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Resistance, Reactance and Impedance, Part 3

PART 3: IMPEDANCE

This is Part 3 of a three-part series on resistance, reactance and impedance. Most of us are familiar with resistance - at least we think we are. You may have found Part 1 pretty basic. But few of us really understand reactance and its relationship to resistance. Part 2 was a lot more interesting! And while most of us sort of understand what impedance is, few really understand the relationship between it and the other two properties. This three-part series will tie them all together.

IMPEDANCE

In Part 1, we talked about resistance. Resistance (R) is pretty simple; there is a single component value, it does not depend on frequency, and there is no phase shift. In Part 2 we talked about reactance. Reactance (X) is much more complicated; there are component values, the "impedance" to current flow depends on frequency, there is a 90 degree phase shift, and the effects are opposite depending on whether you are talking about inductors or capacitors.

Impedance (Z) is the general case of what happens when we combine resistance and reactance in a single circuit. The general expression for impedance is:

 Z = R + jX     Equation 1

where "j" is the imaginary operator equal to the square root of minus one (as introduced in Part 2.)

It is possible for impedance to be purely resistive (if the reactance term is zero), and it is not improper to refer to a purely resistive circuit as having an impedance. Indeed, the reactance term is zero at resonance for a series RLC circuit, at which point the impedance is purely resistive. Similarly, it is possible for an impedance to be purely reactive (if the resistance term is zero), and it is not improper to refer to a purely reactive circuit as having an impedance. But impedance is what we get when we combine the effects of both resistance and reactance.

There are two different contexts in which we commonly refer to impedance. One is to the impedance of a circuit at a specific frequency, at which point it has a specific value and a specific phase shift. The other is to refer to an impedance function over a range of frequencies. Figure 4 in Part 2 of this series plots an impedance function (actually a reactance function, or an impedance function when the resistance term is absent) when combining an inductor and a capacitor in either series or parallel.

At a Frequency

Figure 1 illustrates a simple series RLC circuit. In Part 2 we talked about the reactance part of a similar circuit with a capacitor whose value was 0.1 uF and an inductor whose value was 5 uH. Let's also add a resistance to this circuit whose value is 90 ohms. Then let's look at the impedance of this circuit at 1 MHz. Based on what we have learned before, we now know that: 

Figure 1. A simple series RLC circuit.

We also know that series reactances add together, so we can readily understand that the impedance of this circuit (at 1.0 MHz) is:

Z = 90 + j29.8 ohms           Equation 3

We can represent this impedance graphically. By convention we draw the resistance along the horizontal axis and the reactance along the vertical axis. (Note: The vertical axis is either plus or minus 90 degrees to the resistance axis, depending on whether the reactance term is positive (inductive) or capacitive (negative)). We plot the impedance as the point on the graph where (in this case) R = 90 and X = 29.8. See Figure 2.

Figure 2. Impedance of our example.

We can represent the impedance as the point where R = 90 and X = 29.8. More commonly we represent the impedance as the vector from the origin to that point. Then, the magnitude of the impedance is the length of the vector and the phase angle associated with that impedance is the angle Θ. That angle is the angle whose tangent (i.e., the arctan, or tan-1) equals the ratio of X/R. Doing those calculations yields:

As a Function of Frequency

Alternatively, we may be interested in how impedance changes with frequency. In an earlier column (Note 1) I showed several impedance curves for power distribution systems. Let's look at the impedance curve for Figure 1. But this time let's change the resistance to .001 ohms. (This makes the curve a little more interesting!) This is not unlike a 0.1 bulk bypass capacitor with an ESR of .001, although admittedly the inductance is a little high at 5 uH for a typical case. An impedance curve is shown in Figure 3.

