adjacent length, Q = opposite length, and S = hypotenuse length. The opposite angle is equal to the circuit's impedance (Z) phase angle.

Calculating power factor

• Poor power factor in an AC circuit may be “corrected”, or re-established at a value close to 1, by adding a parallel reactance opposite the effect of the load's reactance. If the load's reactance is inductive in nature (which is almost always will be), parallel capacitance is what is needed to correct poor power factor.

Chapter 12

AC METERING CIRCUITS

AC voltmeters and ammeters

• Polarized (DC) meter movements must use devices called diodes to be able to indicate AC quantities.
• Electromechanical meter movements, whether electromagnetic or electrostatic, naturally provide the average value of a measured AC quantity. These instruments may be ranged to indicate RMS value, but only if the shape of the AC waveform is precisely known beforehand!
• So-called true RMS meters use different technology to provide indications representing the actual RMS (rather than skewed average or peak) of an AC waveform.

Frequency and phase measurement

• Some frequency meters work on the principle of mechanical resonance, indicating frequency by relative oscillation among a set of uniquely tuned “reeds” shaken at the measured frequency.
• Other frequency meters use electric resonant circuits (LC tank circuits, usually) to indicate frequency. One or both components is made to be adjustable, with an accurately calibrated adjustment knob, and a sensitive meter is read for maximum voltage or current at the point of resonance.
• Frequency can be measured in a comparative fashion, as is the case when using a CRT to generate Lissajous figures. Reference frequency signals can be made with a high degree of accuracy by oscillator circuits using quartz crystals as resonant devices. For ultra precision, atomic clock signal standards (based on the resonant frequencies of individual atoms) can be used.

Power measurement

• AC bridge circuits work on the same basic principle as DC bridge circuits: that a balanced ratio of impedances (rather than resistances) will result in a “balanced” condition as indicated by the null-detector device.
• Null detectors for AC bridges may be sensitive electromechanical meter movements, oscilloscopes (CRT's), headphones (amplified or unamplified), or any other device capable of registering very small AC voltage levels. Like DC null detectors, its only required point of calibration accuracy is at zero.
• AC bridge circuits can be of the “symmetrical” type where an unknown impedance is balanced by a standard impedance of similar type on the same side (top or bottom) of the bridge. Or, they can be “nonsymmetrical,” using parallel impedances to balance series impedances, or even capacitances balancing out inductances.
• AC bridge circuits often have more than one adjustment, since both impedance magnitude and phase angle must be properly matched to balance.
• Some impedance bridge circuits are frequency-sensitive while others are not. The frequency-sensitive types may be used as frequency measurement devices if all component values are accurately known.
• A Wagner earth or Wagner ground is a voltage divider circuit added to AC bridges to help reduce errors due to stray capacitance coupling the null detector to ground.

 

 

Chapter 13

TRANSMISSION LINES

A 50-ohm cable?

• In an electric circuit, the effects of electron motion travel approximately at the speed of light, although electrons within the conductors do not travel anywhere near that velocity.

Characteristic impedance

• A transmission line is a pair of parallel conductors exhibiting certain characteristics due to distributed capacitance and inductance along its length.
• When a voltage is suddenly applied to one end of a transmission line, both a voltage “wave” and a current “wave” propagate along the line at nearly light speed.
• If a DC voltage is applied to one end of an infinitely long transmission line, the line will draw current from the DC source as though it were a constant resistance.
• The characteristic impedance (Z₀) of a transmission line is the resistance it would exhibit if it were infinite in length. This is entirely different from leakage resistance of the dielectric separating the two conductors, and the metallic resistance of the wires themselves. Characteristic impedance is purely a function of the capacitance and inductance distributed along the line's length, and would exist even if the dielectric were perfect (infinite parallel resistance) and the wires superconducting (zero series resistance).
• Velocity factor is a fractional value relating a transmission line's propagation speed to the speed of light in a vacuum. Values range between 0.66 and 0.80 for typical two-wire lines and coaxial cables. For any cable type, it is equal to the reciprocal (1/x) of the square root of the relative permittivity of the cable's insulation.

Finite-length transmission lines

• Characteristic impedance is also known as surge impedance, due to the temporarily resistive behavior of any length transmission line.
• A finite-length transmission line will appear to a DC voltage source as a constant resistance for some short time, then as whatever impedance the line is terminated with. Therefore, an open-ended cable simply reads “open” when measured with an ohmmeter, and “shorted” when its end is short-circuited.
• A transient (“surge”) signal applied to one end of an open-ended or short-circuited transmission line will “reflect” off the far end of the line as a secondary wave. A signal traveling on a transmission line from source to load is called an incident wave; a signal “bounced” off the end of a transmission line, traveling from load to source, is called a reflected wave.
• Reflected waves will also appear in transmission lines terminated by resistors not precisely matching the characteristic impedance.
• A finite-length transmission line may be made to appear infinite in length if terminated by a resistor of equal value to the line's characteristic impedance. This eliminates all signal reflections.
• A reflected wave may become re-reflected off the source-end of a transmission line if the source's internal impedance does not match the line's characteristic impedance. This re-reflected wave will appear, of course, like another pulse signal transmitted from the source.

“Long” and “short” transmission lines

• Coaxial cabling is sometimes used in DC and low-frequency AC circuits as well as in high-frequency circuits, for the excellent immunity to induced “noise” that it provides for signals.
• When the period of a transmitted voltage or current signal greatly exceeds the propagation time for a transmission line, the line is considered electrically short. Conversely, when the propagation time is a large fraction or multiple of the signal's period, the line is considered electrically long.
• A signal's wavelength is the physical distance it will propagate in the timespan of one period. Wavelength is calculated by the formula λ=v/f, where “λ” is the wavelength, “v” is the propagation velocity, and “f” is the signal frequency.
• A rule-of-thumb for transmission line “shortness” is that the line must be at least 1/4 wavelength before it is considered “long.”
• In a circuit with a “short” line, the terminating (load) impedance dominates circuit behavior. The source effectively sees nothing but the load's impedance, barring any resistive losses in the transmission line.
• In a circuit with a “long” line, the line's own characteristic impedance dominates circuit behavior. The ultimate example of this is a transmission line of infinite length: since the signal will never reach the load impedance, the source only “sees” the cable's characteristic impedance.
• When a transmission line is terminated by a load precisely matching its impedance, there are no reflected waves and thus no problems with line length.

