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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 ∠ 203o (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 180o 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: XL = 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! ZTotal = Z1 + Z2 + . . . Zn
  • A purely resistive impedance will always have a phase angle of exactly 0o (ZR = R Ω ∠ 0o).
  • A purely inductive impedance will always have a phase angle of exactly +90o (ZL = XL Ω ∠ 90o).
  • 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 0o and +90o. The circuit current will have a phase angle somewhere between 0o and -90o.
  • 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! ZTotal = 1/(1/Z1 + 1/Z2 + . . . 1/Zn)
  • 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 0o and +90o. The circuit current will have a phase angle somewhere between 0o and -90o.
  • 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: XC = 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! ZTotal = Z1 + Z2 + . . . Zn
  • 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 0o (ZR = R Ω ∠ 0o).
  • A purely capacitive impedance will always have a phase angle of exactly -90o (ZC = XC Ω ∠ -90o).
  • 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 0o and -90o.
  • 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! ZTotal = 1/(1/Z1 + 1/Z2 + . . . 1/Zn)
  • 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 0o and -90o. The circuit current will have a phase angle somewhere between 0o and +90o.
  • 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 90o (actually, -90o) 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 90o 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 0o and 90o 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: ZTotal = Z1 + Z2 + . . . Zn
  • 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: Rparallel = 1 / (1/R1 + 1/R2 + . . . 1/Rn). 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: Xparallel = 1/(1/X1 + 1/X2 + . . . 1/Xn). 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 = Pstored/Pdissipated = 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.

 

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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 90o 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 90o.
  • 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, 180o 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 180o 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 120o 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 90o 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 0o and 90o.

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: Zload 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.

 

 

 

 

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