2.2 Impedance and admittance concepts

Impedance and Related Concepts

Impedance and admittance describe how AC circuit components interact with sinusoidal signals. Impedance captures the opposition to current flow, while admittance captures how easily current flows. Together, they give you the tools to analyze any AC circuit using phasor methods and complex algebra.

Understanding Impedance and Resistance

Impedance ($Z$) is the AC generalization of resistance. It's a complex quantity measured in ohms ($\Omega$) that accounts for both energy dissipation and energy storage in a circuit element.

$Z = R + jX$

• The real part ($R$) is resistance, the same quantity you know from DC circuits. It dissipates energy as heat and doesn't depend on frequency.
• The imaginary part ($X$) is reactance, which represents energy storage in inductors and capacitors. Reactance does depend on frequency.

Because $Z$ is complex, it carries two pieces of information: how much the component reduces the current's amplitude, and how much it shifts the current's phase relative to the voltage.

Reactance and Frequency Dependence

Reactance ($X$) is what makes AC analysis different from DC. It arises whenever a component stores energy in a magnetic field (inductors) or an electric field (capacitors).

• Inductive reactance is positive and grows with frequency:

$X_L = 2\pi f L = \omega L$

At higher frequencies, an inductor opposes current more strongly because the rate of change of current increases.

• Capacitive reactance is negative (by convention) and shrinks in magnitude with frequency:

$X_C = -\frac{1}{2\pi f C} = -\frac{1}{\omega C}$

At higher frequencies, a capacitor passes current more easily because it charges and discharges faster.

From the rectangular form $Z = R + jX$, you can extract:

• Magnitude: $|Z| = \sqrt{R^2 + X^2}$
• Phase angle: $\theta = \tan^{-1}\!\left(\frac{X}{R}\right)$

A positive phase angle means the impedance is inductive (current lags voltage). A negative phase angle means it's capacitive (current leads voltage).

Applications and Examples

• Power systems use impedance models for load balancing, fault current calculations, and protective relay coordination.
• Audio equipment relies on impedance matching between amplifiers and speakers (e.g., matching an 8 $\Omega$ speaker to an amplifier's output impedance) to maximize power transfer and minimize distortion.
• RF and antenna design depends on impedance matching (often to 50 $\Omega$) to ensure maximum power transfer and minimize signal reflections on transmission lines.
• Biomedical sensors measure tissue impedance across a range of frequencies to characterize cell structure and fluid content.

Admittance and Related Concepts

Admittance Fundamentals

Admittance ($Y$) is the reciprocal of impedance. Where impedance tells you how much a component opposes current, admittance tells you how readily it allows current to flow.

$Y = \frac{1}{Z}$

It's measured in siemens (S) and is also a complex number:

$Y = G + jB$

Admittance is especially useful when analyzing parallel circuits. Just as you add impedances in series, you add admittances in parallel, which makes the math much cleaner.

• Magnitude: $|Y| = \frac{1}{|Z|}$
• Phase angle: $\angle Y = -\angle Z$

Conductance and Susceptance Components

The two parts of admittance have distinct physical meanings:

• Conductance ($G$) is the real part. It represents the component of current that's in phase with the voltage (the part doing real work). For a purely resistive element, $G = \frac{1}{R}$, measured in siemens.
• Susceptance ($B$) is the imaginary part. It represents the component of current that's 90° out of phase with the voltage (reactive current). For a purely reactive element, $B = -\frac{1}{X}$, also in siemens.

One common mistake: for a general impedance $Z = R + jX$, conductance is not simply $1/R$. You need to compute the full reciprocal:

$Y = \frac{1}{R + jX} = \frac{R - jX}{R^2 + X^2}$

So $G = \frac{R}{R^2 + X^2}$ and $B = \frac{-X}{R^2 + X^2}$. The simple reciprocal relationships ($G = 1/R$, $B = 1/X$) only hold when $R$ and $X$ appear alone, not together in the same element.

Practical Applications

• Power factor correction uses admittance to determine how much capacitive susceptance to add in parallel to cancel inductive susceptance from motors and transformers.
• Microwave circuit design often works with admittance (via the Smith chart's admittance coordinates) because components are frequently connected in parallel.
• Electronic filter design uses admittance to shape frequency response, since parallel combinations of components are easier to handle in the admittance domain.

AC Circuit Analysis

Ohm's Law in AC Circuits

Ohm's law extends directly to AC circuits when you use phasors and complex impedance:

$\mathbf{V} = \mathbf{I} \, Z$

Here $\mathbf{V}$ and $\mathbf{I}$ are phasor quantities (complex numbers encoding both magnitude and phase), and $Z$ is the complex impedance. This single equation handles both the amplitude relationship and the phase shift simultaneously.

You can rearrange it just like the DC version:

• $\mathbf{I} = \frac{\mathbf{V}}{Z}$
• $\mathbf{V} = \mathbf{I} \, Z$

This applies to individual components, series/parallel combinations, or entire networks.

Phasor Representation and Complex Algebra

Phasors convert sinusoidal time-domain signals into complex numbers, making AC analysis into algebra instead of differential equations.

A sinusoidal voltage $v(t) = V_m \cos(\omega t + \theta)$ is represented by the phasor:

$\mathbf{V} = V_m \angle \theta \quad \text{or equivalently} \quad \mathbf{V} = V_m \, e^{j\theta}$

With phasors, you can:

• Add/subtract phasors (easiest in rectangular form) to apply Kirchhoff's voltage and current laws
• Multiply/divide phasors (easiest in polar form) to apply Ohm's law

All of Kirchhoff's laws hold in phasor form, so KVL and KCL work exactly as in DC analysis once you convert to phasors.

Power Calculations in AC Circuits

AC power has three components because voltage and current can be out of phase:

• Real (active) power $P = |V||I|\cos\theta$ in watts (W). This is the power actually consumed.
• Reactive power $Q = |V||I|\sin\theta$ in volt-amperes reactive (VAR). This is power that oscillates between source and reactive elements.
• Apparent power $S = |V||I|$ in volt-amperes (VA). This is the total power the source must supply.

These three form the power triangle, a right triangle where:

$S^2 = P^2 + Q^2$

In complex form, apparent power is $\mathbf{S} = \mathbf{V}\mathbf{I}^*$, where $\mathbf{I}^*$ is the complex conjugate of the current phasor. The real part of $\mathbf{S}$ gives $P$, and the imaginary part gives $Q$.

The power factor is the ratio of real to apparent power:

$PF = \frac{P}{S} = \cos\theta$

A power factor of 1 means all power is real (purely resistive load). Values less than 1 indicate reactive power is present, which is why utilities and engineers care about power factor correction.