Electrical Engineering Equation Descriptions

**Table of Contents**

Charges and the Electric Field

**Charges and the Electric Field**

All electrical phenomena have their origin in the stable, fundamental particles that carry electric charge---protons and electrons. They carry an equal and opposite charges of 1.6×10^{-19} C (coulombs); by convention, protons have the positive charge and electrons the negative. Each particle is said to generate an electric field, represented by field lines, as in Figure 1.

Work is done whenever a charge *q* is moved along one of the field lines within the vector field E:

It is also important to note that φ(r) is the potential of the charge at position r. When the charge moves perpendicular to the field line, . It is therefore on an equipotential surface, to which field lines are everywhere normal. In a vacuum, the system is conservative; if, in Equation 1 r_{1}=r_{2}, then φ(r_{1})- φ(r_{2})=0 and no work is done. In materials, however, this is generally not the case.

Electrical engineering is often concerned with the *movement* of charge, such as the flow of electrons through a circuit, or through a power cable to your computer. The amount of charge that passes through a given point in a given time is measured as **current**, the SI unit for which is the ampere (abbreviated A) which equals 1 coulomb/second. Typically, the *direction* of current is defined as the direction of the net flow of positive charge (i.e. the *opposite* direction of electron flow).

Ohm’s law (2) is the simplest formation of the relationship between electrical potential and current flow through a resistive medium.

V = I R (2)

Electrical **resistance** is a measure of how much an object (typically a wire or specifically designed resistor) resists the flow of current. The SI unit for resistance is the ohm, represented by the Greek letter Ω (omega). It is directly proportional to the length of the wire and inversely proportional to the cross sectional area. A shorter, wider wire offers less resistance to current in the same way that a short, wide road offers easier travel from Point A to Point B. The Greek letter ρ (rho) represents the resistivity of a material - how much it resists the flow of electrons based on its molecular structure. Ergo, different metals have different specific electrical resistance.

Re-arrange the first part of Ohm’s Law:

I = V/R.

Imagine pouring a bucket of water onto a paddle wheel, such as Figure 1. If you double the distance between the bucket and the ground, you would double the change in potential energy, and thus the flow of current over the wheel would double. If you changed the shape or material of the wheel so that it resisted the flow of water half as much, the current would also double. In this example, the water represents the electrical current and the paddle wheel represents the resistor across which the current flows. If, in an actual circuit, one were to control the current across a resistor, instead of the voltage, the voltage drop across the resistor would increase as the current increased, and the reverse. This would be exactly akin to the water-pressure building up in a reservoir in order to drive the wheel more strongly. Thus a return to V=IR.

Resistance also varies with temperature---sometimes non-linearly, as in the case of superconductors. For most applications, however, it is sufficient to approximate the change in temperature as a linear change about the resistance R_{0} at a reference temperature T_{0}, usually 20ºC. The resulting expression is

R_{T} = R_{0} + R_{0} α (T-T0) = R_{0} + R_{0} α ΔT,

where α is a material-specific coefficient. It is usually positive; vibrations in the lattice scatter electrons, increasing resistivity, and higher temperatures increase scattering.

Q: Given a 24 gauge speaker hook-up wire, how long would the wire have to be to offer a resistance of 1 milliohm? Assume the wire is made of copper (specific electrical resistance ρ of 1.68×10^{-8} Ω-m).

A:

After consulting the standard engineering tables, we find that 24ga wire has a cross sectional area of 0.820 mm^{2}. Thus

From which we may infer that the signal—and therefore sound quality—over such wire is not likely to degrade in any home theater application.

Q: Find the current across a 100Ω resistor given P = 10W.

A:

A circuit is simply an interconnection of electrical devices, and a *series* circuit implies a singular, direct path from the **source **to the **ground**. An ideal currentsource supplies a constant (with respect to time) *I*_{s} **[1]**amperes of current in the direction of the arrow symbol and will provide whatever voltage is required by the circuit’s components. Likewise, an ideal voltage source supplies a constant *v*_{s} volts of voltage across its terminals and will provide whatever current is required by the circuit’s components. Ideal ground is a common voltage reference point where the relative voltage is defined as 0, and acts as an infinite sink that can absorb an unlimited amount of current without changing its potential. In real life, sources are not constant, especially when they are introduced or removed from a circuit (turning a device on or off) and ground offers some minimal resistance.

