AC Theory: Inductors, Capacitors, and Reactance – Study Notes

AC Theory Fundamentals

Introduction to AC Theory

Alternating Current (AC) theory is essential for understanding the behavior of electrical circuits containing inductors and capacitors. These components respond differently to AC compared to DC, leading to unique properties such as reactance and phase relationships.

Inductors in AC Circuits

Inductance: Definition and Properties

• Inductance is the property of a coil (inductor) that opposes changes in current.

• Unit of Inductance: The SI unit is the henry (H).

• Symbol: Inductance is represented by the letter L in formulas.

• Lenz’s Law: An inductor opposes a change in current by inducing a voltage.

• Factors Affecting Inductance:

  • Core material (increases inductance)

  • Number of turns (increases inductance)

  • Core area (increases inductance)

  • Core length (increasing length decreases inductance)

Inductor Behavior in AC Circuits

• Phase Relationship: The induced voltage in an inductor leads the applied voltage by 90 degrees.

• Current-Voltage Relationship: Inductor current lags voltage by 90 degrees.

• Inductive Reactance ($X_L$): Opposition to AC current, given by: $X_L = 2\pi f L$ where $f$ is frequency (Hz), $L$ is inductance (H).

• Series and Parallel Inductors:

  • Series: $L_{total} = L_1 + L_2 + ... + L_n$ (total inductance increases)

  • Parallel: $\frac{1}{L_{total}} = \frac{1}{L_1} + \frac{1}{L_2} + ... + \frac{1}{L_n}$ (total inductance decreases)

• Current Limiting: The property of an inductor to limit current is called inductive reactance.

Inductor Calculations and Examples

• Example: Calculate $X_L$ for a 10 H coil at 60 Hz: $X_L = 2\pi \times 60 \times 10 = 3,770 \ \Omega$

• Example: Find current for $V = 120$ V, $X_L = 240 \ \Omega$: $I = \frac{V}{X_L} = \frac{120}{240} = 0.5$ A

Capacitors in AC Circuits

Capacitance: Definition and Properties

• Capacitance is the ability of a device to store electrical charge.

• Unit of Capacitance: The SI unit is the farad (F).

• Conversion: $1 \mu F = 1 \times 10^{-6}$ F; $47 \mu F = 4.7 \times 10^{-5}$ F

Charging and Discharging Capacitors

• Time Constant ($\tau$): $\tau = R \times C$ (in seconds), where $R$ is resistance ($\Omega$), $C$ is capacitance (F).

• Charging Curve:

  • After 1 time constant: 63% charged

  • Fully charged: after 5 time constants (99.3%)

  • Percentage after $n$ time constants: $Q = Q_{max}(1 - e^{-n})$

• Example: Calculate $\tau$ for $R = 100$ k$\Omega$, $C = 10 \mu$F: $\tau = 100,000 \times 10 \times 10^{-6} = 1$ s

• Example: Time to fully charge a 220 $\mu$F capacitor in series with 50 k$\Omega$: $\tau = 50,000 \times 220 \times 10^{-6} = 11$ s; Fully charged in $5\tau = 55$ s

Capacitor Behavior in DC and AC Circuits

• DC Behavior:

  • After fully charged, a capacitor acts as an open circuit.

  • After steady state, an inductor acts as a short circuit.

• Types of Capacitors:

  • Motor running capacitors: Usually oil-filled or polypropylene types.

  • Electrolytic capacitors: Damaged by reversed polarity.

Capacitive Reactance and Calculations

• Capacitive Reactance ($X_C$): Opposition to AC current, given by: $X_C = \frac{1}{2\pi f C}$ where $f$ is frequency (Hz), $C$ is capacitance (F).

• Phase Relationship: In a purely capacitive circuit, current leads voltage by 90 degrees.

• Series and Parallel Capacitors:

  • Series: $\frac{1}{C_{total}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}$

  • Parallel: $C_{total} = C_1 + C_2 + ... + C_n$

• Example: Total capacitance of 2 $\mu$F, 4 $\mu$F, 6 $\mu$F in series: $\frac{1}{C_{total}} = \frac{1}{2} + \frac{1}{4} + \frac{1}{6} = 0.5 + 0.25 + 0.1667 = 0.9167$ $C_{total} = 1.09 \mu$F

• Example: $X_C$ for 150 $\mu$F at 60 Hz: $X_C = \frac{1}{2\pi \times 60 \times 150 \times 10^{-6}} = 17.7 \ \Omega$

• Example: Find current for $V = 120$ V, $X_C = 20 \ \Omega$: $I = \frac{V}{X_C} = \frac{120}{20} = 6$ A

Summary Table: Inductors vs. Capacitors in AC Circuits

Property	Inductor	Capacitor
Unit	Henry (H)	Farad (F)
Symbol	L	C
Reactance Formula	$X_L = 2\pi f L$	$X_C = \frac{1}{2\pi f C}$
Phase Relationship	Current lags voltage by 90°	Current leads voltage by 90°
Series Combination	Adds: $L_{total} = L_1 + L_2 + ...$	Inverse sum: $\frac{1}{C_{total}} = \frac{1}{C_1} + ...$
Parallel Combination	Inverse sum: $\frac{1}{L_{total}} = \frac{1}{L_1} + ...$	Adds: $C_{total} = C_1 + C_2 + ...$
DC Steady State	Short circuit	Open circuit

Additional info:

• Motor running capacitors are typically non-polarized and designed for continuous AC operation.

• Electrolytic capacitors are polarized and must be connected with correct polarity to avoid damage.

• Time constant ($\tau$) is a key concept for both charging and discharging capacitors in RC circuits.