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Impedance

Module by: Don Johnson

Summary: Introducing what is the impedance, and why and where to use it.

The major consequence of assuming complex exponential voltage and currents is that the ratio Z=VI Z V I for each element does not depend on time. This quantity is known as the element's impedance.

Figure 1
Subfigure 1.1: Resistor: Z R =R Z R R Subfigure 1.2: Capacitor: Z C =1ⅈ2πfC Z C 1 2 f C Subfigure 1.3: Inductor: Z L =ⅈ2πfL Z L 2 f L
, Subfigure 1.1 (resistor.png), Subfigure 1.2 (capacitor.png), Subfigure 1.3 (inductor.png)

The impedance is, in general, a complex-valued, frequency-dependent quantity. For example, the magnitude of the capacitor's impedance is inversely related to frequency, and has a phase of -π2 2 . This observation means that if the current is a complex exponential and has constant amplitude, the amplitude of the voltage decreases with frequency.

Let's consider Kirchoff's circuit laws. When voltages around a loop are all complex exponentials of the same frequency, we have

∑n v n =∑n V n ⅇⅈ2πft=0 n n v n n n V n 2 f t 0 (1)
which means
∑n v n =0 n n v n 0 (2)
the complex amplitudes of the voltages obey KVL. We can easily imagine that the complex amplitudes of the currents obey KCL.

What we have discovered is that source(s) equaling a complex exponential of the same frequency forces all circuit variables to be complex exponentials of the same frequency. Consequently, the ratio of voltage to current for each element equals the ratio of their complex amplitudes, which depends only on the source's frequency and element values. This situation occurs because the circuit elements are linear and time-invariant. For example, suppose we had a circuit element where the voltage equaled the square of the current: vt=Ki2t v t K i t 2 . If it=Iⅇⅈ2πft i t I 2 f t , vt=KI2ⅇⅈ2π2ft v t K I 2 2 2 f t , meaning that voltage and current no longer had the same frequency and that their ratio was time-dependent.

Because for linear circuit elements the complex amplitude of voltage is proportional to the complex amplitude of current— V=ZI V Z I —circuit elements behave as if they were resistors, where instead of resistance, we use impedance. Because complex amplitudes for voltage and current also obey Kirchoff's laws, we can solve circuits using voltage and current divider and the series and parallel combination rules by considering the elements to be impedances.

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