Formula Sheet: AC Circuit Analysis Part 1
Comprehensive formula sheet extracted from FAD1022 Tutorial 6 — AC Circuit Analysis Part 1
1. Reactance
1.1 Capacitive Reactance
$$X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$$
| Variable | Description | SI Unit |
|---|---|---|
| $X_C$ | Capacitive reactance | $\Omega$ (Ohms) |
| $\omega$ | Angular frequency | $\text{rad s}^{-1}$ |
| $f$ | Frequency | $\text{Hz}$ |
| $C$ | Capacitance | $\text{F}$ (Farads) |
Key behavior: $X_C \propto \dfrac{1}{f}$. Higher frequency $\rightarrow$ lower reactance. DC ($f=0$) $\rightarrow$ $X_C \rightarrow \infty$ (open circuit).
1.2 Inductive Reactance
$$X_L = \omega L = 2\pi f L$$
| Variable | Description | SI Unit |
|---|---|---|
| $X_L$ | Inductive reactance | $\Omega$ |
| $L$ | Inductance | $\text{H}$ (Henries) |
Key behavior: $X_L \propto f$. Higher frequency $\rightarrow$ higher reactance. DC ($f=0$) $\rightarrow$ $X_L = 0$ (short circuit).
2. Impedance
2.1 Definition
$$Z = \frac{V_{rms}}{I_{rms}} = \frac{V_0}{I_0}$$
| Variable | Description | SI Unit |
|---|---|---|
| $Z$ | Impedance | $\Omega$ |
2.2 RL Series Circuit
$$Z = \sqrt{R^2 + X_L^2}$$
2.3 RC Series Circuit
$$Z = \sqrt{R^2 + X_C^2}$$
2.4 RLC Series Circuit (General)
$$Z = \sqrt{R^2 + (X_L - X_C)^2}$$
| Variable | Description | SI Unit |
|---|---|---|
| $R$ | Resistance | $\Omega$ |
| $X_L$ | Inductive reactance | $\Omega$ |
| $X_C$ | Capacitive reactance | $\Omega$ |
Note: Impedance is the AC analogue of resistance. It accounts for both resistive and reactive opposition to current flow.
3. Phase Angle
3.1 Definition
$$\tan\phi = \frac{X_L - X_C}{R}$$
$$\phi = \arctan\left(\frac{X_L - X_C}{R}\right)$$
| Variable | Description | SI Unit |
|---|---|---|
| $\phi$ | Phase angle | rad or degrees |
3.2 RL Series Circuit
$$\tan\phi = \frac{X_L}{R}$$
Result: $\phi > 0$. Current lags voltage (inductive behavior).
3.3 RC Series Circuit
$$\tan\phi = -\frac{X_C}{R}$$
Or equivalently:
$$\tan|\phi| = \frac{X_C}{R}$$
Result: $\phi < 0$. Current leads voltage (capacitive behavior).
3.4 RLC Series Circuit Cases
| Condition | Phase Angle | Behavior |
|---|---|---|
| $X_L > X_C$ | $\phi > 0$ | Inductive — current lags voltage |
| $X_C > X_L$ | $\phi < 0$ | Capacitive — current leads voltage |
| $X_L = X_C$ | $\phi = 0$ | Resistive — current and voltage in phase (resonance) |
3.5 Phase Relationships Summary
$$\cos\phi = \frac{R}{Z}, \quad \sin\phi = \frac{X_L - X_C}{Z}$$
4. Current and Voltage in AC Circuits
4.1 RMS Current
$$I_{rms} = \frac{V_{rms}}{Z}$$
4.2 Peak Current
$$I_0 = \sqrt{2} , I_{rms} = \frac{V_0}{Z}$$
4.3 Peak Voltage
$$V_0 = \sqrt{2} , V_{rms}$$
4.4 Voltage Across Individual Components
Resistor:
$$V_R = I_{rms} R = I_0 R \sin(\omega t)$$
$V_R$ is in phase with current.
Capacitor:
$$V_C = I_{rms} X_C = I_0 X_C \sin(\omega t - \tfrac{\pi}{2})$$
$V_C$ lags current by $90°$ ($\pi/2$ rad).
