Impedance
When alone in an AC circuit, inductors, capacitors, and resistors all impede current. How do they behave when all three occur together? Interestingly, their individual resistances in ohms do not simply add. Because inductors and capacitors behave in opposite ways, they partially to totally cancel each other’s effect. Figure 6.48 shows an RLC series circuit with an AC voltage source, the behavior of which is the subject of this section. The crux of the analysis of an RLC circuit is the frequency dependence of and and the effect they have on the phase of voltage versus current (established in the preceding section). These give rise to the frequency dependence of the circuit, with important resonance features that are the basis of many applications, such as radio tuners.
The combined effect of resistance inductive reactance and capacitive reactance is defined to be impedance—an AC analogue to resistance in a DC circuit. Current, voltage, and impedance in an RLC circuit are related by the following AC version of Ohm’s law:
6.63
Here, is the peak current, the peak source voltage, and is the impedance of the circuit. The units of impedance are ohms, and its effect on the circuit is as you might expect: the greater the impedance, the smaller the current. To get an expression for in terms of and we will now examine how the voltages across the various components are related to the source voltage. Those voltages are labeled and in Figure 6.48.
Conservation of charge requires current to be the same in each part of the circuit at all times, so that we can say the currents in and are equal and in phase. But we know from the preceding section that the voltage across the inductor leads the current by one-fourth of a cycle, the voltage across the capacitor follows the current by one-fourth of a cycle, and the voltage across the resistor is exactly in phase with the current. Figure 6.49 shows these relationships in one graph, as well as showing the total voltage around the circuit where all four voltages are the instantaneous values. According to Kirchhoff’s loop rule, the total voltage around the circuit is also the voltage of the source.
You can see from Figure 6.49 that while is in phase with the current, leads by and follows by Thus, and are out of phase (crest to trough) and tend to cancel, although not completely unless they have the same magnitude. Since the peak voltages are not aligned (not in phase), the peak voltage of the source does not equal the sum of the peak voltages across and The actual relationship is
6.64
where and are the peak voltages across and respectively. Now, using Ohm’s law and definitions from Reactance, Inductive and Capacitive, we substitute into the above, as well as and yielding
6.65 cancels to yield an expression for
6.66
which is the impedance of an RLC series AC circuit. For circuits without a resistor, take for those without an inductor, take and for those without a capacitor, take
Example 6.12 Calculating Impedance and Current
An RLC series circuit has a resistor, a 3.00 mH inductor, and a
capacitor. (a) Find the circuit’s impedance at 60.0 Hz and 10.0 kHz, noting that these frequencies and the values for
and
are the same as in Example 6.10 and Example 6.11. (b) If the voltage source has what is at each frequency?
Strategy
For each frequency, we use to find the impedance and then Ohm’s law to find current. We can take advantage of the results of the previous two examples rather than calculate the reactances again.
Solution for (a)
At 60.0 Hz, the values of the reactances were found in Example 6.10 to be and in Example 6.11 to be Entering these and the given for resistance into yields
6.67 Similarly, at 10.0 kHz, and so that
6.68
Discussion for (a)
In both cases, the result is nearly the same as the largest value, and the impedance is definitely not the sum of the individual values. It is clear that dominates at high frequency and dominates at low frequency.
Solution for (b)
The current can be found using the AC version of Ohm’s law in Equation
6.69
at 60.0 Hz.
Finally, at 10.0 kHz, we find
6.70
at 10.0 kHz.
Discussion for (a)
The current at 60.0 Hz is the same (to three digits) as that found for the capacitor alone in Example 6.11. The capacitor dominates at low frequency. The current at 10.0 kHz is only slightly different from that found for the inductor alone in Example 6.10. The inductor dominates at high frequency.