By knowing that a circuit’s elements are in series, we can reach several conclusions. First, the current is uniform through all elements of the circuit. This is a necessary consequence of the conservation of charge; what enters the series must either exit or remain trapped inside. The second conclusion is that the total circuit resistance is equal to the sum of all resistors in the circuit. The proof runs by observing that the total drop in potential must equal the individual potential drops:

Q: Consider a series circuit where V_{s} = 50V, R_{1} = 50Ω, R_{2} = 100Ω, and R_{3} is unknown. If the current is measured to be 0.1A, what must the value of R_{3} be?

A:

What about when the path from the source to the ground is *not* singular? In this case, we say that circuit elements are in parallel. Resistors can only be considered in parallel if they have a common connection to a conductor. In Figure 4, R_{1,} R_{2}, and R_{3} are all in parallel.

Current flows through the path of least resistance, so the amount of current through a branch is inversely proportional to the resistance of that branch, i.e. the greatest amount of current flows through the smallest resistor’s path. Harking back to the water analogy of before, the common source is like a great reservoir feeding a number of water-wheels. Water will flow toward the least resistive wheel until the amount of power released is as much as the water level in the reservoir can support—but by construction, the level in the reservoir is not impacted by this draw. Another wheel will also be able to draw from the same potential source, and the water-current to another wheel will increase until it, too, has as much water as the reservoir’s level will admit. Obviously the total outflow from the reservoir has increased with the addition of the second wheel. In both cases, though, the amount of water flowing across each is a matter of the level of the reservoir and the resistance of the wheel. Returning to the electrical situation of Figure 4, the potential at the source is constant, so the current across any of the resistors will be that necessary to have the same voltage drop across each. It follows that the incorporation of parallel resistors decreases the effective resistance, as total current increases whilst holding the same potential. This is expressed mathematically, using Ohm’s law and the constant potential, as

It is good practice to check that the calculated total resistance is less than the value of the largest parallel resistor, since each additional parallel resistor allows current to flow more easily from the source to ground.

Q: Consider a parallel circuit where i_{s} = 10A, R_{1} = 50Ω and R_{2} = 100Ω. What resistance value would R_{3} have to be so that the voltage across all three parallel resistors is 10V?

An inductor is a circuit element, usually a coiled wire, which resists *change* in electrical current. When current flows through an inductor, a magnetic field is created due to the nature of its shape. Figure 3 shows how a magnetic field is induced perpendicular to the flow of current, following the right-hand rule. When the current changes, the magnetic field induces a voltage across the inductor that resists the change in current. The letter L stands for inductance, the ratio of induced voltage to the rate of change of current, the standard unit for which is the henry (abbreviated H).

where L is inductance, I is current, t is time.

Q: Find the total inductance of three inductors in parallel where L_{1} = 10H, L_{2} = 20H, and L_{3} = 30H.

A:

A capacitor is an electronic component used to store energy in an electric field. The simplest form—the parallel plate capacitor—consists of two unconnected parallel surfaces; a cartoon of this style is shown in the figure below. Recall formula (1), in which the potential between two points was the line integral of the electric field between those points. When a voltage V is applied across an uncharged capacitor, charges accumulate on both sides until the electric field between the two plates satisfies. This may be simplified: in the parallel place capacitor, the electric field is effectively constant between the two plates; fringing effects at the edges are only a minor correction and may be neglected. Therefore the electric field moves out of the integral and the equation simplifies to

V = E d.

Of course, there are a wide range of capacitors available, matched to needs, availability, and cost,[2]but above all the most important characteristic of a capacitor is its capacitance—the ratio of the total electric charge on each terminal (Q) to the voltage difference across the terminals (V). Capacitance is measure in farads (abbreviated F), equal to one coulomb/volt, though as a practical matter the majority of capacitors on the market are in the micro- and picofarad ranges. With this definition, it is easy to derive the capacitance of the parallel plate capacitor.