Standing waves and resonance

• Standing waves are waves of voltage and current which do not propagate (i.e. they are stationary), but are the result of interference between incident and reflected waves along a transmission line.
• A node is a point on a standing wave of minimum amplitude.
• An antinode is a point on a standing wave of maximum amplitude.
• Standing waves can only exist in a transmission line when the terminating impedance does not match the line's characteristic impedance. In a perfectly terminated line, there are no reflected waves, and therefore no standing waves at all.
• At certain frequencies, the nodes and antinodes of standing waves will correlate with the ends of a transmission line, resulting in resonance.
• The lowest-frequency resonant point on a transmission line is where the line is one quarter-wavelength long. Resonant points exist at every harmonic (integer-multiple) frequency of the fundamental (quarter-wavelength).
• Standing wave ratio, or SWR, is the ratio of maximum standing wave amplitude to minimum standing wave amplitude. It may also be calculated by dividing termination impedance by characteristic impedance, or vice versa, which ever yields the greatest quotient. A line with no standing waves (perfectly matched: Z_load to Z₀) has an SWR equal to 1.
• Transmission lines may be damaged by the high maximum amplitudes of standing waves. Voltage antinodes may break down insulation between conductors, and current antinodes may overheat conductors.

Impedance transformation

• A transmission line with standing waves may be used to match different impedance values if operated at the correct frequency(ies).
• When operated at a frequency corresponding to a standing wave of 1/4-wavelength along the transmission line, the line's characteristic impedance necessary for impedance transformation must be equal to the square root of the product of the source's impedance and the load's impedance.

Waveguides

• Waveguides are metal tubes functioning as “conduits” for carrying electromagnetic waves. They are practical only for signals of extremely high frequency, where the signal wavelength approaches the cross-sectional dimensions of the waveguide.
• Wave propagation through a waveguide may be classified into two broad categories: TE (Transverse Electric), or TM (Transverse Magnetic), depending on which field (electric or magnetic) is perpendicular (transverse) to the direction of wave travel. Wave travel along a standard, two-conductor transmission line is of the TEM (Transverse Electric and Magnetic) mode, where both fields are oriented perpendicular to the direction of travel. TEM mode is only possible with two conductors and cannot exist in a waveguide.
• A dead-ended waveguide serving as a resonant element in a microwave circuit is called a cavity resonator.
• A cavity resonator with an open end functions as a unidirectional antenna, sending or receiving RF energy to/from the direction of the open end.

 

 

 

 

جزوه زبان فنی برق بخش سوم

Chapter 5

SERIES AND PARALLEL CIRCUITS

What are "series" and "parallel" circuits?

• In a series circuit, all components are connected end-to-end, forming a single path for electrons to flow.
• In a parallel circuit, all components are connected across each other, forming exactly two sets of electrically common points.
• A "branch" in a parallel circuit is a path for electric current formed by one of the load components (such as a resistor).

Simple series circuits

• Components in a series circuit share the same current: I_Total = I₁ = I₂ = . . . I_n
• Total resistance in a series circuit is equal to the sum of the individual resistances: R_Total = R₁ + R₂ + . . . R_n
• Total voltage in a series circuit is equal to the sum of the individual voltage drops: E_Total = E₁ + E₂ + . . . E_n

Simple parallel circuits

• Components in a parallel circuit share the same voltage: E_Total = E₁ = E₂ = . . . E_n
• Total resistance in a parallel circuit is less than any of the individual resistances: R_Total = 1 / (1/R₁ + 1/R₂ + . . . 1/R_n)
• Total current in a parallel circuit is equal to the sum of the individual branch currents: I_Total = I₁ + I₂ + . . . I_n.

Conductance

• Conductance is the opposite of resistance: the measure of how easy is it for electrons to flow through something.
• Conductance is symbolized with the letter "G" and is measured in units of mhos or Siemens.
• Mathematically, conductance equals the reciprocal of resistance: G = 1/R

Power calculations

• Power is additive in any configuration of resistive circuit: P_Total = P₁ + P₂ + . . . P_n

Correct use of Ohm's Law

• Apply Ohm's Law to vertical columns in the table.
• Apply rules of series/parallel to horizontal rows in the table.
• Check your calculations by working "backwards" to try to arrive at originally given values (from your first calculated answers), or by solving for a quantity using more than one method (from different given values).

Component failure analysis

• To determine what would happen in a circuit if a component fails, re-draw that circuit with the equivalent resistance of the failed component in place and re-calculate all values.
• The ability to intuitively determine what will happen to a circuit with any given component fault is a crucial skill for any electronics troubleshooter to develop. The best way to learn is to experiment with circuit calculations and real-life circuits, paying close attention to what changes with a fault, what remains the same, and why!
• A shorted component is one whose resistance has dramatically decreased.
• An open component is one whose resistance has dramatically increased. For the record, resistors tend to fail open more often than fail shorted, and they almost never fail unless physically or electrically overstressed (physically abused or overheated).

Building simple resistor circuits

• A solderless breadboard is a device used to quickly assemble temporary circuits by plugging wires and components into electrically common spring-clips arranged underneath rows of holes in a plastic board.
• Soldering is a low-temperature welding process utilizing a lead/tin or tin/silver alloy to bond wires and component leads together, usually with the components secured to a fiberglass board.
• Wire-wrapping is an alternative to soldering, involving small-gauge wire tightly wrapped around component leads rather than a welded joint to connect components together.
• A terminal strip, also known as a barrier strip or terminal block is another device used to mount components and wires to build circuits. Screw terminals or heavy spring clips attached to metal bars provide connection points for the wire ends and component leads, these metal bars mounted separately to a piece of nonconducting material such as plastic, bakelite, or ceramic.

