Chapter 12

PHYSICS OF CONDUCTORS AND INSULATORS

Introduction

• Electrical conductivity of a material is determined by the configuration of electrons in that materials atoms and molecules (groups of bonded atoms).
• All normal conductors possess resistance to some degree.
• Electrons flowing through a conductor with (any) resistance will produce some amount of voltage drop across the length of that conductor.

Conductor size

• Electrons flow through large-diameter wires easier than small-diameter wires, due to the greater cross-sectional area they have in which to move.
• Rather than measure small wire sizes in inches, the unit of "mil" (1/1000 of an inch) is often employed.
• The cross-sectional area of a wire can be expressed in terms of square units (square inches or square mils), circular mils, or "gauge" scale.
• Calculating square-unit wire area for a circular wire involves the circle area formula:
• Calculating circular-mil wire area for a circular wire is much simpler, due to the fact that the unit of "circular mil" was sized just for this purpose: to eliminate the "pi" and the d/2 (radius) factors in the formula.
• There are π (3.1416) square mils for every 4 circular mils.
• The gauge system of wire sizing is based on whole numbers, larger numbers representing smaller-area wires and vice versa. Wires thicker than 1 gauge are represented by zeros: 0, 00, 000, and 0000 (spoken "single-ought," "double-ought," "triple-ought," and "quadruple-ought."
• Very large wire sizes are rated in thousands of circular mils (MCM's), typical for busbars and wire sizes beyond 4/0.
• Busbars are solid bars of copper or aluminum used in high-current circuit construction. Connections made to busbars are usually welded or bolted, and the busbars are often bare (uninsulated), supported away from metal frames through the use of insulating standoffs.

Conductor ampacity

• Wire resistance creates heat in operating circuits. This heat is a potential fire ignition hazard.
• Skinny wires have a lower allowable current ("ampacity") than fat wires, due to their greater resistance per unit length, and consequently greater heat generation per unit current.
• The National Electrical Code (NEC) specifies ampacities for power wiring based on allowable insulation temperature and wire application.

Fuses

• A fuse is a small, thin conductor designed to melt and separate into two pieces for the purpose of breaking a circuit in the event of excessive current.
• A circuit breaker is a specially designed switch that automatically opens to interrupt circuit current in the event of an overcurrent condition. They can be "tripped" (opened) thermally, by magnetic fields, or by external devices called "protective relays," depending on the design of breaker, its size, and the application.
• Fuses are primarily rated in terms of maximum current, but are also rated in terms of how much voltage drop they will safely withstand after interrupting a circuit.
• Fuses can be designed to blow fast, slow, or anywhere in between for the same maximum level of current.
• The best place to install a fuse in a grounded power system is on the ungrounded conductor path to the load. That way, when the fuse blows there will only be the grounded (safe) conductor still connected to the load, making it safer for people to be around.

Specific resistance

• Conductor resistance increases with increased length and decreases with increased cross-sectional area, all other factors being equal.
• Specific Resistance ("ρ") is a property of any conductive material, a figure used to determine the end-to-end resistance of a conductor given length and area in this formula: R = ρl/A
• Specific resistance for materials are given in units of Ω-cmil/ft or Ω-meters (metric). Conversion factor between these two units is 1.66243 x 10^-9 Ω-meters per Ω-cmil/ft, or 1.66243 x 10^-7 Ω-cm per Ω-cmil/ft.
• If wiring voltage drop in a circuit is critical, exact resistance calculations for the wires must be made before wire size is chosen.

Temperature coefficient of resistance

• Most conductive materials change specific resistance with changes in temperature. This is why figures of specific resistance are always specified at a standard temperature (usually 20^o or 25^o Celsius).
• The resistance-change factor per degree Celsius of temperature change is called the temperature coefficient of resistance. This factor is represented by the Greek lower-case letter "alpha" (α).
• A positive coefficient for a material means that its resistance increases with an increase in temperature. Pure metals typically have positive temperature coefficients of resistance. Coefficients approaching zero can be obtained by alloying certain metals.
• A negative coefficient for a material means that its resistance decreases with an increase in temperature. Semiconductor materials (carbon, silicon, germanium) typically have negative temperature coefficients of resistance.
• The formula used to determine the resistance of a conductor at some temperature other than what is specified in a resistance table is as follows:
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Superconductivity

• Superconductors are materials which have absolutely zero electrical resistance.
• All presently known superconductive materials need to be cooled far below ambient temperature to superconduct. The maximum temperature at which they do so is called the transition temperature.

