Impedance

In DC circuits the relationships of various voltages and currents are described by a set of linear algebraic equations, while in AC circuits they are described by a set of linear differential equations.

The resistance $R$ of a resistor is defined by Ohm's law as the ratio of voltage $v$ across a resistor and current $i$ through the resistor: $R=v/i$. This concept is generalized to that of impedance $Z$ of any element (L, C, as well R) in AC circuits, defined as the frequency response function of the element with the current through the element as the input and the voltage across the element as the output:

$$Z=\frac{V}{I}=\frac{v_m e^{j\phi}}{i_m e^{j\psi}}=\frac{v_m}{i_m} e^{j(\phi-\psi)}$$ (480)

This is the generalized Ohm's law, which represents impedance $Z$ as the ration of the phasor voltage $V$ across an element and the phasor current $I$ through it. In other words, impedance $Z$ represents (a) the phase difference between $V$ and $I$ as well as (b) the ratio of their amplitudes.

• Inductor $L$: Assume the current through $L$ is $I e^{j\omega t}$, then the voltage across L is:

  $$V e^{j\omega t}=L\frac{d}{dt} [I e^{j\omega t}]=j\omega L I e^{j\omega t} \;\;\;\; \;\;\;\;\;\;Z_L=\frac{V}{I}=\frac{j\omega LI}{I}=j\omega L$$ (481)

• Capacitor $C$: Assume the voltage across $C$ is $V e^{j\omega t}$, then the current through C is

  $$I e^{j\omega t}=C\frac{d}{dt} [V e^{j\omega t}]=j\omega C V e^{j\omega t} \;\;\;\; \;\;\;\;\;\;Z_C=\frac{V}{I}=\frac{V}{j\omega CV}=\frac{1}{j\omega C}$$ (482)

• Resistor $R$: the voltage across R and current through R are in phase, therefore:

  $$Z_R=\frac{V}{I}=\frac{v_me^{j\phi}}{i_me^{j\phi}}=\frac{v_m}{i_m}=R$$ (483)

All familiar laws such as Ohm's law, KCL and KCL, current divider and voltage divider, can be generalized and applied to the analysis of AC circuit containing elements such as L, C as well as R.