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# Complex Numbers

Module by: Scott Starks. E-mail the author

Summary: This module is part of a collection of modules that address engineering applications of PreCalculus. The collection is intended for use by students enrolled in a special section of MATH 1508 (PreCalculus) for preengineers at the University of Texas at El Paso.

## Complex Numbers

### Introduction

It is essential that engineers master the concept of complex numbers because the important role that complex numbers play in a variety of application areas. In this module applications in the field of electric circuits are provided.

### Alternating Current (AC) Electric Circuits

Earlier we introduced a number of components that are typically found in common electric circuits. These included voltage sources, current sources and resistors. We also observed that the behavior of an electric circuit could be predicted by using several laws from Physics, including Ohm’s Law and Kirchoff’s Laws.

In this laboratory exercise, we will introduce two additional components of electric circuits: the inductor and the capacitor. These elements are typically found in electric circuits which involve sinusoidally varying voltage or current sources. These circuits are called alternating current or AC circuits. AC circuits abound in the physical world. The voltage and current that power household appliances comes from AC sources.

Figure 1 shows the plot for a sinusoidally varying waveform that represents the output of an AC voltage source. Such a waveform could also be used to represent the current that is supplied by an AC current source. It is important to note that the waveform has a repetitive or periodic nature.

In the figure, we note that the amount of time that occurs between successive maxima of the sinusoidal waveform is equal to the period. The angular frequency of the waveform is denoted by the symbol ω and is defined in terms of the period by the equation

ω = 2 π T rad / s ω = 2 π T rad / s size 12{ω= { {2π} over {T} }  ital "rad"/s} {}
(1)

If we denote the amplitude as Vmax, then we can express the sinusoidal waveform for the voltage mathematically as

v ( t ) = V max cos ( ω t + θ v ) v ( t ) = V max cos ( ω t + θ v ) size 12{v $$t$$ =V rSub { size 8{"max"} } "cos" $$ωt+θ rSub { size 8{v} }$$ } {}
(2)

Here the instantaneous value of the voltage is measured in the units volts. The term θv is called the phase angle of the sinusoidal waveform. It is measured in degrees. Its usage and importance in the analysis of AC circuits will be discussed later in the course during the study of trigonometry.

Inductors and capacitors are found in circuits of all types and designs, so their understanding is critical to the education of an engineer or scientist. One important distinction between resistors and these two new components (inductors and capacitors) is that they are analyzed using different mathematic techniques. In the case of a resistor, it was quite easy to determine the relationship between the current, voltage and resistance present in a circuit by means of simple algebra. In the case of the inductor and the capacitor, we will see that we must expand our knowledge of mathematics particulary in the are of complex numbers to analyze circuits that contain inductors and capacitors.

### Inductors

An inductor is an electrical component that stores energy in the form of a magnetic field. In its simplest form, an inductor consistsof a wire loop or coil. Figure 2 depicts an inductor next to a coin to show its relative size and structure.

The inductance of the component is directly proportional to the number of turns present in the wire that makes up the coil. Inductance also depends on the radius of the coil and on the type of material around which the coil is wound. The standard unit of inductance is the Henry (H).

The schematic symbol for an inductor is shown in Figure 3.

### Capacitor

A capacitor is an electrical device consisting of two conducting plates separated by an electrical insulator (the dielectric), designed to hold an electric charge. Charge builds up when a voltage is applied across the plates, creating an electric field between them. Current can flow through a capacitor only as the voltage across it is changing, not when it is constant. Capacitors are used in power supplies, amplifiers, signal processors, oscillators, and logic gates.

The standard unit of capacitance is the farad (F). Typical capacitance values are small. Common capacitors have values of capacitance that are expressed in units of microfarads (µF). Figure 4 shows a photograph of a several different capacitors.

The standard symbol for a capacitor is shown in Figure 5.

### Impedance

In the case of electric circuits that are driven by a sinusoidally varying voltage source, the impedance serves to restrict the flow of current. Like the resistance, impedance is measured in ohms (Ω). However, the impedance differs from resistance in that the impedance is a complex quantity. Because the impedance is a complex quantity, we will represent the impedance as a complex number

Z = R + j X Z = R + j X size 12{Z=R+jX} {}
(3)

The real part of Z as stated in equation (3) is R and the imaginary part is X.

Resistors, inductors and capacitors serve to contribute to the impedance present in a sinusoidally varying electric circuit. The impedance of a resistor is merely the value of its resistance.

The impedance of an inductor (ZL) can be easily computed via the relationship

Z L = j ω L Z L = j ω L size 12{Z rSub { size 8{L} } =jωL} {}
(4)

The impedance of an inductor is measured in the units Ω. The term ω is equal to the angular frequency of the sinusoidally varying source voltage. Examination of equation (4) indicates that as the angular frequency increase, so too does the impedance of the inductor. At very high frequencies an inductor will essentially inhibit all flow of current through itself.