Note in this figure we have plotted both impedance and phase angle. As noted in Part 2, if the impedance curve is downward sloping to the right the impedance is capacitive and the phase angle is negative. If the impedance is upward sloping to the right the impedance is inductive and the phase angle is positive. Over most of the frequency range the phase angle is either minus or plus 90 degrees. At the specific point of resonance, where the reactance term is zero (in fact, that is one definition of resonance) the phase angle is zero. But the phase angle curve shoots through that point fairly quickly, transitioning from minus 90 degrees to plus 90 degrees. The specific point of resonance is a very specific point. We may be able to calculate that point to an arbitrary level of accuracy, but to design a circuit with a specific resonance point is not an easy trick.

Figure 3. Impedance and phase angle curves as a function of frequency.

Figure 4. Expansion of Figure 3 around the resonance point.

Figure 4 expands the horizontal axis around the resonant point. The purpose of Figure 4 is to illustrate that while the phase angle curve is a smooth curve that does transition smoothly from minus 90 degrees to plus 90 degrees, it does so very quickly. This is not uncommon with impedance curves, especially those with small values of resistance. Larger values of R would result in broader transition regions.

Interpretation of Phase Shift

The interpretation of phase shift in a real circuit can be quite subtle. It means that voltage either leads or lags current through a circuit. But there are some aspects of phase shift that can be confusing. For example, there is never a phase shift between voltage and current through a resistor. There is always a 90 degree phase shift (plus or minus) through a reactive component or a set of reactive components. Even if there is a capacitor and an inductor in series and the frequency is the resonant frequency of the circuit, there is still a minus 90 degree phase shift through the capacitor and a plus 90 degree phase shift through the inductor.

This is true even though the phase shift through the series combination is (exactly) zero (because they cancel at that specific frequency.) It is only when we look at a circuit that contains both resistance and reactance that we can get phase shifts between zero and (plus or minus) 90 degrees.

UltraCAD has put together an extensive (and free) simulation regarding phase shift; a screen shot of that simulation is shown in Figure 5. Go to our Web page for more information on how to access and use the simulation (see Note 2).

Figure 5. UltraCAD's Phase Shift Simulator.Series and Parallel Combinations of Impedance

Impedance combines in series and parallel combinations just like resistance and reactance does. Impedances in series combine by addition. That is, if:

Z1 = R1 + j X1, andZ2 = R2 + j X2,

then the series combination of Z1 and Z2 is:

The phase angle for this combination is the angle whose tangent is the ratio of the reactance term divided by the resistance term, or:

Impedances in parallel combine using the parallel combination formula:

Calculation of Zparallel can be quite complicated, especially for any practical problem. For those who are interested in a simple calculational tool for working with impedance and impedance combinations, UltraCAD has introduced a basic calculator just for this purpose, shown in Figure 6. See Note 3.

Figure 6. UltraCAD's new Impedance Tool can help simplify many impedance calculations.

Power Considerations

In Part 1 of this series we discussed how resistance dissipates power, and how i2R heating causes traces to heat up (among other things.) In Part 2 we said that there is no comparable dissipation with reactance. Capacitors and inductors absorb energy in creating the electric and magnetic fields when current flows. But when these fields collapse, all the energy (at least for pure reactances) is returned to the circuit.

When we have a combination of resistance and reactance (i.e., when we have an impedance) power can be (and is) dissipated through the resistance term (but not through the reactance term.) See the discussion on power dissipation in Part 1.

Notes: 

• 1. See my column on Calculating PDS Impedance.
• 2. Learn more about this simulation at www.ultracad.com. Click on the Simulations tab.
• 3. Learn more about this tool at www.ultracad.com. Click on the Calculators and Tools tab

Douglas Brooks has an MS/EE from Stanford University and a PhD from the University of Washington. He has spent most of his career in the electronics industry in positions of engineering, marketing, general management, and as CEO of several companies. He has owned UltraCAD Design Inc. since 1992. He is the author of numerous articles in several disciplines, and has written articles and given seminars all over the world on Signal Integrity issues since founding UltraCAD. His book, Printed Circuit Board Design and Signal Integrity Issues was published by Prentice Hall in 2003. Visit his Web site at www.ultracad.com.