Insulator breakdown voltage

• With a high enough applied voltage, electrons can be freed from the atoms of insulating materials, resulting in current through that material.
• The minimum voltage required to "violate" an insulator by forcing current through it is called the breakdown voltage, or dielectric strength.
• The thicker a piece of insulating material, the higher the breakdown voltage, all other factors being equal.
• Specific dielectric strength is typically rated in one of two equivalent units: volts per mil, or kilovolts per inch.

Chapter 13

CAPACITORS

Electric fields and capacitance

• Capacitors react against changes in voltage by supplying or drawing current in the direction necessary to oppose the change.
• When a capacitor is faced with an increasing voltage, it acts as a load: drawing current as it absorbs energy (current going in the negative side and out the positive side, like a resistor).
• When a capacitor is faced with a decreasing voltage, it acts as a source: supplying current as it releases stored energy (current going out the negative side and in the positive side, like a battery).
• The ability of a capacitor to store energy in the form of an electric field (and consequently to oppose changes in voltage) is called capacitance. It is measured in the unit of the Farad (F).
• Capacitors used to be commonly known by another term: condenser (alternatively spelled "condensor").

Capacitors and calculus

• Capacitances diminish in series.
• Capacitances add in parallel.

Practical considerations

Capacitors, like all electrical components, have limitations which must be respected for the sake of reliability and proper circuit operation.

Working voltage: Since capacitors are nothing more than two conductors separated by an insulator (the dielectric), you must pay attention to the maximum voltage allowed across it. If too much voltage is applied, the "breakdown" rating of the dielectric material may be exceeded, resulting in the capacitor internally short-circuiting.

Polarity: Some capacitors are manufactured so they can only tolerate applied voltage in one polarity but not the other. This is due to their construction: the dielectric is a microscopically thin layer of insulation deposited on one of the plates by a DC voltage during manufacture. These are called electrolytic capacitors, and their polarity is clearly marked.

Reversing voltage polarity to an electrolytic capacitor may result in the destruction of that super-thin dielectric layer, thus ruining the device. However, the thinness of that dielectric permits extremely high values of capacitance in a relatively small package size. For the same reason, electrolytic capacitors tend to be low in voltage rating as compared with other types of capacitor construction.

Equivalent circuit: Since the plates in a capacitor have some resistance, and since no dielectric is a perfect insulator, there is no such thing as a "perfect" capacitor. In real life, a capacitor has both a series resistance and a parallel (leakage) resistance interacting with its purely capacitive characteristics:

Fortunately, it is relatively easy to manufacture capacitors with very small series resistances and very high leakage resistances!

Physical Size: For most applications in electronics, minimum size is the goal for component engineering. The smaller components can be made, the more circuitry can be built into a smaller package, and usually weight is saved as well. With capacitors, there are two major limiting factors to the minimum size of a unit: working voltage and capacitance. And these two factors tend to be in opposition to each other. For any given choice in dielectric materials, the only way to increase the voltage rating of a capacitor is to increase the thickness of the dielectric. However, as we have seen, this has the effect of decreasing capacitance. Capacitance can be brought back up by increasing plate area. but this makes for a larger unit. This is why you cannot judge a capacitor's rating in Farads simply by size. A capacitor of any given size may be relatively high in capacitance and low in working voltage, vice versa, or some compromise between the two extremes. Take the following two photographs for example:

This is a fairly large capacitor in physical size, but it has quite a low capacitance value: only 2 µF. However, its working voltage is quite high: 2000 volts! If this capacitor were re-engineered to have a thinner layer of dielectric between its plates, at least a hundredfold increase in capacitance might be achievable, but at a cost of significantly lowering its working voltage. Compare the above photograph with the one below. The capacitor shown in the lower picture is an electrolytic unit, similar in size to the one above, but with very different values of capacitance and working voltage:

The thinner dielectric layer gives it a much greater capacitance (20,000 µF) and a drastically reduced working voltage (35 volts continuous, 45 volts intermittent).

