23.11 Reactance, Inductive and Capacitive

AC Circuits and Reactance

AC circuits behave very differently depending on whether they contain resistors, inductors, capacitors, or some combination of all three. Each component has its own relationship between voltage and current, and understanding these relationships is what reactance is all about. Reactance is the opposition to AC current flow caused by inductors and capacitors, and it depends on frequency.

Voltage and Current Patterns in RLC Circuits

The key to AC circuits is that voltage and current don't always peak at the same time. Each component type creates a different phase relationship:

• Resistor (R): Voltage and current are in phase. They rise and fall together, just like in a DC circuit.
• Inductor (L): Current lags voltage by 90°. The voltage peaks first, then the current follows a quarter-cycle later. (Remember: "ELI" — voltage E leads current I in an inductor L.)
• Capacitor (C): Current leads voltage by 90°. The current peaks first, then the voltage follows. (Remember: "ICE" — current I leads voltage E in a capacitor C.)

When R, L, and C are combined in a circuit, the overall phase relationship depends on the relative sizes of each component's effect. At the resonance frequency, inductive and capacitive reactances are equal and cancel each other out, leaving a purely resistive circuit where voltage and current are back in phase.

Phasor diagrams are a useful tool for visualizing these relationships. Each component's voltage is drawn as a rotating arrow (phasor), and the vector sum gives you the total circuit voltage.

Reactance Calculation for Inductors and Capacitors

Reactance is measured in ohms ($\Omega$), just like resistance, but it changes with frequency.

Inductive reactance ($X_L$) is the opposition to current flow from an inductor:

$X_L = 2\pi fL$

• $f$ = frequency of the AC signal (Hz)
• $L$ = inductance (H)

$X_L$ is directly proportional to both frequency and inductance. At higher frequencies, the inductor opposes current more because the magnetic field has to change more rapidly.

Capacitive reactance ($X_C$) is the opposition to current flow from a capacitor:

$X_C = \frac{1}{2\pi fC}$

• $f$ = frequency of the AC signal (Hz)
• $C$ = capacitance (F)

$X_C$ is inversely proportional to both frequency and capacitance. At higher frequencies, the capacitor charges and discharges so quickly that it barely impedes the current at all. At low frequencies (or DC, where $f = 0$), $X_C$ goes to infinity, meaning the capacitor blocks current entirely.

Notice the opposite behavior: as frequency increases, $X_L$ goes up while $X_C$ goes down. This is why there's a specific frequency where they're equal (resonance).

Current-Voltage Relationships in AC Circuits

For single-component circuits, the current calculation looks like Ohm's law, but with reactance replacing resistance:

• Resistor only: $I = \frac{V}{R}$ (voltage and current in phase)
• Inductor only: $I = \frac{V}{X_L}$ (current lags voltage by 90°)
• Capacitor only: $I = \frac{V}{X_C}$ (current leads voltage by 90°)

For a circuit with R, L, and C combined, you can't just add R, $X_L$, and $X_C$ directly because they're out of phase with each other. Instead, you use impedance ($Z$), which is the vector sum:

1. Calculate impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$

2. Calculate current: $I = \frac{V}{Z}$

3. Calculate the phase angle between voltage and current: $\phi = \tan^{-1}\left(\frac{X_L - X_C}{R}\right)$

If $\phi$ is positive, the circuit behaves more like an inductor (current lags voltage). If $\phi$ is negative, it behaves more like a capacitor (current leads voltage). If $\phi = 0$, you're at resonance.

Circuit Performance Characteristics

• Quality factor (Q) measures how sharp the resonance peak is. A high Q means the circuit responds strongly at the resonance frequency but drops off quickly on either side. For a series RLC circuit, $Q = \frac{1}{R}\sqrt{\frac{L}{C}}$. Lower resistance gives a higher Q.
• Bandwidth is the range of frequencies around resonance where the circuit still responds effectively (typically defined as the range where power is at least half its peak value). A higher Q means a narrower bandwidth.
• Power factor equals $\cos\phi$ and tells you what fraction of the apparent power actually does useful work. A power factor of 1 (purely resistive circuit) means all the power is used; a power factor near 0 means most of the energy is just sloshing back and forth between the inductor and capacitor without being consumed.