9.2 Impedance and Admittance

Impedance and Admittance

Impedance and admittance describe how circuit elements behave with alternating current. They package resistance, inductance, and capacitance effects into complex numbers, which lets you analyze AC circuits using the same DC techniques you already know (mesh analysis, node voltage, Thévenin/Norton equivalents).

These two concepts are the foundation of sinusoidal steady-state analysis. Once you can express every element as an impedance or admittance, an AC circuit problem looks and solves just like a DC one, except with complex arithmetic.

Impedance and Admittance

Fundamental Concepts

Impedance ($Z$) is the total opposition a circuit element presents to alternating current, measured in ohms ($\Omega$). Think of it as the AC generalization of resistance: it accounts not only for energy dissipation (resistance) but also for energy storage in electric and magnetic fields (reactance).

Admittance ($Y$) is the reciprocal quantity, measuring how easily AC flows through an element. It's measured in siemens (S).

Both quantities depend on frequency, which is what makes AC analysis different from DC.

• Impedance in rectangular form: $Z = R + jX$, where $R$ is resistance (real part) and $X$ is reactance (imaginary part).
• Admittance in rectangular form: $Y = G + jB$, where $G$ is conductance (real part) and $B$ is susceptance (imaginary part).
• They're related by $Y = \frac{1}{Z}$, so you can always convert between the two.

A common point of confusion: $G$ and $B$ are not simply $1/R$ and $1/X$ when both $R$ and $X$ are nonzero. You have to invert the full complex number. For $Z = R + jX$:

$G = \frac{R}{R^2 + X^2}, \quad B = \frac{-X}{R^2 + X^2}$

Complex Number Representation

You'll constantly switch between two forms:

• Rectangular form ($a + jb$): best for addition and subtraction (series impedances, parallel admittances).
• Polar form ($|Z| \angle \theta$): best for multiplication and division (voltage/current ratios, power calculations).

To convert from rectangular to polar:

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

$\theta = \tan^{-1}\!\left(\frac{X}{R}\right)$

To convert from polar to rectangular:

$R = |Z|\cos\theta, \quad X = |Z|\sin\theta$

Euler's formula ties these together: $e^{j\theta} = \cos\theta + j\sin\theta$. This means polar form can also be written as $|Z|e^{j\theta}$.

The $j$-operator ($j = \sqrt{-1}$) is what separates resistive components (real axis) from reactive components (imaginary axis). Phasor diagrams plot these on the complex plane so you can visualize magnitude and phase relationships between voltages and currents.

Impedance of Circuit Elements

Resistors

$Z_R = R$

The impedance of a resistor is purely real and doesn't change with frequency. Voltage and current are in phase (0° phase difference), meaning they reach their peaks at the same instant.

• Admittance: $Y_R = \frac{1}{R}$
• Power dissipation: $P = I^2 R = \frac{V^2}{R}$

Inductors

$Z_L = j\omega L$

An inductor's impedance is purely imaginary and positive, meaning it grows linearly with frequency. At DC ($\omega = 0$), the impedance is zero (a short circuit). At very high frequencies, it approaches an open circuit.

The phase angle is +90°: voltage leads current by 90°. A helpful mnemonic is ELI (voltage E leads current I in an inductor L).

• Admittance: $Y_L = \frac{1}{j\omega L} = \frac{-j}{\omega L}$
• Energy stored in the magnetic field: $E = \frac{1}{2}LI^2$

Capacitors

$Z_C = \frac{1}{j\omega C} = \frac{-j}{\omega C}$

A capacitor's impedance is purely imaginary and negative, and it decreases as frequency increases. At DC, the impedance is infinite (an open circuit). At very high frequencies, it approaches a short circuit. This is the opposite behavior of an inductor.

The phase angle is −90°: current leads voltage by 90°. The mnemonic here is ICE (current I leads voltage E in a capacitor C).

• Admittance: $Y_C = j\omega C$
• Energy stored in the electric field: $E = \frac{1}{2}CV^2$

Non-Ideal Components

Real components aren't perfect. Here are the most common deviations:

• Inductors have wire resistance, so a more realistic model is $Z = R_L + j\omega L$ (a series R-L combination).
• Capacitors have leakage resistance, typically modeled as a resistor in parallel with the ideal capacitor.
• Skin effect causes the effective resistance of conductors to increase at high frequencies because current crowds toward the surface.
• Both inductors and capacitors have a self-resonant frequency beyond which they stop behaving as expected (an inductor starts looking capacitive, and vice versa).

Series and Parallel Combinations

Series Connections

When elements are in series, their impedances add directly:

$Z_{\text{total}} = Z_1 + Z_2 + \cdots + Z_n$

This is the same rule as resistors in series, just with complex numbers. Use rectangular form for the addition.

Voltage divider for AC circuits works the same way as DC:

$V_k = \frac{Z_k}{Z_{\text{total}}} \cdot V_{\text{source}}$

Series resonance occurs in an RLC circuit when the inductive reactance equals the capacitive reactance ($\omega L = \frac{1}{\omega C}$). At resonance, the imaginary parts cancel and the total impedance is purely resistive ($Z = R$), meaning current is maximized.

The quality factor for a series RLC circuit is:

$Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 RC}$

where $\omega_0$ is the resonant frequency. A higher $Q$ means a sharper, more selective resonance peak.

Parallel Connections

When elements are in parallel, their admittances add directly:

$Y_{\text{total}} = Y_1 + Y_2 + \cdots + Y_n$

Or equivalently, in terms of impedance:

$\frac{1}{Z_{\text{total}}} = \frac{1}{Z_1} + \frac{1}{Z_2} + \cdots + \frac{1}{Z_n}$

For just two impedances in parallel, the familiar product-over-sum formula applies:

$Z_{\text{total}} = \frac{Z_1 \cdot Z_2}{Z_1 + Z_2}$

Current divider for AC:

$I_k = \frac{Y_k}{Y_{\text{total}}} \cdot I_{\text{source}}$

Parallel resonance occurs when the susceptances of the inductor and capacitor cancel. At resonance, the total admittance is minimized (purely real), so impedance is maximized.

The quality factor for a parallel RLC circuit is:

$Q = R\sqrt{\frac{C}{L}}$

Notice this is the inverse relationship compared to series: in a series circuit, lower $R$ gives higher $Q$; in a parallel circuit, higher $R$ gives higher $Q$.

Complex Circuit Analysis

For circuits that aren't simple series or parallel combinations, you have several tools:

• Series-parallel conversion: You can convert a series R-X combination into an equivalent parallel R-X combination (and vice versa) to simplify mixed networks. The conversion uses the $Q$ of the branch.
• Delta-wye ($\Delta$-Y) transformations: These work the same as in DC circuits, but with complex impedances.
• Superposition: Valid for circuits with multiple AC sources, but all sources must be at the same frequency for a single phasor analysis. Different frequencies require separate analyses.
• Thévenin and Norton equivalents: You can reduce any linear AC network to a voltage source with a series impedance (Thévenin) or a current source with a parallel admittance (Norton).
• Mesh current and node voltage methods: Apply exactly as in DC, substituting impedances for resistances and using phasor voltages/currents.

The key takeaway: once every element is expressed as an impedance or admittance, every technique from DC circuit analysis carries over directly. The only new skill is complex arithmetic.