Chapter 6

DIVIDER CIRCUITS AND KIRCHHOFF'S LAWS

Voltage divider circuits

• Series circuits proportion, or divide, the total supply voltage among individual voltage drops, the proportions being strictly dependent upon resistances: E_Rn = E_Total (R_n / R_Total)
• A potentiometer is a variable-resistance component with three connection points, frequently used as an adjustable voltage divider.

Kirchhoff's Voltage Law (KVL)

• Kirchhoff's Voltage Law (KVL): "The algebraic sum of all voltages in a loop must equal zero"

Current divider circuits

Parallel circuits proportion, or "divide," the total circuit current among individual branch currents, the proportions being strictly dependent upon resistances: I_n = I_Total (R_Total / R_n)

Kirchhoff's Current Law (KCL)

• Kirchhoff's Current Law (KCL): "The algebraic sum of all currents entering and exiting a node must equal zero"

 

Chapter 7

SERIES-PARALLEL COMBINATION CIRCUITS

What is a series-parallel circuit?

 

• The rules of series and parallel circuits must be applied selectively to circuits containing both types of interconnections.

Analysis technique

• To analyze a series-parallel combination circuit, follow these steps:
• Reduce the original circuit to a single equivalent resistor, re-drawing the circuit in each step of reduction as simple series and simple parallel parts are reduced to single, equivalent resistors.
• Solve for total resistance.
• Solve for total current (I=E/R).
• Determine equivalent resistor voltage drops and branch currents one stage at a time, working backwards to the original circuit configuration again.

Re-drawing complex schematics

• Wires in diagrams and in real circuits can be lengthened, shortened, and/or moved without affecting circuit operation.
• To simplify a convoluted circuit schematic, follow these steps:
• Trace current from one side of the battery to the other, following any single path ("loop") to the battery. Sometimes it works better to start with the loop containing the most components, but regardless of the path taken the result will be accurate. Mark polarity of voltage drops across each resistor as you trace the loop. Draw those components you encounter along this loop in a vertical schematic.
• Mark traced components in the original diagram and trace remaining loops of components in the circuit. Use polarity marks across traced components as guides for what connects where. Document new components in loops on the vertical re-draw schematic as well.
• Repeat last step as often as needed until all components in original diagram have been traced.

Component failure analysis

• For any single component failure (open or shorted), the total resistance will always change in the same direction (either increase or decrease) as the resistance change of the failed component.
• When a component fails shorted, its resistance always decreases. Also, the current through it will increase, and the voltage across it may drop. I say "may" because in some cases it will remain the same (case in point: a simple parallel circuit with an ideal power source).
• When a component fails open, its resistance always increases. The current through that component will decrease to zero, because it is an incomplete electrical path (no continuity). This may result in an increase of voltage across it. The same exception stated above applies here as well: in a simple parallel circuit with an ideal voltage source, the voltage across an open-failed component will remain unchanged.

Building series-parallel resistor circuits

• Circuits built on terminal strips can be difficult to lay out, but when built they are robust enough to be considered permanent, yet easy to modify.
• It is bad practice to secure more than two wire ends and/or component leads under a single terminal screw or clip on a terminal strip. Try to arrange connecting wires so as to avoid this condition.
• Whenever possible, build your circuits with clarity and ease of understanding in mind. Even though component and wiring layout is usually of little consequence in DC circuit function, it matters significantly for the sake of the person who has to modify or troubleshoot it later.

Chapter 8

DC METERING CIRCUITS

What is a meter?

• A "movement" is the display mechanism of a meter.
• Electromagnetic movements work on the principle of a magnetic field being generated by electric current through a wire. Examples of electromagnetic meter movements include the D'Arsonval, Weston, and iron-vane designs.
• Electrostatic movements work on the principle of physical force generated by an electric field between two plates.
• Cathode Ray Tubes (CRT's) use an electrostatic field to bend the path of an electron beam, providing indication of the beam's position by light created when the beam strikes the end of the glass tube.

Voltmeter design

Extended voltmeter ranges are created for sensitive meter movements by adding series "multiplier" resistors to the movement circuit, providing a precise voltage division ratio.

Voltmeter impact on measured circuit

• An ideal voltmeter has infinite resistance.
• Too low of an internal resistance in a voltmeter will adversely affect the circuit being measured.
• Vacuum tube voltmeters (VTVM's), transistor voltmeters, and potentiometric circuits are all means of minimizing the load placed on a measured circuit. Of these methods, the potentiometric ("null-balance") technique is the only one capable of placing zero load on the circuit.
• A null detector is a device built for maximum sensitivity to small voltages or currents. It is used in potentiometric voltmeter circuits to indicate the absence of voltage between two points, thus indicating a condition of balance between an adjustable voltage source and the voltage being measured.

Ammeter design

• Ammeter ranges are created by adding parallel "shunt" resistors to the movement circuit, providing a precise current division.
• Shunt resistors may have high power dissipations, so be careful when choosing parts for such meters!
• Shunt resistors can be used in conjunction with high-resistance voltmeters as well as low-resistance ammeter movements, producing accurate voltage drops for given amounts of current. Shunt resistors should be selected for as low a resistance value as possible to minimize their impact upon the circuit under test.

Ammeter impact on measured circuit

• An ideal ammeter has zero resistance.
• A "clamp-on" ammeter measures current through a wire by measuring the strength of the magnetic field around it rather than by becoming part of the circuit, making it an ideal ammeter.
• Clamp-on meters make for quick and safe current measurements, because there is no conductive contact between the meter and the circuit.

Ohmmeter design

• Ohmmeters contain internal sources of voltage to supply power in taking resistance measurements.
• An analog ohmmeter scale is "backwards" from that of a voltmeter or ammeter, the movement needle reading zero resistance at full-scale and infinite resistance at rest.
• Analog ohmmeters also have logarithmic scales, "expanded" at the low end of the scale and "compressed" at the high end to be able to span from zero to infinite resistance.
• Analog ohmmeters are not precision instruments.
• Ohmmeters should never be connected to an energized circuit (that is, a circuit with its own source of voltage). Any voltage applied to the test leads of an ohmmeter will invalidate its reading.

