Impedance

The impedance $(Z)$ of an RLC circuit is the effective resistance of the all the components in the circuit. It arises from combining the reactance $(X)$ of the capacitors and inductors with the resistance $(R)$ of the resistors in quadrature.

$$\large Z = \sqrt{ R^2 + X^2}$$

Circuits with at least two out of resistors, inductors, and capacitors connected to an alternating current source cannot be studied the same way as direct current circuits. Since the capacitor is active when charge is stored on its plates, the resistor is active when current flows through it, and the inductor is active when the current through it changes, the elements are active at different points in the cycle of the AC source. By expressing everything as a resistance-equivalent impedance, Ohm's law can once again be used for the circuit.

Ohm's law for an AC circuit $$\large V=IZ$$

When calculating the impedance of a circuit, it is helpful to first, calculate the reactance of each individual component before plugging everything into $Z = \sqrt{R^2+X^2}.$

Find the impedance of an RC circuit with components $R = 10 \Omega$ and $C = 500 \mu\text{F}$ attached to an AC source with driving frequency $\omega_d = 200 \frac{\text{rad}}{\text{s}}.$

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(1) Calculate individual reactances.

In this case, only the capacitor with contribute reactance to the circuit according to

$$\begin{aligned} X_C &= \frac{1}{\omega C} \\ &= \frac{1}{(200)(500\times 10^{-6})} \\ &= 10 \Omega. \end{aligned}$$

(2) Calculate impedance.

$$\begin{aligned} Z &= \sqrt{R^2+X^2} \\ &= \sqrt{10^2 + 10^2} \\ &= 10\sqrt{2} \Omega \end{aligned}$$

While it may seem oddly specific, the RLC circuit is the first circuit type encountered in physics that cannot be solved without considerations of impedance, since the components all operate in different phases.

The voltage of each component in the RLC circuit can be expressed in terms of Ohm's Law $(V=IR)$ with the understanding that reactance $(X)$ is the effective resistance of the capacitor $(C)$ and inductor $(L)$.

$$V_{\text{Resistor}} = IR$$

$$V_{\text{Inductor}} = I X_L$$

$$V_{\text{Capacitor}} = I X_C$$

Since the RLC circuit is attached to an alternating current source, the current in these equations is not constant, but oscillates like

$$i(t) = I \cos(\omega t).$$

Hence the voltage of each component as a function of t is

$$\begin{aligned} V_R(t) &= IR \cos(\omega t) \\ \end{aligned}$$

$$\begin{aligned} V_L(t) &= L \frac{dI}{dt} \\ &= L \big(-\omega I \sin(\omega t) \big) \\ &= I \omega L \cos \big(\omega t - \frac{\pi}{2} \big) \\ &= I X_L \cos \big(\omega t - \frac{\pi}{2} \big) \end{aligned}$$

$$\begin{aligned} V_C(t) &= \frac1C \int I(t) dt \\ &= \frac1C \big(\frac{I}{\omega} \sin(\omega t) \big) \\ &= \frac{I}{\omega C} \cos \big(\omega t + \frac{\pi}{2} \big) \\ &= I X_C \cos \big(\omega t + \frac{\pi}{2} \big) \\ \end{aligned}$$

The amplitudes of these voltage functions are

$$V_{\text{Resistor}} = IR$$

$$V_{\text{Inductor}} = I X_L$$

$$V_{\text{Capacitor}} = I X_C$$

Since the voltage functions are all $\frac{\pi}{2}$ out of phase, the total voltage of the circuit cannot be computed by simply adding them together. The inductor and capacitor voltages are $\pi$ out of phase, so they will always interfere destructively, and the resultant magnitude is thus

$$V_L - V_C.$$

Since the resistor voltage is $\frac{\pi}{2}$ out of phase with each of these voltages, its magnitude must be be combined in quadrature.

$$V_{\text{Circuit} }^2 = V_R^2 + ( V_L - V_C )^2$$

Since $V_{\text{Circuit}}$ is more often called EMF $( \varepsilon )$ and in light of the equations for the voltage amplitudes above,

$$\begin{aligned} \varepsilon &= \sqrt{V_R^2 + ( V_L - V_C )^2} \\ &= \sqrt{(IR)^2 + ( I X_L - I X_C )^2} \\ &= I \sqrt{R^2 + ( X_L - X_C )^2} \\ \end{aligned}$$

In light of Ohm's Law, the square root term must be dimensionally identical to resistance, and is it contains all the components of the circuit, it is called the impedance.

Impedance of RLC circuit attached to sinusoidal AC source

$$Z = \sqrt{R^2 + ( X_L - X_C )^2}$$