Inductor:
$$V_L = I_{rms} X_L = I_0 X_L \sin(\omega t + \tfrac{\pi}{2})$$
$V_L$ leads current by $90°$ ($\pi/2$ rad).
| Variable | Description | SI Unit |
|---|---|---|
| $V_R$ | RMS voltage across resistor | $\text{V}$ |
| $V_C$ | RMS voltage across capacitor | $\text{V}$ |
| $V_L$ | RMS voltage across inductor | $\text{V}$ |
| $I_{rms}$ | RMS circuit current | $\text{A}$ |
5. Phasor Relationships
5.1 Source Voltage from Component Voltages
$$V_{rms} = \sqrt{V_R^2 + (V_L - V_C)^2}$$
5.2 For RL Circuit
$$V_{rms} = \sqrt{V_R^2 + V_L^2}$$
5.3 For RC Circuit
$$V_{rms} = \sqrt{V_R^2 + V_C^2}$$
5.4 Phase Angle from Voltages
$$\tan\phi = \frac{V_L - V_C}{V_R}$$
For RL: $\tan\phi = \dfrac{V_L}{V_R}$
For RC: $\tan\phi = -\dfrac{V_C}{V_R}$
6. Power in AC Circuits (General)
6.1 Average (Real) Power
$$P_{avg} = I_{rms} V_{rms} \cos\phi = I_{rms}^2 R$$
6.2 Power Factor
$$\text{Power Factor} = \cos\phi = \frac{R}{Z}$$
| Variable | Description |
|---|---|
| $\cos\phi$ | Power factor (dimensionless, $0 \leq \cos\phi \leq 1$) |
Note: Only the resistive component dissipates power. Reactances store and release energy but do not dissipate it.
6.3 Reactive Power
$$Q = I_{rms} V_{rms} \sin\phi$$
| Variable | Description | SI Unit |
|---|---|---|
| $Q$ | Reactive power | $\text{VAR}$ (Volt-Amperes Reactive) |
6.4 Apparent Power
$$S = I_{rms} V_{rms} = I_{rms}^2 Z$$
| Variable | Description | SI Unit |
|---|---|---|
| $S$ | Apparent power | $\text{VA}$ (Volt-Amperes) |
7. Frequency Dependence Summary
7.1 Capacitive Reactance vs Frequency
$$X_C(f) = \frac{1}{2\pi f C}$$
- Doubling $f$ → $X_C$ halves
- Halving $f$ → $X_C$ doubles
7.2 Inductive Reactance vs Frequency
$$X_L(f) = 2\pi f L$$
- Doubling $f$ → $X_L$ doubles
- Halving $f$ → $X_L$ halves
7.3 Impedance vs Frequency (RL Circuit)
$$Z = \sqrt{R^2 + (2\pi f L)^2}$$
- Increasing $f$ → $Z$ increases → $I$ decreases
7.4 Impedance vs Frequency (RC Circuit)
$$Z = \sqrt{R^2 + \left(\frac{1}{2\pi f C}\right)^2}$$
- Increasing $f$ → $Z$ decreases → $I$ increases
8. Summary of Key Relationships
| Concept | Formula |
|---|---|
| Capacitive reactance | $X_C = \dfrac{1}{\omega C} = \dfrac{1}{2\pi f C}$ |
| Inductive reactance | $X_L = \omega L = 2\pi f L$ |
| Impedance (RL) | $Z = \sqrt{R^2 + X_L^2}$ |
| Impedance (RC) | $Z = \sqrt{R^2 + X_C^2}$ |
| Impedance (RLC) | $Z = \sqrt{R^2 + (X_L - X_C)^2}$ |
| Phase angle (general) | $\tan\phi = \dfrac{X_L - X_C}{R}$ |
| Phase angle (RL) | $\tan\phi = \dfrac{X_L}{R}$ (current lags) |
| Phase angle (RC) | $\tan\phi = -\dfrac{X_C}{R}$ (current leads) |
| RMS current | $I_{rms} = \dfrac{V_{rms}}{Z}$ |
| Peak current | $I_0 = \sqrt{2} , I_{rms}$ |
| Voltage across resistor | $V_R = I_{rms} R$ |
| Voltage across capacitor | $V_C = I_{rms} X_C$ |
| Voltage across inductor | $V_L = I_{rms} X_L$ |
| Source voltage | $V = \sqrt{V_R^2 + (V_L - V_C)^2}$ |
| Average power | $P_{avg} = I_{rms} V_{rms} \cos\phi = I_{rms}^2 R$ |
| Power factor | $\cos\phi = \dfrac{R}{Z}$ |
Related Concepts
- AC Circuits
- AC Circuit
- Capacitive Reactance
- Inductive Reactance
- Impedance
- Phase Angle
- Phasor Diagram
- RL Circuit
- RC Circuit
- Reactance
- Power Factor
- Resonance