High voltage ohmmeters

• Bridge circuits rely on sensitive null-voltage meters to compare two voltages for equality.
• A Wheatstone bridge can be used to measure resistance by comparing the unknown resistor against precision resistors of known value, much like a laboratory scale measures an unknown weight by comparing it against known standard weights.
• A Kelvin Double bridge is a variant of the Wheatstone bridge used for measuring very low resistances. Its additional complexity over the basic Wheatstone design is necessary for avoiding errors otherwise incurred by stray resistances along the current path between the low-resistance standard and the resistance being measured.

Wattmeter design

Power in an electric circuit is the product (multiplication) of voltage and current, so any meter designed to measure power must account for both of these variables.

A special meter movement designed especially for power measurement is called the dynamometer movement, and is similar to a D'Arsonval or Weston movement in that a lightweight coil of wire is attached to the pointer mechanism. However, unlike the D'Arsonval or Weston movement, another (stationary) coil is used instead of a permanent magnet to provide the magnetic field for the moving coil to react against. The moving coil is generally energized by the voltage in the circuit, while the stationary coil is generally energized by the current in the circuit. A dynamometer movement connected in a circuit looks something like this:

 

The top (horizontal) coil of wire measures load current while the bottom (vertical) coil measures load voltage. Just like the lightweight moving coils of voltmeter movements, the (moving) voltage coil of a dynamometer is typically connected in series with a range resistor so that full load voltage is not applied to it. Likewise, the (stationary) current coil of a dynamometer may have precision shunt resistors to divide the load current around it. With custom-built dynamometer movements, shunt resistors are less likely to be needed because the stationary coil can be constructed with as heavy of wire as needed without impacting meter response, unlike the moving coil which must be constructed of lightweight wire for minimum inertia.

 

• REVIEW:
• Wattmeters are often designed around dynamometer meter movements, which employ both voltage and current coils to move a needle.

Creating custom calibration resistances

Often in the course of designing and building electrical meter circuits, it is necessary to have precise resistances to obtain the desired range(s). More often than not, the resistance values required cannot be found in any manufactured resistor unit and therefore must be built by you.

One solution to this dilemma is to make your own resistor out of a length of special high-resistance wire. Usually, a small "bobbin" is used as a form for the resulting wire coil, and the coil is wound in such a way as to eliminate any electromagnetic effects: the desired wire length is folded in half, and the looped wire wound around the bobbin so that current through the wire winds clockwise around the bobbin for half the wire's length, then counter-clockwise for the other half. This is known as a bifilar winding. Any magnetic fields generated by the current are thus canceled, and external magnetic fields cannot induce any voltage in the resistance wire coil:

 

As you might imagine, this can be a labor-intensive process, especially if more than one resistor must be built! Another, easier solution to the dilemma of a custom resistance is to connect multiple fixed-value resistors together in series-parallel fashion to obtain the desired value of resistance. This solution, although potentially time-intensive in choosing the best resistor values for making the first resistance, can be duplicated much faster for creating multiple custom resistances of the same value:

A disadvantage of either technique, though, is the fact that both result in a fixed resistance value. In a perfect world where meter movements never lose magnetic strength of their permanent magnets, where temperature and time have no effect on component resistances, and where wire connections maintain zero resistance forever, fixed-value resistors work quite well for establishing the ranges of precision instruments. However, in the real world, it is advantageous to have the ability to calibrate, or adjust, the instrument in the future.

It makes sense, then, to use potentiometers (connected as rheostats, usually) as variable resistances for range resistors. The potentiometer may be mounted inside the instrument case so that only a service technician has access to change its value, and the shaft may be locked in place with thread-fastening compound (ordinary nail polish works well for this!) so that it will not move if subjected to vibration.

However, most potentiometers provide too large a resistance span over their mechanically-short movement range to allow for precise adjustment. Suppose you desired a resistance of 8.335 kΩ +/- 1 Ω, and wanted to use a 10 kΩ potentiometer (rheostat) to obtain it. A precision of 1 Ω out of a span of 10 kΩ is 1 part in 10,000, or 1/100 of a percent! Even with a 10-turn potentiometer, it will be very difficult to adjust it to any value this finely. Such a feat would be nearly impossible using a standard 3/4 turn potentiometer. So how can we get the resistance value we need and still have room for adjustment?

The solution to this problem is to use a potentiometer as part of a larger resistance network which will create a limited adjustment range. Observe the following example:

Here, the 1 kΩ potentiometer, connected as a rheostat, provides by itself a 1 kΩ span (a range of 0 Ω to 1 kΩ). Connected in series with an 8 kΩ resistor, this offsets the total resistance by 8,000 Ω, giving an adjustable range of 8 kΩ to 9 kΩ. Now, a precision of +/- 1 Ω represents 1 part in 1000, or 1/10 of a percent of potentiometer shaft motion. This is ten times better, in terms of adjustment sensitivity, than what we had using a 10 kΩ potentiometer.

If we desire to make our adjustment capability even more precise -- so we can set the resistance at 8.335 kΩ with even greater precision -- we may reduce the span of the potentiometer by connecting a fixed-value resistor in parallel with it:

Now, the calibration span of the resistor network is only 500 Ω, from 8 kΩ to 8.5 kΩ. This makes a precision of +/- 1 Ω equal to 1 part in 500, or 0.2 percent. The adjustment is now half as sensitive as it was before the addition of the parallel resistor, facilitating much easier calibration to the target value. The adjustment will not be linear, unfortunately (halfway on the potentiometer's shaft position will not result in 8.25 kΩ total resistance, but rather 8.333 kΩ). Still, it is an improvement in terms of sensitivity, and it is a practical solution to our problem of building an adjustable resistance for a precision instrument!

