FAD1022 L17-L21 — AC Series Circuits — Formula Sheet
Comprehensive formula sheet for AC series circuits: RL, RC, RLC, resonance, and power analysis.
1. Fundamental AC Quantities
| Symbol | Description | Unit |
|---|---|---|
| $f$ | Frequency | Hz |
| $\omega$ | Angular frequency | rad/s |
| $T$ | Period | s |
| $V_{rms}$ | Root-mean-square voltage | V |
| $I_{rms}$ | Root-mean-square current | A |
| $V_{peak}$ | Peak (amplitude) voltage | V |
| $I_{peak}$ | Peak (amplitude) current | A |
Angular Frequency
$$\omega = 2\pi f$$
- $f$ — frequency (Hz)
2. Reactance
2.1 Inductive Reactance
$$X_L = \omega L = 2\pi f L$$
- $X_L$ — inductive reactance ($\Omega$)
- $\omega$ — angular frequency (rad/s)
- $f$ — frequency (Hz)
- $L$ — inductance (H)
2.2 Capacitive Reactance
$$X_C = \frac{1}{\omega C} = \frac{1}{2\pi f C}$$
- $X_C$ — capacitive reactance ($\Omega$)
- $\omega$ — angular frequency (rad/s)
- $f$ — frequency (Hz)
- $C$ — capacitance (F)
2.3 Net Reactance (RLC Circuits)
$$X = X_L - X_C$$
- $X$ — net reactance ($\Omega$)
3. RL Series Circuit (Lecture 17)
3.1 Current Relationship
$$I_T = I_R = I_L$$
- Current is the same through all series components.
3.2 Total Voltage
$$V_T = \sqrt{V_R^2 + V_L^2}$$
- $V_T$ — total supply voltage (V)
- $V_R$ — voltage across resistor (V)
- $V_L$ — voltage across inductor (V)
3.3 Component Voltages
$$V_R = I_{rms} R$$
$$V_L = I_{rms} X_L$$
3.4 Impedance
$$Z = \sqrt{R^2 + X_L^2}$$
- $Z$ — impedance ($\Omega$)
- $R$ — resistance ($\Omega$)
- $X_L$ — inductive reactance ($\Omega$)
3.5 Phase Angle
$$\theta = \tan^{-1}\left(\frac{V_L}{V_R}\right) = \tan^{-1}\left(\frac{X_L}{R}\right)$$
- $\theta$ — phase angle by which voltage leads current (° or rad)
- $\theta$ is positive
3.6 RMS Current
$$I_{rms} = \frac{V_{rms}}{Z}$$
3.7 Power in Pure Inductive Circuit
$$P_{avg} = 0 \text{ W}$$
- Pure inductors consume zero average power.
4. RC Series Circuit (Lecture 18)
4.1 Current Relationship
$$I_T = I_R = I_C$$
4.2 Total Voltage
$$V_T = \sqrt{V_R^2 + V_C^2}$$
- $V_C$ — voltage across capacitor (V)
4.3 Component Voltages
$$V_R = I_{rms} R$$
$$V_C = I_{rms} X_C$$
4.4 Impedance
$$Z = \sqrt{R^2 + X_C^2}$$
4.5 Phase Angle
$$\theta = \tan^{-1}\left(\frac{-X_C}{R}\right) = \tan^{-1}\left(\frac{V_C}{V_R}\right)$$
- $\theta$ — phase angle (° or rad)
- Current leads voltage by $|\theta|$
- $\theta$ is negative
4.6 RMS Current
$$I_{rms} = \frac{V_{rms}}{Z}$$
4.7 Finding Capacitance from Phase Angle
$$\tan|\theta| = \frac{X_C}{R} \quad \Rightarrow \quad X_C = R \tan|\theta|$$
$$C = \frac{1}{2\pi f X_C}$$
5. RLC Series Circuit (Lecture 19)
5.1 Current Relationship
$$I_T = I_R = I_L = I_C$$
5.2 Phasor Voltage Sum
$$\vec{V}_T = \vec{V}_R + \vec{V}_L + \vec{V}_C$$
- $V_L$ and $V_C$ are $180°$ out of phase and partially cancel.
