5.8.7. Impedance

Table of Contents

Opposition to AC beyond simple resistance

In direct current circuits, a resistor opposes current with resistance, $R$. In alternating current circuits, current and voltage can also be affected by capacitors and inductors. Their opposition to current depends on frequency and on the phase difference between voltage and current. The general quantity that describes this opposition is called impedance.

Impedance plays the same role in AC circuits that resistance plays in DC circuits, but it is more general. It includes both ordinary resistance and the effects of energy storage in electric and magnetic fields.

Impedance is the total opposition a circuit offers to alternating current.
It is usually written as $Z$ and measured in ohms, $\Omega$.

Why resistance is not enough in AC

A resistor converts electrical energy into thermal energy, so voltage and current stay in phase. But a capacitor stores energy in an electric field, and an inductor stores energy in a magnetic field. Because of this energy storage, current and voltage do not always reach their maximum values at the same time.

This means that in AC circuits, opposition to current has two parts. One part dissipates energy, resistance. The other part stores and releases energy, reactance.

Resistance and reactance

Reactance is the AC opposition produced by capacitors and inductors.

For an inductor, the inductive reactance is

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

For a capacitor, the capacitive reactance is

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

Here, $f$ is the frequency and $\omega = 2\pi f$ is the angular frequency.

Inductive reactance increases with frequency. Capacitive reactance decreases with frequency. This is an important difference.

Inductive reactance:
$$
X_L = \omega L
$$
Capacitive reactance:
$$
X_C = \frac{1}{\omega C}
$$
Both are measured in ohms.

Impedance in simple AC components

For a circuit containing only a resistor,

$$
Z = R
$$

For a circuit containing only an inductor,

$$
Z = X_L
$$

For a circuit containing only a capacitor,

$$
Z = X_C
$$

However, when resistance and reactance appear together, impedance must account for both magnitude and phase.

Impedance in series RLC circuits

In a series circuit with a resistor, inductor, and capacitor, the net reactance is

$$
X = X_L - X_C
$$

The impedance magnitude is

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

This shows that resistance and reactance combine differently from ordinary numbers in simple addition, because they affect current in different ways.

For a series RLC circuit,
$$
Z = \sqrt{R^2 + (X_L - X_C)^2}
$$
where
$$
X_L = \omega L, \qquad X_C = \frac{1}{\omega C}
$$

Phase angle and impedance

Because voltage and current may be out of phase, impedance is associated with a phase angle $\phi$. For a series RLC circuit,

$$
\tan \phi = \frac{X_L - X_C}{R}
$$

If $X_L > X_C$, the circuit is overall inductive. If $X_C > X_L$, it is overall capacitive. If $X_L = X_C$, the reactive effects cancel.

A positive phase angle means voltage leads current, which is inductive behavior. A negative phase angle means current leads voltage, which is capacitive behavior.

Complex form of impedance

A more complete mathematical description uses complex numbers. In that form,

$$
Z = R + i(X_L - X_C)
$$

where $i = \sqrt{-1}$.

This form is very useful because it keeps both magnitude and phase together in one expression. The real part represents resistance, and the imaginary part represents reactance.

For the individual components,

$$
Z_R = R
$$

$$
Z_L = i\omega L
$$

$$
Z_C = -\frac{i}{\omega C}
$$

For many beginner problems, only the magnitude $|Z|$ is used, but the complex form explains why phase differences appear.

Complex impedance:
$$
Z = R + i(X_L - X_C)
$$
Magnitude:
$$
|Z| = \sqrt{R^2 + (X_L - X_C)^2}
$$

AC version of Ohm's law

In AC circuits, Ohm's law takes the form

$$
V = IZ
$$

If using magnitudes only, this becomes

$$
V_{\text{rms}} = I_{\text{rms}} Z
$$

This is often the most practical equation for finding current or voltage in AC circuit problems.

AC Ohm's law:
$$
V_{\text{rms}} = I_{\text{rms}} Z
$$

Frequency dependence

A key feature of impedance is that it depends on frequency whenever inductors or capacitors are present.

The table below summarizes this behavior.

Component	Opposition in AC	Formula	Effect of increasing frequency
Resistor	Resistance	$R$	No change
Inductor	Inductive reactance	$X_L = \omega L$	Increases
Capacitor	Capacitive reactance	$X_C = \frac{1}{\omega C}$	Decreases

This is why the same circuit can behave differently at low and high frequencies.

Special case, resonance

When a series RLC circuit satisfies

$$
X_L = X_C
$$

the net reactance is zero, so

$$
Z = R
$$

At this point, the impedance is minimum for that circuit, and the current is maximum for a given applied voltage. This condition is called resonance, which is discussed in its own chapter.

Geometric interpretation

Impedance can be visualized using a right triangle. Resistance forms one side, reactance forms the other, and impedance is the hypotenuse.

Rendering scientific sketch…

Impedance triangle

From this triangle,

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

and

$$
\tan\phi = \frac{X_L - X_C}{R}
$$

A short example

Suppose a series circuit has

$$
R = 30\,\Omega, \qquad X_L = 40\,\Omega, \qquad X_C = 10\,\Omega
$$

Then the net reactance is

$$
X = X_L - X_C = 40 - 10 = 30\,\Omega
$$

So the impedance is

$$
Z = \sqrt{30^2 + 30^2} = \sqrt{1800} \approx 42.4\,\Omega
$$

If the RMS voltage is $120\,\text{V}$, then the RMS current is

$$
I_{\text{rms}} = \frac{V_{\text{rms}}}{Z} = \frac{120}{42.4} \approx 2.83\,\text{A}
$$

Physical meaning

Impedance tells you how hard it is for alternating current to flow. Resistance permanently removes energy from the electrical system, usually as heat. Reactance temporarily stores energy and returns it later. Because of this, impedance reflects both energy loss and energy storage.

This is why two circuits can have the same current-limiting effect in magnitude, but behave differently in timing and energy transfer.

Key formulas

Important impedance relations:
$$
X_L = \omega L
$$
$$
X_C = \frac{1}{\omega C}
$$
$$
Z = \sqrt{R^2 + (X_L - X_C)^2}
$$
$$
\tan\phi = \frac{X_L - X_C}{R}
$$
$$
V_{\text{rms}} = I_{\text{rms}} Z
$$

Impedance is one of the central ideas in AC circuit analysis, because it unifies resistance, inductive effects, and capacitive effects into one single quantity.

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