Chapter 9

ELECTRICAL INSTRUMENTATION SIGNALS

Analog and digital signals

 

• A signal is any kind of detectable quantity used to communicate information.
• An analog signal is a signal that can be continuously, or infinitely, varied to represent any small amount of change.
• Pneumatic, or air pressure, signals used to be used predominately in industrial instrumentation signal systems. This has been largely superseded by analog electrical signals such as voltage and current.
• A live zero refers to an analog signal scale using a non-zero quantity to represent 0 percent of real-world measurement, so that any system malfunction resulting in a natural "rest" state of zero signal pressure, voltage, or current can be immediately recognized.

Voltage signal systems

• DC voltage can be used as an analog signal to relay information from one location to another.
• A major disadvantage of voltage signaling is the possibility that the voltage at the indicator (voltmeter) will be less than the voltage at the signal source, due to line resistance and indicator current draw. This drop in voltage along the conductor length constitutes a measurement error from transmitter to indicator.

Current signal systems

• A current source is a device (usually constructed of several electronic components) that outputs a constant amount of current through a circuit, much like a voltage source (ideal battery) outputting a constant amount of voltage to a circuit.
• A current "loop" instrumentation circuit relies on the series circuit principle of current being equal through all components to insure no signal error due to wiring resistance.
• The most common analog current signal standard in modern use is the "4 to 20 milliamp current loop."

Tachogenerators

• The Seebeck Effect is the production of a voltage between two dissimilar, joined metals that is proportional to the temperature of that junction.
• In any thermocouple circuit, there are two equivalent junctions formed between dissimilar metals. The junction placed at the site of intended measurement is called the measurement junction, while the other (single or equivalent) junction is called the reference junction.
• Two thermocouple junctions can be connected in opposition to each other to generate a voltage signal proportional to differential temperature between the two junctions. A collection of junctions so connected for the purpose of generating electricity is called a thermopile.
• When electrons flow through the junctions of a thermopile, heat energy is transferred from one set of junctions to the other. This is known as the Peltier Effect.
• Multiple thermocouple junctions can be connected in parallel with each other to generate a voltage signal representing the average temperature between the junctions. "Swamping" resistors may be connected in series with each thermocouple to help maintain equality between the junctions, so the resultant voltage will be more representative of a true average temperature.
• It is imperative that current in a thermocouple circuit be kept as low as possible for good measurement accuracy. Also, all related wire connections should be clean and tight. Mere millivolts of drop at any place in the circuit will cause substantial measurement errors.

pH measurement

• All pH electrodes have a finite life, and that lifespan depends greatly on the type and severity of service. In some applications, a pH electrode life of one month may be considered long, and in other applications the same electrode(s) may be expected to last for over a year.
• Because the glass (measurement) electrode is responsible for generating the pH-proportional voltage, it is the one to be considered suspect if the measurement system fails to generate sufficient voltage change for a given change in pH (approximately 59 millivolts per pH unit), or fails to respond quickly enough to a fast change in test liquid pH.
• If a pH measurement system "drifts," creating offset errors, the problem likely lies with the reference electrode, which is supposed to provide a zero-voltage connection with the measured solution.
• Because pH measurement is a logarithmic representation of ion concentration, there is an incredible range of process conditions represented in the seemingly simple 0-14 pH scale. Also, due to the nonlinear nature of the logarithmic scale, a change of 1 pH at the top end (say, from 12 to 13 pH) does not represent the same quantity of chemical activity change as a change of 1 pH at the bottom end (say, from 2 to 3 pH). Control system engineers and technicians must be aware of this dynamic if there is to be any hope of controlling process pH at a stable value.
• The following conditions are hazardous to measurement (glass) electrodes: high temperatures, extreme pH levels (either acidic or alkaline), high ionic concentration in the liquid, abrasion, hydrofluoric acid in the liquid (HF acid dissolves glass!), and any kind of material coating on the surface of the glass.
• Temperature changes in the measured liquid affect both the response of the measurement electrode to a given pH level (ideally at 59 mV per pH unit), and the actual pH of the liquid. Temperature measurement devices can be inserted into the liquid, and the signals from those devices used to compensate for the effect of temperature on pH measurement, but this will only compensate for the measurement electrode's mV/pH response, not the actual pH change of the process liquid!
• pH is a representation of hydrogen ion activity in a liquid. It is the negative logarithm of the amount of hydrogen ions (in moles) per liter of liquid. Thus: 10^-11 moles of hydrogen ions in 1 liter of liquid = 11 pH. 10^-5.3 moles of hydrogen ions in 1 liter of liquid = 5.3 pH.
• The basic pH scale extends from 0 (strong acid) to 7 (neutral, pure water) to 14 (strong caustic). Chemical solutions with pH levels below zero and above 14 are possible, but rare.
• pH can be measured by measuring the voltage produced between two special electrodes immersed in the liquid solution.
• One electrode, made of a special glass, is called the measurement electrode. It's job it to generate a small voltage proportional to pH (ideally 59.16 mV per pH unit).
• The other electrode (called the reference electrode) uses a porous junction between the measured liquid and a stable, neutral pH buffer solution (usually potassium chloride) to create a zero-voltage electrical connection to the liquid. This provides a point of continuity for a complete circuit so that the voltage produced across the thickness of the glass in the measurement electrode can be measured by an external voltmeter.
• The extremely high resistance of the measurement electrode's glass membrane mandates the use of a voltmeter with extremely high internal resistance, or a null-balance voltmeter, to measure the voltage.

Strain gauges

• A strain gauge is a thin strip of metal designed to measure mechanical load by changing resistance when stressed (stretched or compressed within its elastic limit).
• Strain gauge resistance changes are typically measured in a bridge circuit, to allow for precise measurement of the small resistance changes, and to provide compensation for resistance variations due to temperature.

Chapter 10

DC NETWORK ANALYSIS

What is network analysis?