Here are some samples of different capacitor types, all smaller than the units shown previously:

The electrolytic and tantalum capacitors are polarized (polarity sensitive), and are always labeled as such. The electrolytic units have their negative (-) leads distinguished by arrow symbols on their cases. Some polarized capacitors have their polarity designated by marking the positive terminal. The large, 20,000 µF electrolytic unit shown in the upright position has its positive (+) terminal labeled with a "plus" mark. Ceramic, mylar, plastic film, and air capacitors do not have polarity markings, because those types are nonpolarized (they are not polarity sensitive).

Capacitors are very common components in electronic circuits. Take a close look at the following photograph -- every component marked with a "C" designation on the printed circuit board is a capacitor:

Some of the capacitors shown on this circuit board are standard electrolytic: C₃₀ (top of board, center) and C₃₆ (left side, 1/3 from the top). Some others are a special kind of electrolytic capacitor called tantalum, because this is the type of metal used to make the plates. Tantalum capacitors have relatively high capacitance for their physical size. The following capacitors on the circuit board shown above are tantalum: C₁₄ (just to the lower-left of C₃₀), C₁₉ (directly below R₁₀, which is below C₃₀), C₂₄ (lower-left corner of board), and C₂₂ (lower-right).

Examples of even smaller capacitors can be seen in this photograph:

The capacitors on this circuit board are "surface mount devices" as are all the resistors, for reasons of saving space. Following component labeling convention, the capacitors can be identified by labels beginning with the letter "C".

Chapter 14

MAGNETISM AND ELECTROMAGNETISM

Permanent magnets

• Lodestone (also called Magnetite) is a naturally-occurring "permanent" magnet mineral. By "permanent," it is meant that the material maintains a magnetic field with no external help. The characteristic of any magnetic material to do so is called retentivity.
• Ferromagnetic materials are easily magnetized.
• Paramagnetic materials are magnetized with more difficulty.
• Diamagnetic materials actually tend to repel external magnetic fields by magnetizing in the opposite direction.

Electromagnetism

• The permeability of a material changes with the amount of magnetic flux forced through it.
• The specific relationship of force to flux (field intensity H to flux density B) is graphed in a form called the normal magnetization curve.
• It is possible to apply so much magnetic field force to a ferromagnetic material that no more flux can be crammed into it. This condition is known as magnetic saturation.
• When the retentivity of a ferromagnetic substance interferes with its re-magnetization in the opposite direction, a condition known as hysteresis occurs.

Electromagnetic induction

• A magnetic field of changing intensity perpendicular to a wire will induce a voltage along the length of that wire. The amount of voltage induced depends on the rate of change of the magnetic field flux and the number of turns of wire (if coiled) exposed to the change in flux.
• Faraday's equation for induced voltage: e = N(dΦ/dt)
• A current-carrying wire will experience an induced voltage along its length if the current changes (thus changing the magnetic field flux perpendicular to the wire, thus inducing voltage according to Faraday's formula). A device built specifically to take advantage of this effect is called an inductor.

Mutual inductance

• Mutual inductance is where the magnetic field generated by a coil of wire induces voltage in an adjacent coil of wire.
• A transformer is a device constructed of two or more coils in close proximity to each other, with the express purpose of creating a condition of mutual inductance between the coils.
• Transformers only work with changing voltages, not steady voltages. Thus, they may be classified as an AC device and not a DC device.

Chapter 15

INDUCTORS

Magnetic fields and inductance

• Inductors react against changes in current by dropping voltage in the polarity necessary to oppose the change.
• When an inductor is faced with an increasing current, it acts as a load: dropping voltage as it absorbs energy (negative on the current entry side and positive on the current exit side, like a resistor).
• When an inductor is faced with a decreasing current, it acts as a source: creating voltage as it releases stored energy (positive on the current entry side and negative on the current exit side, like a battery).
• The ability of an inductor to store energy in the form of a magnetic field (and consequently to oppose changes in current) is called inductance. It is measured in the unit of the Henry (H).
• Inductors used to be commonly known by another term: choke. In large power applications, they are sometimes referred to as reactors.

Inductors and calculus

• Inductances add in series.
• Inductances diminish in parallel.

