Summary

Important Concepts

AC circuits are driven by an emf

ε = ε_{0} cos ω t

that oscillates with angular frequency $ω = 2 π f$ .

Phasors can be used to represent the oscillating emf, current, and voltage.

The figure illustrates a phasor diagram.

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Basic circuit elements

Element	$i$ and $v$	Resistance/reactance	$I$ and $V$	Power
Resistor	In phase	$R$ is fixed	$V = I R$	$I_{rms} V_{rms}$
Capacitor	$i$ leads $v$ by $90 °$	$X_{C} = 1 / ω C$	$V = I X_{C}$	0
Inductor	$i$ lags $v$ by $90 °$	$X_{L} = ω L$	$V = I X_{L}$	0

For many purposes, especially calculating power, the root-mean-square (rms) quantities

V_{rms} = V / \sqrt{2} I_{rms} = I / \sqrt{2} ε_{rms} = ε_{0} / \sqrt{2}

are equivalent to the corresponding DC quantities.

Key Skills

Using phasor diagrams

Start with a phasor ( $v$ or $i$ ) common to two or more circuit elements.
The sum of instantaneous quantities is vector addition.
Use the Pythagorean theorem to relate peak quantities.

The figure illustrates a phasor diagram of an R C circuit.

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For an $R C$ circuit, shown here,

\begin{array}{l} v_{R} + v_{C} = ε \\ V_{R}^{2} + V_{C}^{2} = ε_{0}^{2} \end{array}

Instantaneous and peak quantities

The instantaneous quantities $v$ and $i$ vary sinusoidally. The peak quantities $V$ and $I$ are the maximum values of $v$ and $i$ . For capacitors and inductors, the peak quantities are related by $V = I X$ , where $X$ is the reactance, but this relationship does not apply to $v$ and $i$ .

Kirchhoff’s loop law says that the sum of the potential differences around a loop is zero.

Charge conservation says that circuit elements in series all have the same instantaneous current $i$ and the same peak current $I$ .

Applications

$R C$ filter circuits

The figure illustrates a circuit diagram of a low pass filter.

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\begin{array}{l} v_{C} = \frac{ε_{​ 0} X_{C}}{\sqrt{R^{2} + X_{C} ​^{2}}} \\ v_{C} \to ε_{​ 0} as ω \to 0 \end{array}

A low-pass filter transmits low frequencies and blocks high frequencies.

The figure illustrates a circuit diagram of a high pass filter.

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\begin{array}{l} v_{R} = \frac{ε_{​ 0} R}{\sqrt{R^{2} + X_{C} ​^{2}}} \\ v_{R} \to ε_{​ 0} as ω \to \infty \end{array}

A high-pass filter transmits high frequencies and blocks low frequencies.

Series $R L C$ circuits

The figure illustrates a series R L C circuit.

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$I = ε_{ 0} / Z$ where $Z$ is the impedance
$Z = \sqrt{R^{2} + {(X_{ L} - X_{ C})}^{2}}$
$V_{R} = I R V_{L} = I X_{ L} V_{C} = I X_{ C}$

When $ω = ω_{0} = 1 / \sqrt{L C}$ (the resonance frequency), the current in the circuit is a maximum $I_{max} = ε_{0} / R$ .

In general, the current $i$ lags behind $ε$ by the phase angle $ϕ = \tan^{- 1} ((X_{L} - X_{C}) / R)$ .

The power supplied by the emf is $P_{source} = I_{rms} ε_{rms} cos ϕ$ , where $cos ϕ$ is called the power factor.

The power lost in the resistor is $P_{R} = I_{rms} V_{rms} = (I_{rms})^{2} R$ .