 

• Some circuit configurations (“networks”) cannot be solved by reduction according to series/parallel circuit rules, due to multiple unknown values.
• Mathematical techniques to solve for multiple unknowns (called “simultaneous equations” or “systems”) can be applied to basic Laws of circuits to solve networks

Branch current method

• Steps to follow for the “Branch Current” method of analysis:
• (1) Choose a node and assume directions of currents.
• (2) Write a KCL equation relating currents at the node.
• (3) Label resistor voltage drop polarities based on assumed currents.
• (4) Write KVL equations for each loop of the circuit, substituting the product IR for E in each resistor term of the equations.
• (5) Solve for unknown branch currents (simultaneous equations).
• (6) If any solution is negative, then the assumed direction of current for that solution is wrong!
• (7) Solve for voltage drops across all resistors (E=IR).

Mesh current method

• Steps to follow for the “Mesh Current” method of analysis:
• (1) Draw mesh currents in loops of circuit, enough to account for all components.
• (2) Label resistor voltage drop polarities based on assumed directions of mesh currents.
• (3) Write KVL equations for each loop of the circuit, substituting the product IR for E in each resistor term of the equation. Where two mesh currents intersect through a component, express the current as the algebraic sum of those two mesh currents (i.e. I₁ + I₂) if the currents go in the same direction through that component. If not, express the current as the difference (i.e. I₁ - I₂).
• (4) Solve for unknown mesh currents (simultaneous equations).
• (5) If any solution is negative, then the assumed current direction is wrong!
• (6) Algebraically add mesh currents to find current in components sharing multiple mesh currents.
• (7) Solve for voltage drops across all resistors (E=IR).

Mesh current by inspection

• Mesh current rules:
• This method assumes electron flow (not conventional current flow) voltage sources. Replace any current source in parallel with a resistor with an equivalent voltage source in series with an equivalent resistance.
• Ignoring current direction or voltage polarity on resistors, draw counterclockwise current loops traversing all components. Avoid nested loops.
• Write voltage-law equations in terms of unknown currents currents: I₁, I₂, and I₃. Equaton 1 coefficient 1, equation 2, coefficient 2, and equation 3 coefficient 3 are the positive sums of resistors around the respective loops.
• All other coefficients are negative, representative of the resistance common to a pair of loops. Equation 1 coefficent 2 is the resistor common to loops 1 and 2, coefficient 3 the resistor common to loops 1 an 3. Repeat for other equations and coefficients.

·           +(sum of R's loop 1)I₁ - (common R loop 1-2)I₂ - (common R loop 1-3)I₃   = E₁

·           -(common R loop 1-2)I₁ + (sum of R's loop 2)I₂ - (common R loop 2-3)I₃   = E₂

·           -(common R loop 1-3)I₁ - (common R loop 2-3)I₂ + (sum of R's loop 3)I₃   = E₃

• The right hand side of the equations is equal to any electron current flow voltage source. A voltage rise with respect to the counterclockwise assumed current is positive, and 0 for no voltage source.
• Solve equations for mesh currents:I₁, I₂, and I₃ . Solve for currents through individual resistors with KCL. Solve for voltages with Ohms Law and KVL.
• The modified mesh-current method avoids having to determine the signs of the equation coefficients by drawing all mesh currents counterclockwise for electron current flow.
• However, we do need to determine the sign of any voltage sources in the loop. The voltage source is positive if the assumed ccw current flows with the battery (source). The sign is negative if the assumed ccw current flows against the battery.
• See rules above for details.

Node voltage method

• Node voltage rules:
• Convert voltage sources in series with a resistor to an equivalent current source with the resistor in parallel.
• Change resistor values to conductances.
• Select a reference node(E₀)
• Assign unknown voltages (E₁)(E₂) ... (E_N)to remaining nodes.
• Write a KCL equation for each node 1,2, ... N. The positive coefficient of the first voltage in the first equation is the sum of conductances connected to the node. The coefficient for the second voltage in the second equation is the sum of conductances connected to that node. Repeat for coefficient of third voltage, third equation, and other equations. These coefficients fall on a diagonal.
• All other coefficients for all equations are negative, representing conductances between nodes. The first equation, second coefficient is the conductance from node 1 to node 2, the third coefficient is the conductance from node 1 to node 3. Fill in negative coefficients for other equations.
• The right hand side of the equations is the current source connected to the respective nodes.
• Solve system of equations for unknown node voltages.

• Given a network of conductances and current sources, the node voltage method of circuit analysis solves for unknown node voltages from KCL equations.
• See rules above for details in writing the equations by inspection.
• The unit of conductance G is the siemens S. Conductance is the reciprocal of resistance: G = 1/R

Introduction to network theorems

• Millman's Theorem treats circuits as a parallel set of series-component branches.
• All voltages entered and solved for in Millman's Theorem are polarity-referenced at the same point in the circuit (typically the bottom wire of the parallel network).

Superposition Theorem

• The Superposition Theorem states that a circuit can be analyzed with only one source of power at a time, the corresponding component voltages and currents algebraically added to find out what they'll do with all power sources in effect.
• To negate all but one power source for analysis, replace any source of voltage (batteries) with a wire; replace any current source with an open (break).

Thevenin's Theorem

• Thevenin's Theorem is a way to reduce a network to an equivalent circuit composed of a single voltage source, series resistance, and series load.
• Steps to follow for Thevenin's Theorem:
• (1) Find the Thevenin source voltage by removing the load resistor from the original circuit and calculating voltage across the open connection points where the load resistor used to be.
• (2) Find the Thevenin resistance by removing all power sources in the original circuit (voltage sources shorted and current sources open) and calculating total resistance between the open connection points.
• (3) Draw the Thevenin equivalent circuit, with the Thevenin voltage source in series with the Thevenin resistance. The load resistor re-attaches between the two open points of the equivalent circuit.
• (4) Analyze voltage and current for the load resistor following the rules for series circuits.