Begin with Gauss’ Law in SI unit form (to convert to cgs units, recal ε_{0}= (4π)^{-1} ), and consider the amount of charge contained in a box-shaped surface about a single planes, which carries a surface charge σ. This box has sides perpendicular to the plate and a face surface area *a*:

q_{encl}= a σ

Symmetry demands that the electric field be uniformly distributed and perpendicular to the surface, in both directions from the plane, so that the divergence theorem gives

Of course, there are two planes in the capacitor, of matched surface charge, so that the field inside the capacitor becomes

This is good enough for a vacuum. However, it is common that materials known as dielectrics are inserted into the gap, where they decrease the field by their own polarization. This is represented as a ratio to the vacuum permittivity; the factor K is the dielectric constant of the particular material and ε_{0} is the aforementioned vacuum permittivity. Thus

The capacitor plates have area *A*, so the surface charge σ can be broken into the total charge and the area, which in turn allows for the application of the definition of capacitance as a function of voltage. Thus

From this derivation it follows immediately that the rules for finding equivalent capacitance are the *opposite* of resistance and inductance. If two identical capacitors are in parallel, the effective area fed by the voltage has doubled. Thus voltage across them is the same, but the total charge Q is twice that of a single capacitor, doubling the capacitance. With series capacitors, the stored charge Q on each capacitor is the same, but there is a voltage drop across each additional capacitor. That is to say,

Q: Find the equivalent capacitance for the circuit shown.

A:

It is also imperative to determine how much energy a capacitor holds. Work is done in moving the charges toward the plates; the work done for each differential fragment of charge is proportional to the distance moved in electric field developed by all the prior charges. Thus

A more useful form would be in terms of capacitance and voltage across it, however. Using the equations developed earlier, this integral can be recast as a function of capacitance and voltage, therefore:

The SI unit for Work is the Joule, which is equal to the product of 1 Volt and 1 Coulomb.

Consider two possible circuits:

Circuit (a) is the simplest RC circuit, and (b) is the simplest LC circuit. It is possible to analyze the behavior of both these systems in terms of charge and their time derivatives—the ODEs alluded to earlier. Beginning with the RC circuit, recall the following: . Kirchhoff’s Voltage Law asserts that a the various voltage changes in a loop must sum to zero, so the total voltage around circuit (a) will be

a first-order ordinary differential equation for charge as a function of time. Rearranging gives us a very familiar form, easily addressed with standard techniques:

A similar analysis follows for the inductor, which is susceptible to the change in the current flow:

Thus

In either case, the time constant characterizes a first-order circuit. It is approximately true that voltage is reduced to one third its original value at t = 1τ, and that voltage drops to 0 after t = 5τ. It can therefore be seen that, for an RC circuit, greater circuit resistance prevents the charge from flowing quickly, whilst a greater capacitance provides more charge to dissipate. Thus an increase in either R or C increases time-constant, or decreases the rate at which the voltage decays. The situation is somewhat reversed in a first order RL circuit: a smaller resistance increases the current flow for a given voltage, to which the inductance reacts, building up more voltage and keeping the potential high. By restricting the current with a large resistance, the system can dissipate more quickly.

A filter is an electrical component that passes or blocks as a function of frequency. A range of frequencies that pass through a particular filter is called a passband, and a range of frequencies blocked is called a stopband. The transition between these two areas is, in reality, far from ideal, but can be characterized by the cutoff frequency, defined as the frequency at which signal strength is equal to 1/√2 the maximum signal gain. A simple low-pass filter can be made with a resistor in series with the load and a capacitor in parallel with the load, while the opposite configuration will produce a simple hi-pass filter. The capacitor acts as a filter due to its tendency to act as an open circuit at low frequencies and a short circuit at high frequencies.

[1] Note that some authors reserve the miniscule *i _{s }*for time-varying currents and the majuscule

[2] The reader is advised to consult *The Art of Electronics *by Horowitz and Hill for an amusing and informative run-down of different types of capacitor.

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