5.3 General Total Voltage
$$V_T = \sqrt{V_R^2 + (V_L - V_C)^2}$$
Note: $(V_L - V_C)$ may be negative; the square eliminates sign issues.
5.4 General Impedance
$$Z = \sqrt{R^2 + (X_L - X_C)^2}$$
5.5 General Phase Angle
$$\theta = \tan^{-1}\left(\frac{X_L - X_C}{R}\right)$$
- $\theta > 0$ → inductive (voltage leads current)
- $\theta < 0$ → capacitive (current leads voltage)
- $\theta = 0$ → resistive / resonance
5.6 Case 1: More Inductive ($X_L > X_C$ or $V_L > V_C$)
$$V_T = \sqrt{V_R^2 + (V_L - V_C)^2}$$
$$\theta = \tan^{-1}\left(\frac{V_L - V_C}{V_R}\right)$$
$$Z = \sqrt{R^2 + (X_L - X_C)^2}$$
$$\theta = \tan^{-1}\left(\frac{X_L - X_C}{R}\right)$$
- Voltage leads current by $\theta$
- $\theta$ is positive
5.7 Case 2: More Capacitive ($X_C > X_L$ or $V_C > V_L$)
$$V_T = \sqrt{V_R^2 + (V_C - V_L)^2}$$
$$\theta = \tan^{-1}\left(\frac{V_C - V_L}{V_R}\right)$$
$$Z = \sqrt{R^2 + (X_C - X_L)^2}$$
$$\theta = \tan^{-1}\left(\frac{X_C - X_L}{R}\right)$$
- Current leads voltage by $\theta$
- $\theta$ is negative
5.8 Component Voltages
$$V_R = I_{rms} R$$
$$V_L = I_{rms} X_L$$
$$V_C = I_{rms} X_C$$
5.9 RMS Current
$$I_{rms} = \frac{V_{rms}}{Z}$$
6. RLC Series Resonance (Lecture 20)
6.1 Resonance Condition
$$X_L = X_C$$
6.2 Resonant Frequency Derivation
$$2\pi f_0 L = \frac{1}{2\pi f_0 C}$$
$$(2\pi f_0 L)(2\pi f_0 C) = 1$$
$$4\pi^2 f_0^2 LC = 1$$
6.3 Resonant Frequency
$$f_0 = \frac{1}{2\pi\sqrt{LC}}$$
- $f_0$ — resonant frequency (Hz)
- $L$ — inductance (H)
- $C$ — capacitance (F)
6.4 Capacitance at Resonance
$$C = \frac{1}{4\pi^2 f_0^2 L}$$
6.5 Properties at Resonance
| Property | Formula |
|---|---|
| Impedance (minimum) | $Z = R$ |
| Current (maximum) | $I_{rms} = \dfrac{V_{rms}}{R}$ |
| Phase angle | $\theta = 0°$ |
| Power factor | $\text{PF} = 1$ (unity) |
| Circuit behavior | Purely resistive |
6.6 Reactance Ratio at Different Frequencies
If $X_L = X_C$ at frequency $f_0$, then at another frequency $f$:
$$\frac{X_C}{X_L} = \left(\frac{f_0}{f}\right)^2$$
7. AC Power and Power Factor (Lecture 21)
7.1 Three Types of Power
| Type | Symbol | Unit | Dissipated By |
|---|---|---|---|
| Average (Real) Power | $P_{ave}$ | W | Resistor only |
| Reactive Power | $P_R$ or $Q$ | VAr | Inductor / Capacitor |
| Apparent Power | $P_A$ or $S$ | VA | Impedance $Z$ |
7.2 Power Factor Definition
$$\text{Power Factor} = \frac{\text{Average Power (W)}}{\text{Apparent Power (VA)}}$$
$$\text{Power Factor} = \cos\phi = \frac{P_{ave}}{P_A}$$