Norton's Theorem

• Norton's Theorem is a way to reduce a network to an equivalent circuit composed of a single current source, parallel resistance, and parallel load.
• Steps to follow for Norton's Theorem:
• (1) Find the Norton source current by removing the load resistor from the original circuit and calculating current through a short (wire) jumping across the open connection points where the load resistor used to be.
• (2) Find the Norton resistance by removing all power sources in the original circuit (voltage sources shorted and current sources open) and calculating total resistance between the open connection points.
• (3) Draw the Norton equivalent circuit, with the Norton current source in parallel with the Norton resistance. The load resistor re-attaches between the two open points of the equivalent circuit.
• (4) Analyze voltage and current for the load resistor following the rules for parallel circuits.

Thevenin-Norton equivalencies

• Thevenin and Norton resistances are equal.
• Thevenin voltage is equal to Norton current times Norton resistance.
• Norton current is equal to Thevenin voltage divided by Thevenin resistance.

Millman's Theorem revisited

You may have wondered where we got that strange equation for the determination of “Millman Voltage” across parallel branches of a circuit where each branch contains a series resistance and voltage source:

Parts of this equation seem familiar to equations we've seen before. For instance, the denominator of the large fraction looks conspicuously like the denominator of our parallel resistance equation. And, of course, the E/R terms in the numerator of the large fraction should give figures for current, Ohm's Law being what it is (I=E/R).

Now that we've covered Thevenin and Norton source equivalencies, we have the tools necessary to understand Millman's equation. What Millman's equation is actually doing is treating each branch (with its series voltage source and resistance) as a Thevenin equivalent circuit and then converting each one into equivalent Norton circuits.

 

Thus, in the circuit above, battery B₁ and resistor R₁ are seen as a Thevenin source to be converted into a Norton source of 7 amps (28 volts / 4 Ω) in parallel with a 4 Ω resistor. The rightmost branch will be converted into a 7 amp current source (7 volts / 1 Ω) and 1 Ω resistor in parallel. The center branch, containing no voltage source at all, will be converted into a Norton source of 0 amps in parallel with a 2 Ω resistor:

Since current sources directly add their respective currents in parallel, the total circuit current will be 7 + 0 + 7, or 14 amps. This addition of Norton source currents is what's being represented in the numerator of the Millman equation:

All the Norton resistances are in parallel with each other as well in the equivalent circuit, so they diminish to create a total resistance. This diminishing of source resistances is what's being represented in the denominator of the Millman's equation:

In this case, the resistance total will be equal to 571.43 milliohms (571.43 mΩ). We can re-draw our equivalent circuit now as one with a single Norton current source and Norton resistance:

Ohm's Law can tell us the voltage across these two components now (E=IR):

Let's summarize what we know about the circuit thus far. We know that the total current in this circuit is given by the sum of all the branch voltages divided by their respective currents. We also know that the total resistance is found by taking the reciprocal of all the branch resistance reciprocals. Furthermore, we should be well aware of the fact that total voltage across all the branches can be found by multiplying total current by total resistance (E=IR). All we need to do is put together the two equations we had earlier for total circuit current and total resistance, multiplying them to find total voltage:

The Millman's equation is nothing more than a Thevenin-to-Norton conversion matched together with the parallel resistance formula to find total voltage across all the branches of the circuit. So, hopefully some of the mystery is gone now!

Maximum Power Transfer Theorem

The Maximum Power Transfer Theorem is not so much a means of analysis as it is an aid to system design. Simply stated, the maximum amount of power will be dissipated by a load resistance when that load resistance is equal to the Thevenin/Norton resistance of the network supplying the power. If the load resistance is lower or higher than the Thevenin/Norton resistance of the source network, its dissipated power will be less than maximum.

This is essentially what is aimed for in stereo system design, where speaker “impedance” is matched to amplifier “impedance” for maximum sound power output. Impedance, the overall opposition to AC and DC current, is very similar to resistance, and must be equal between source and load for the greatest amount of power to be transferred to the load. A load impedance that is too high will result in low power output. A load impedance that is too low will not only result in low power output, but possibly overheating of the amplifier due to the power dissipated in its internal (Thevenin or Norton) impedance.

Taking our Thevenin equivalent example circuit, the Maximum Power Transfer Theorem tells us that the load resistance resulting in greatest power dissipation is equal in value to the Thevenin resistance (in this case, 0.8 Ω):

With this value of load resistance, the dissipated power will be 39.2 watts:

If we were to try a lower value for the load resistance (0.5 Ω instead of 0.8 Ω, for example), our power dissipated by the load resistance would decrease:

 

Power dissipation increased for both the Thevenin resistance and the total circuit, but it decreased for the load resistor. Likewise, if we increase the load resistance (1.1 Ω instead of 0.8 Ω, for example), power dissipation will also be less than it was at 0.8 Ω exactly:

If you were designing a circuit for maximum power dissipation at the load resistance, this theorem would be very useful. Having reduced a network down to a Thevenin voltage and resistance (or Norton current and resistance), you simply set the load resistance equal to that Thevenin or Norton equivalent (or vice versa) to ensure maximum power dissipation at the load. Practical applications of this might include stereo amplifier design (seeking to maximize power delivered to speakers) or electric vehicle design (seeking to maximize power delivered to drive motor).

• REVIEW:
• The Maximum Power Transfer Theorem states that the maximum amount of power will be dissipated by a load resistance if it is equal to the Thevenin or Norton resistance of the network supplying power.

Δ-Y and Y-Δ conversions

In many circuit applications, we encounter components connected together in one of two ways to form a three-terminal network: the “Delta,” or Δ (also known as the “Pi,” or π) configuration, and the “Y” (also known as the “T”) configuration.

It is possible to calculate the proper values of resistors necessary to form one kind of network (Δ or Y) that behaves identically to the other kind, as analyzed from the terminal connections alone. That is, if we had two separate resistor networks, one Δ and one Y, each with its resistors hidden from view, with nothing but the three terminals (A, B, and C) exposed for testing, the resistors could be sized for the two networks so that there would be no way to electrically determine one network apart from the other. In other words, equivalent Δ and Y networks behave identically.