Practical considerations

Inductors, like all electrical components, have limitations which must be respected for the sake of reliability and proper circuit operation.

Rated current: Since inductors are constructed of coiled wire, and any wire will be limited in its current-carrying capacity by its resistance and ability to dissipate heat, you must pay attention to the maximum current allowed through an inductor.

Equivalent circuit: Since inductor wire has some resistance, and circuit design constraints typically demand the inductor be built to the smallest possible dimensions, there is no such thing as a "perfect" inductor. Inductor coil wire usually presents a substantial amount of series resistance, and the close spacing of wire from one coil turn to another (separated by insulation) may present measurable amounts of stray capacitance to interact with its purely inductive characteristics. Unlike capacitors, which are relatively easy to manufacture with negligible stray effects, inductors are difficult to find in "pure" form. In certain applications, these undesirable characteristics may present significant engineering problems.

Inductor size: Inductors tend to be much larger, physically, than capacitors are for storing equivalent amounts of energy. This is especially true considering the recent advances in electrolytic capacitor technology, allowing incredibly large capacitance values to be packed into a small package. If a circuit designer needs to store a large amount of energy in a small volume and has the freedom to choose either capacitors or inductors for the task, he or she will most likely choose a capacitor. A notable exception to this rule is in applications requiring huge amounts of either capacitance or inductance to store electrical energy: inductors made of superconducting wire (zero resistance) are more practical to build and safely operate than capacitors of equivalent value, and are probably smaller too.

Interference: Inductors may affect nearby components on a circuit board with their magnetic fields, which can extend significant distances beyond the inductor. This is especially true if there are other inductors nearby on the circuit board. If the magnetic fields of two or more inductors are able to "link" with each others' turns of wire, there will be mutual inductance present in the circuit as well as self-inductance, which could very well cause unwanted effects. This is another reason why circuit designers tend to choose capacitors over inductors to perform similar tasks: capacitors inherently contain their respective electric fields neatly within the component package and therefore do not typically generate any "mutual" effects with other components.

Chapter 16

RC AND L/R TIME CONSTANTS

Electrical transients

• Capacitors act somewhat like secondary-cell batteries when faced with a sudden change in applied voltage: they initially react by producing a high current which tapers off over time.
• A fully discharged capacitor initially acts as a short circuit (current with no voltage drop) when faced with the sudden application of voltage. After charging fully to that level of voltage, it acts as an open circuit (voltage drop with no current).
• In a resistor-capacitor charging circuit, capacitor voltage goes from nothing to full source voltage while current goes from maximum to zero, both variables changing most rapidly at first, approaching their final values slower and slower as time goes on.

Inductor transient response

• A fully "discharged" inductor (no current through it) initially acts as an open circuit (voltage drop with no current) when faced with the sudden application of voltage. After "charging" fully to the final level of current, it acts as a short circuit (current with no voltage drop).
• In a resistor-inductor "charging" circuit, inductor current goes from nothing to full value while voltage goes from maximum to zero, both variables changing most rapidly at first, approaching their final values slower and slower as time goes on.

Voltage and current calculations

• Universal Time Constant Formula:
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• To analyze an RC or L/R circuit, follow these steps:
• (1): Determine the time constant for the circuit (RC or L/R).
• (2): Identify the quantity to be calculated (whatever quantity whose change is directly opposed by the reactive component. For capacitors this is voltage; for inductors this is current).
• (3): Determine the starting and final values for that quantity.
• (4): Plug all these values (Final, Start, time, time constant) into the universal time constant formula and solve for change in quantity.
• (5): If the starting value was zero, then the actual value at the specified time is equal to the calculated change given by the universal formula. If not, add the change to the starting value to find out where you're at.

Why L/R and not LR?

• To analyze an RC or L/R circuit more complex than simple series, convert the circuit into a Thevenin equivalent by treating the reactive component (capacitor or inductor) as the "load" and reducing everything else to an equivalent circuit of one voltage source and one series resistor. Then, analyze what happens over time with the universal time constant formula.

Solving for unknown time

• To determine the time it takes for an RC or L/R circuit to reach a certain value of voltage or current, you'll have to modify the universal time constant formula to solve for time instead of change.
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• The mathematical function for reversing an exponent of "e" is the natural logarithm (ln), provided on any scientific calculator.