Ohm's Law and Impedance

by Harold Hallikainen

San Luis Obispo, Calif. Thus far in this series, we've looked at various analysis techniques (Ohm's law, Kirchoff's laws, Thevenin equivalents, Norton equivalents, etc.) and their application to DC circuits. We've also determined the instantaneous, average, and RMS voltages for a sine wave. Let's now apply the analysis techniques to circuits with sinusoidal AC voltage sources (which we'll just call AC voltage sources).

With direct current, we were concerned with sign (direction) and magnitude. With alternating current, we shall concern ourselves with phase and magnitude. We will measure phase in degrees and magnitude in the appropriate units (volts, amps, ohms, etc.). Let's define a voltage source as being 1 / 0 and another as 1 / -45. While each is 1 volt, the second voltage lags (occurs later than) the first voltage by 45 degrees. If both signals are 1 KHz, the period of the signal is 1 mS (the period is calculated as 1/f, so in this case, it's 1/1e3). The phase is how many 360ths of the period one waveform leads or lags the other. In this example, the second voltage source lags the first by (45/360)*1mS=125uS. Figure 1 shows the resulting waveforms as viewed on an oscilloscope or using the circuit simulation program pSpice (available from MicroSim at http://www.microsim.com). Note that the negative sign in the phase of the source indicates that the signal lags the reference waveform, while a positive sign in the phase of a source indicates the signal leads the reference waveform.

Impedance

Impedance is similar to resistance except that impedance includes real and imaginary parts if using rectangular coordinates or a magnitude and a phase if using polar coordinates.

Looking first at rectangular coordinates, we find that Z=R+jX where Z is the impedance, R is the real part of the impedance, which corresponds to resistance, j is the square root of -1, and X is the imaginary part of the impedance, which corresponds to reactance. Note that mathematicians use i (standing for imaginary) to represent the square root of -1. In electronics, we use I to represent current (probably originally electric intensity), so we jumped up the alphabet a bit to use j. Figure 2 shows a resistance and reactance in series. The total impedance of this series circuit is R+jX.

The impedance of an inductor is Z_L=sL, where Z_L is the impedance, s is the complex frequency of the signal driving the circuit, and L is the inductance measured in Henries. For a continuous sine wave, the complex frequency is jw where w (omega) is the frequency in radians per second. Since there are 2*pi radians in one cycle, the complex frequency can also be expressed as j*2*pi*f where f is the frequency in Hz. Putting it all together, we find that Z_L=j*2*pi*f*L. This impedance is purely imaginary, since there is no real part. An equivalent expression for Z_L would be 0+j*2*pi*f*L where 0 is the real part and 2*pi*f*L is the imaginary part. This imaginary part is the reactance of an inductor. X_L=2*pi*f*L.

The impedance of a capacitor is Z_C=1/sC, where Z_C is the impedance, s is the complex frequency, and C is the capacitance measured in Farads. Substituting for s, as above, we get Z_C=1/(j*2*pi*f*C). If we multiply this expression by the number one in the form of j/j, we get Z_C=j/(j²*2*pi*f*C). Since j is the square root of -1, j²=-1. Substituting and moving the -1 to the numerator, we find Z_C=-j*(1/(2*pi*f*C)). If we define the reactance of the capacitor to be X_C=1/(2*pi*f*C), then Z_C=0-jX_C.

Generalizing, we may define an impedance in rectangular coordinates as Z=R+jX where R is the resistance and X is the reactance, with the understanding that inductive reactance is positive and capacitive reactance is negative.

Figure 3 shows the addition of resistive and reactive components in a complex impedance. We "go out" R units, then "up" X units to arrive at the point representing the impedance. One of my students once explained that the j in the impedance expression stands for "jump." He reasoned that we "go out" R units, then "jump up" (or down, if negative) X units. It works!

In figure 3 we see that besides taking the "rectangular route" of going out R units and up X units, we could take the more direct polar route where we go out some distance in some direction to directly get to the point representing Z. You can brush up on your trigonometry for the next article in the series, where we'll look at rectangular to polar conversions. For now, let's just look at a few special cases.

Above, we determined that the impedance was 0+jX_L. We have "gone out" 0 ohms along the X-axis and "jumped up" X_L ohms. We can also express Z_L in polar notation by noting that we have "gone out" X_L ohms at an angle of 90 degrees (measuring counterclockwise from the positive X axis). Therefore, Z_L=X_L/ 90 in polar notation.

Similarly, we can determine the impedance of each of the other components listed in table 1.

Multiplication, Division, and Ohm's Law

We'll start complex number arithmetic by looking at multiplication and division. To multiply two complex numbers in polar form, multiply the magnitudes and add the phase angles. To divide two complex numbers, divide the magnitudes and subtract the phase angles. We'll look at this in more detail next month, but for now, let's try an example.

Let's say we want to determine the current through a 1 mH inductor driven by a 1 volt, 1 KHz sine wave. Our modified Ohm's Law states that I=V/Z. Here, V=1/ 0. That is, 1 volt at zero degrees (we'll define it as our reference phase). The impedance of the inductor (Z_L) is X_L / 90 from table 1. X_L=2*pi*f*L = 2*pi*1e3*1e-3 = 6.283 ohms. Putting it all together, we find

The current through the inductor is 159 mA at an angle of -90 degrees. Note that the magnitude of the current is 1/6.283 while the phase of the current is 0 - 90. The negative phase angle tells us that the current lags the applied voltage (which we defined as being at zero degrees), just as we would expect the current in an inductor to lag the voltage.

Next time, we'll look at addition and subtraction of complex numbers, allowing us to analyze series and parallel circuits. We'll also use some of the free circuit analysis and math software that's out there to analyze a 90 degree T network, perhaps the one at the base of your AM tower. 'til next time... Stay Tuned!

Harold Hallikainen designs transmitter control and lighting equipment for Dove Systems, a manufacturer serving the broadcast and entertainment industries. He also teaches electronics at Cuesta College and is an avid contra dancer. He can be reached at +1 805 541 0200 (voice), +1 805 541 0201 (fax), harold@hallikainen.com (email), and http://hallikainen.com (World Wide Web, where an archive of these articles is maintained).