$$\text{Power Factor} = \frac{R}{Z}$$
- $\phi$ — phase angle between voltage and current
- Ideal value: 1 (unity)
7.3 Power Triangle
$$P_A^2 = P_{ave}^2 + P_R^2$$
$$P_A = \sqrt{P_{ave}^2 + P_R^2}$$
7.4 Pure Resistive Circuit (PRC)
$$P = VI \quad \text{(instantaneous)}$$
$$P_{ave} = V_{rms} I_{rms}$$
7.5 Pure Capacitive Circuit (PCC)
$$P_{ave} = 0 \text{ W}$$
$$P_r = I_{rms}^2 X_C$$
7.6 Pure Inductive Circuit (PLC)
$$P_{ave} = 0 \text{ W}$$
$$P_r = I_{rms}^2 X_L$$
7.7 RL Circuit Power
$$P_{ave} = I_{rms}^2 R$$
$$P_R = I_{rms}^2 X_L$$
$$P_A = I_{rms}^2 Z = \sqrt{P_{ave}^2 + P_R^2}$$
7.8 RC Circuit Power
$$P_{ave} = I_{rms}^2 R$$
$$P_R = I_{rms}^2 X_C$$
$$P_A = I_{rms}^2 Z = \sqrt{P_{ave}^2 + P_R^2}$$
7.9 RLC Circuit Power
$$P_{ave} = I_{rms}^2 R$$
$$P_R = I_{rms}^2 (X_L - X_C)$$
$$P_A = I_{rms}^2 Z \approx \sqrt{P_{ave}^2 + P_R^2}$$
Note: In RLC, net reactive power is $|X_L - X_C|$.
7.10 General Power Formulas
$$P_{ave} = V_{rms} I_{rms} \cos\phi$$
$$P_R = V_{rms} I_{rms} \sin\phi$$
$$P_A = V_{rms} I_{rms}$$
8. Phase Angle Summary
| Circuit | Phase Angle | Sign | Leading Signal |
|---|---|---|---|
| Pure Resistive | $\theta = 0°$ | — | In phase |
| Pure Inductive | $\theta = +90°$ | Positive | Voltage leads current |
| Pure Capacitive | $\theta = -90°$ | Negative | Current leads voltage |
| RL Series | $\theta = \tan^{-1}\left(\dfrac{X_L}{R}\right)$ | Positive | Voltage leads current |
| RC Series | $\theta = \tan^{-1}\left(\dfrac{-X_C}{R}\right)$ | Negative | Current leads voltage |
| RLC Series | $\theta = \tan^{-1}\left(\dfrac{X_L - X_C}{R}\right)$ | ± | Depends on $X_L$ vs $X_C$ |
| RLC at Resonance | $\theta = 0°$ | — | In phase |
9. Impedance Summary
| Circuit | Impedance Formula |
|---|---|
| Pure Resistive | $Z = R$ |
| Pure Inductive | $Z = X_L$ |
| Pure Capacitive | $Z = X_C$ |
| RL Series | $Z = \sqrt{R^2 + X_L^2}$ |
| RC Series | $Z = \sqrt{R^2 + X_C^2}$ |
| RLC Series | $Z = \sqrt{R^2 + (X_L - X_C)^2}$ |
| RLC at Resonance | $Z = R$ |
10. Quick Reference: CIVIL Mnemonic
Capacitor: I leads V
Inductor: V leads I
| Component | Current vs Voltage |
|---|---|
| Capacitor ($C$) | Current leads voltage by $90°$ |
| Inductor ($L$) | Voltage leads current by $90°$ |
| Resistor ($R$) | Current and voltage are in phase |
11. Conditions for Unity Power Factor
For an RLC circuit to achieve $\text{PF} = 1$:
- $$X_L = X_C$$
- $$Z = R$$
- $$\phi = 0°$$
Formula sheet compiled from FAD1022 L17-L21 lecture notes.