The impedance of a capacitor (Zc) is given by the equation

Z C = j 1 ω C Z C = j 1 ω C size 12{Z rSub { size 8{C} } = - j left ( { {1} over {ωC} } right )} {}
(5)

The impedance of a capacitance is measured in the units Ω. Once again, the term ω is equal to the angular frequency of the sinusoidally varying source voltage. Examination of equation (5) indicates that as the angular frequency increases, the impedance of the capacitor decreases. At very high frequencies a capacitor will behave as a short circuit. That is, its effect at very high frequencies is to allow current to flow through it in an unimpeded manner.

### Series RL Circuit

Just as resistors can be combined using series and parallel connections, so too can impedances. In the case of series connections, impedances are merely added. One distinction is that the addition is performed using complex arithmetic. Let us consider the RL circuit shown in Figure 6.

The series impedance is equal to the sum of the resistance with the impedance of the inductor. Suppose that for this circuit the value of R is 10 Ω and that the value for the inductance is 100 mH. Suppose that the frequency (ω) of the source voltage is 100 rad/sec. For this specification of values, we can compute the impedance of the series connection

Z = 10 + j ( 100 ) ( 100 × 10 3 ) Ω = 10 + j 10 Ω Z = 10 + j ( 100 ) ( 100 × 10 3 ) Ω = 10 + j 10 Ω size 12{Z="10"+j $$"100"$$ $$"100" times "10" rSup { size 8{ - 3} }$$  %OMEGA ="10"+j"10" %OMEGA } {}
(6)

The square of the magnitude of the impedance can be obtained by use of the complex conjugate.

Z 2 = Z × Z ( 10 + j 10 ) ( 10 j 10 ) = 100 + 100 = 200 Z 2 = Z × Z ( 10 + j 10 ) ( 10 j 10 ) = 100 + 100 = 200 size 12{ lline Z rline rSup { size 8{2} } =Z times Z*= $$"10"+j"10"$$ $$"10" - j"10"$$ ="100"+"100"="200"} {}
(7)

So we calculate the magnitude of the impedance to be

Z = 14 . 17 Ω Z = 14 . 17 Ω size 12{ lline Z rline ="14" "." "17" %OMEGA } {}
(8)

An important property of AC circuits that contain an AC source voltage along with resistors, inductors and capacitors is that if the current i(t) will take the form of a sinusoid

i ( t ) = I max cos ( ω t + θ i ) i ( t ) = I max cos ( ω t + θ i ) size 12{i $$t$$ =I rSub { size 8{"max"} } "cos" $$ωt+θ rSub { size 8{i} }$$ } {}
(9)

The instantaneous value of the current is measured in Amps. The angular frequency of the current sinusoid (ω) will be the same as that of the sinusoid that represents the supply voltage. In addition, electric circuits involving resistors, capacitors and inductors will contribute to a change in the phase angle (θi) of the current sinuosoid. In general the phase angle of the voltage sinusoid (θv) will differ from that of the current sinusoid (θi). Once again, the discussion of how the phase angle of the current can be computed will be deferred until our later discussion of trigonometry.

Once we know the magnitude of the impedance, we can use it to calculate the amplitude of the sinusoidally varying current, Imax. This is accomplished by the following formula.

I max = V max Z I max = V max Z size 12{I rSub { size 8{"max"} } = { {V rSub { size 8{"max"} } } over { lline Z rline } } } {}
(10)

So the amplitude of the current is Vmax/141.7. It is interesting to note that this formula is similar to Ohm’s Law. The differences lie in the fact that the magnitude of the impedance appears instead of the resistance. Also, the amplitudes of the sinusoidally varying current and voltage appear.

The formula expressed above is useful in determining the amplitude of the current for a wide range of sinusoidally varying AC circuits. These circuits can combine resistors with inductors and capacitors to create a wide range of design options for electrical engineers. The following exercises illustrate the application of complex numbers to the analysis of AC circuits.

### Exercises

Consider the series RC circuit shown below

Suppose that the sinusoidally varying source voltage is given as v(t)=20cos(10t)V.v(t)=20cos(10t)V. size 12{v $$t$$ ="20""cos" $$"10"t$$ `V "." } {}

1. What are the amplitude and the radian frequency of the source voltage?
2. If the value of the capacitance is C = 100 µF, what is the impedance of the capacitor?
3. If the value for the resistor is R = 5 KΩ, what is the series impedance of the circuit?
4. Find the magnitude of the series impedance?
5. What is the amplitude of the sinusoidally varying current, i(t)?

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