There are several equations used to convert one network to the other:

Δ and Y networks are seen frequently in 3-phase AC power systems (a topic covered in volume II of this book series), but even then they're usually balanced networks (all resistors equal in value) and conversion from one to the other need not involve such complex calculations. When would the average technician ever need to use these equations?

A prime application for Δ-Y conversion is in the solution of unbalanced bridge circuits, such as the one below:

Solution of this circuit with Branch Current or Mesh Current analysis is fairly involved, and neither the Millman nor Superposition Theorems are of any help, since there's only one source of power. We could use Thevenin's or Norton's Theorem, treating R₃ as our load, but what fun would that be?

If we were to treat resistors R₁, R₂, and R₃ as being connected in a Δ configuration (R_ab, R_ac, and R_bc, respectively) and generate an equivalent Y network to replace them, we could turn this bridge circuit into a (simpler) series/parallel combination circuit:

After the Δ-Y conversion . . .

If we perform our calculations correctly, the voltages between points A, B, and C will be the same in the converted circuit as in the original circuit, and we can transfer those values back to the original bridge configuration.

 

 

 

Resistors R₄ and R₅, of course, remain the same at 18 Ω and 12 Ω, respectively. Analyzing the circuit now as a series/parallel combination, we arrive at the following figures:

We must use the voltage drops figures from the table above to determine the voltages between points A, B, and C, seeing how the add up (or subtract, as is the case with voltage between points B and C):

Now that we know these voltages, we can transfer them to the same points A, B, and C in the original bridge circuit:

Voltage drops across R₄ and R₅, of course, are exactly the same as they were in the converted circuit.

At this point, we could take these voltages and determine resistor currents through the repeated use of Ohm's Law (I=E/R):

A quick simulation with SPICE will serve to verify our work:[spi]

unbalanced bridge circuit  

v1 1 0 

r1 1 2 12      

r2 1 3 18      

r3 2 3 6

r4 2 0 18      

r5 3 0 12      

.dc v1 10 10 1 

.print dc v(1,2) v(1,3) v(2,3) v(2,0) v(3,0)   

.end   

 

v1            v(1,2)      v(1,3)      v(2,3)      v(2)        v(3)           

1.000E+01     4.706E+00   5.294E+00   5.882E-01   5.294E+00   4.706E+00

 

The voltage figures, as read from left to right, represent voltage drops across the five respective resistors, R₁ through R₅. I could have shown currents as well, but since that would have required insertion of “dummy” voltage sources in the SPICE netlist, and since we're primarily interested in validating the Δ-Y conversion equations and not Ohm's Law, this will suffice.

• REVIEW:
• “Delta” (Δ) networks are also known as “Pi” (π) networks.
• “Y” networks are also known as “T” networks.
• Δ and Y networks can be converted to their equivalent counterparts with the proper resistance equations. By “equivalent,” I mean that the two networks will be electrically identical as measured from the three terminals (A, B, and C).
• A bridge circuit can be simplified to a series/parallel circuit by converting half of it from a Δ to a Y network. After voltage drops between the original three connection points (A, B, and C) have been solved for, those voltages can be transferred back to the original bridge circuit, across those same equivalent points.

Chapter 11

BATTERIES AND POWER SYSTEMS

Electron activity in chemical reactions

• Atoms bound together by electrons are called molecules.
• Ionic bonds are molecular unions formed when an electron-deficient atom (a positive ion) joins with an electron-excessive atom (a negative ion).
• Electrochemical reactions involve the transfer of electrons between atoms. This transfer can be harnessed to form an electric current.
• A cell is a device constructed to harness such chemical reactions to generate electric current.
• A cell is said to be discharged when its internal chemical reserves have been depleted through use.
• A secondary cell's chemistry can be reversed (recharged) by forcing current backwards through it.
• A primary cell cannot be practically recharged.
• Lead-acid cell charge can be assessed with an instrument called a hydrometer, which measures the density of the electrolyte liquid. The denser the electrolyte, the stronger the acid concentration, and the greater charge state of the cell.

Battery construction

• A battery is a cluster of cells connected together for greater voltage and/or current capacity.
• Cells connected together in series (polarities aiding) results in greater total voltage.
• Physical cell size impacts cell resistance, which in turn impacts the ability for the cell to supply current to a circuit. Generally, the larger the cell, the less its internal resistance.
• Cells connected together in parallel results in less total resistance, and potentially greater total current.

Battery ratings

• The amp-hour is a unit of battery energy capacity, equal to the amount of continuous current multiplied by the discharge time, that a battery can supply before exhausting its internal store of chemical energy.
• An amp-hour battery rating is only an approximation of the battery's charge capacity, and should be trusted only at the current level or time specified by the manufacturer. Such a rating cannot be extrapolated for very high currents or very long times with any accuracy.
• Discharged batteries lose voltage and increase in resistance. The best check for a dead battery is a voltage test under load.

Special-purpose batteries

• mercury standard cells are special types of batteries which were once used as voltage calibration standards before the advent of precision semiconductor reference devices.
• A fuel cell is a kind of battery that uses a combustible fuel and oxidizer as reactants to generate electricity. They are promising sources of electrical power in the future, "burning" fuels with very low emissions.
• A solar cell uses ambient light energy to motivate electrons from one electrode to the other, producing voltage (and current, providing an external circuit).
• A chemical detection cell is a special type of voltaic cell which produces voltage proportional to the concentration of an applied substance (usually a specific gas in ambient air).

Practical considerations

• Connecting batteries in series increases voltage, but does not increase overall amp-hour capacity.
• All batteries in a series bank must have the same amp-hour rating.
• Connecting batteries in parallel increases total current capacity by decreasing total resistance, and it also increases overall amp-hour capacity.
• All batteries in a parallel bank must have the same voltage rating.
• Batteries can be damaged by excessive cycling and overcharging.
• Water-based electrolyte batteries are capable of generating explosive hydrogen gas, which must not be allowed to accumulate in an area.

 

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