Formula Sheet — AC Circuit Analysis Part 2
Source: FAD1022 Tutorial 7 — AC Circuit Analysis Part 2
1. Reactance
Inductive Reactance
$$X_L = \omega L = 2\pi f L$$
| Symbol | Meaning | Units |
|---|---|---|
| $X_L$ | Inductive reactance | $\Omega$ (ohms) |
| $\omega$ | Angular frequency | rad/s |
| $f$ | Frequency | Hz |
| $L$ | Inductance | H (henries) |
Capacitive Reactance
$$X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$$
| Symbol | Meaning | Units |
|---|---|---|
| $X_C$ | Capacitive reactance | $\Omega$ (ohms) |
| $\omega$ | Angular frequency | rad/s |
| $f$ | Frequency | Hz |
| $C$ | Capacitance | F (farads) |
2. Impedance
Series RLC Circuit Impedance
$$Z = \sqrt{R^2 + (X_L - X_C)^2}$$
| Symbol | Meaning | Units |
|---|---|---|
| $Z$ | Impedance | $\Omega$ (ohms) |
| $R$ | Resistance | $\Omega$ (ohms) |
| $X_L$ | Inductive reactance | $\Omega$ (ohms) |
| $X_C$ | Capacitive reactance | $\Omega$ (ohms) |
Impedance from Voltage and Current
$$Z = \frac{V_{rms}}{I_{rms}}$$
| Symbol | Meaning | Units |
|---|---|---|
| $Z$ | Impedance | $\Omega$ (ohms) |
| $V_{rms}$ | RMS voltage | V (volts) |
| $I_{rms}$ | RMS current | A (amperes) |
3. Voltage Relationships in Series RLC Circuit
Voltage Across Resistor
$$V_R = I_{rms} R$$
Voltage Across Inductor
$$V_L = I_{rms} X_L$$
Voltage Across Capacitor
$$V_C = I_{rms} X_C$$
Total RMS Voltage
$$V_{rms} = I_{rms} Z = \sqrt{V_R^2 + (V_L - V_C)^2}$$
| Symbol | Meaning | Units |
|---|---|---|
| $V_R$ | Voltage across resistor | V (volts) |
| $V_L$ | Voltage across inductor | V (volts) |
| $V_C$ | Voltage across capacitor | V (volts) |
| $V_{rms}$ | Total RMS voltage | V (volts) |
| $I_{rms}$ | RMS current | A (amperes) |
| $R$ | Resistance | $\Omega$ (ohms) |
| $X_L$ | Inductive reactance | $\Omega$ (ohms) |
| $X_C$ | Capacitive reactance | $\Omega$ (ohms) |
| $Z$ | Impedance | $\Omega$ (ohms) |
4. Phase Angle
Phase Angle from Reactances and Resistance
$$\tan\phi = \frac{X_L - X_C}{R}$$
$$\phi = \arctan\left(\frac{X_L - X_C}{R}\right)$$
Phase Angle from Impedance
$$\cos\phi = \frac{R}{Z}$$
$$\sin\phi = \frac{X_L - X_C}{Z}$$
| Symbol | Meaning | Units |
|---|---|---|
| $\phi$ | Phase angle between voltage and current | degrees (°) or radians (rad) |
| $X_L$ | Inductive reactance | $\Omega$ (ohms) |
| $X_C$ | Capacitive reactance | $\Omega$ (ohms) |
| $R$ | Resistance | $\Omega$ (ohms) |
| $Z$ | Impedance | $\Omega$ (ohms) |
Note: $\phi > 0$ means voltage leads current (inductive circuit). $\phi < 0$ means current leads voltage (capacitive circuit).
5. Power in AC Circuits
Average (Real) Power
$$P_{avg} = I_{rms}^2 R = V_{rms} I_{rms} \cos\phi$$
Reactive Power
$$Q = I_{rms}^2 (X_L - X_C) = V_{rms} I_{rms} \sin\phi$$
Apparent Power
$$S = V_{rms} I_{rms} = I_{rms}^2 Z$$
Power Relationships
$$S = \sqrt{P_{avg}^2 + Q^2}$$
| Symbol | Meaning | Units |
|---|---|---|
| $P_{avg}$ | Average (real) power | W (watts) |
| $Q$ | Reactive power | VAr (volt-amperes reactive) |
| $S$ | Apparent power | VA (volt-amperes) |
| $I_{rms}$ | RMS current | A (amperes) |
| $V_{rms}$ | RMS voltage | V (volts) |
| $R$ | Resistance | $\Omega$ (ohms) |
| $X_L$ | Inductive reactance | $\Omega$ (ohms) |
| $X_C$ | Capacitive reactance | $\Omega$ (ohms) |
| $Z$ | Impedance | $\Omega$ (ohms) |
| $\phi$ | Phase angle | degrees (°) or radians (rad) |
6. Power Factor
Definition
$$\text{PF} = \cos\phi = \frac{R}{Z} = \frac{P_{avg}}{S}$$
| Symbol | Meaning | Units |
|---|---|---|
| $\text{PF}$ | Power factor | dimensionless |
| $\phi$ | Phase angle | degrees (°) or radians (rad) |
| $R$ | Resistance | $\Omega$ (ohms) |
| $Z$ | Impedance | $\Omega$ (ohms) |
| $P_{avg}$ | Average power | W (watts) |
| $S$ | Apparent power | VA (volt-amperes) |
Note: At resonance, $\text{PF} = 1$ (unity power factor).
7. Resonance in RLC Circuits
Resonant Frequency
$$f_0 = \frac{1}{2\pi\sqrt{LC}}$$
| Symbol | Meaning | Units |
|---|---|---|
| $f_0$ | Resonant frequency | Hz |
| $L$ | Inductance | H (henries) |
| $C$ | Capacitance | F (farads) |
Conditions at Resonance
-
Inductive reactance equals capacitive reactance: $$X_L = X_C$$
-
Impedance is at minimum (equal to resistance): $$Z = R$$
-
Phase angle is zero: $$\phi = 0°$$
-
Power factor is unity: $$\text{PF} = \cos(0°) = 1$$
-
Total reactance is zero: $$X_L - X_C = 0$$
| Symbol | Meaning | Units |
|---|---|---|
| $X_L$ | Inductive reactance | $\Omega$ (ohms) |
| $X_C$ | Capacitive reactance | $\Omega$ (ohms) |
| $Z$ | Impedance | $\Omega$ (ohms) |
| $R$ | Resistance | $\Omega$ (ohms) |
| $\phi$ | Phase angle | degrees (°) |
8. Derived Circuit Quantities
RMS Current
$$I_{rms} = \frac{V_{rms}}{Z} = \frac{P_{avg}}{V_{rms} \cos\phi}$$
Resistance from Power and Current
$$R = \frac{P_{avg}}{I_{rms}^2}$$
Resistance from Impedance and Phase Angle
$$R = Z \cos\phi$$
Reactance Difference from Impedance and Phase Angle
$$X_L - X_C = Z \sin\phi$$
| Symbol | Meaning | Units |
|---|---|---|
| $I_{rms}$ | RMS current | A (amperes) |
| $V_{rms}$ | RMS voltage | V (volts) |
| $Z$ | Impedance | $\Omega$ (ohms) |
| $P_{avg}$ | Average power | W (watts) |
| $\phi$ | Phase angle | degrees (°) or radians (rad) |
| $R$ | Resistance | $\Omega$ (ohms) |
| $X_L$ | Inductive reactance | $\Omega$ (ohms) |
| $X_C$ | Capacitive reactance | $\Omega$ (ohms) |
9. Summary Relationships
| Quantity | Formula | Units |
|---|---|---|
| Impedance | $Z = \sqrt{R^2 + (X_L - X_C)^2}$ | $\Omega$ |
| Phase angle | $\tan\phi = \dfrac{X_L - X_C}{R}$ | ° or rad |
| Average power | $P_{avg} = I_{rms}^2 R = V_{rms} I_{rms} \cos\phi$ | W |
| Reactive power | $Q = I_{rms}^2(X_L - X_C) = V_{rms} I_{rms} \sin\phi$ | VAr |
| Apparent power | $S = V_{rms} I_{rms} = I_{rms}^2 Z$ | VA |
| Power factor | $\text{PF} = \cos\phi = \dfrac{R}{Z} = \dfrac{P_{avg}}{S}$ | — |
| Resonant frequency | $f_0 = \dfrac{1}{2\pi\sqrt{LC}}$ | Hz |
Related Concepts
- AC Circuits
- RLC Circuit
- Series RLC Circuit
- Resonance
- Resonant Frequency
- Impedance
- Power Factor
- Average Power
- Reactive Power
- Apparent Power
- Phase Angle
- Phasor Diagram