FAD1022 L14-L16 — AC Analysis — Formula Sheet
This formula sheet contains all equations and key relationships from Lectures 14–16 on Alternating Current (AC), Phasor Diagrams, and Reactance.
1. AC Signal Fundamentals (L14)
1.1 Instantaneous Current & Voltage
The general equations for sinusoidal AC current and voltage:
$$I(t) = I_0 \sin(\omega t)$$
$$V(t) = V_0 \sin(\omega t)$$
| Variable | Description | Unit |
|---|---|---|
| $I(t)$ | Instantaneous current | A (Ampere) |
| $V(t)$ | Instantaneous voltage | V (Volt) |
| $I_0$ | Peak (maximum) current | A |
| $V_0$ | Peak (maximum) voltage | V |
| $\omega$ | Angular frequency | rad/s |
| $t$ | Time | s |
[!note] AC signals can also be expressed as cosine functions: $I(t) = I_0 \cos(\omega t)$, etc.
1.2 Angular Frequency
$$\omega = \frac{2\pi}{T} = 2\pi f$$
| Variable | Description | Unit |
|---|---|---|
| $\omega$ | Angular frequency | rad/s |
| $T$ | Period (time for one complete cycle) | s |
| $f$ | Frequency (cycles per second) | Hz |
2. Average & RMS Values (L14)
2.1 Average Value of AC
For a full cycle, the average value of a sinusoidal AC signal is:
$$I_{\text{avg (full cycle)}} = 0$$
(Positive and negative halves cancel out over a complete cycle.)
2.2 Root Mean Square (RMS) Value
The RMS value represents the effective DC-equivalent value that delivers the same power.
For current:
$$I_{\text{rms}} = \frac{I_{\max}}{\sqrt{2}} = 0.707, I_{\max}$$
For voltage:
$$V_{\text{rms}} = \frac{V_{\max}}{\sqrt{2}} = 0.707, V_{\max}$$
| Variable | Description | Unit |
|---|---|---|
| $I_{\text{rms}}$ | RMS current | A |
| $V_{\text{rms}}$ | RMS voltage | V |
| $I_{\max}$ / $I_0$ | Maximum (peak) current | A |
| $V_{\max}$ / $V_0$ | Maximum (peak) voltage | V |
2.3 Power in AC Circuits
$$P = V_{\text{rms}}, I_{\text{rms}}$$
| Variable | Description | Unit |
|---|---|---|
| $P$ | Average power | W (Watt) |
| $V_{\text{rms}}$ | RMS voltage | V |
| $I_{\text{rms}}$ | RMS current | A |
Derived from power:
$$I_{\text{rms}} = \frac{P}{V_{\text{rms}}}$$
$$I_{\max} = I_{\text{rms}} \times \sqrt{2}$$
3. Phasor Diagrams & Phase Angle (L15)
3.1 General Sinusoidal Form with Phase Angle
$$A(t) = A_m \sin(\omega t + \phi)$$
| Variable | Description | Unit |
|---|---|---|
| $A(t)$ | Instantaneous value of the quantity | (varies) |
| $A_m$ | Amplitude (maximum value) | (varies) |
| $\omega$ | Angular frequency | rad/s |
| $t$ | Time | s |
| $\phi$ | Phase angle | rad (or degrees) |
3.2 Phase Shift Equations
In-phase (no shift):
$$A(t) = A_m \sin(\omega t), \quad \phi = 0^\circ$$
Leading (left shift, positive $\phi$):
$$A(t) = A_m \sin(\omega t + \phi)$$
Lagging (right shift, negative $\phi$):
$$A(t) = A_m \sin(\omega t - \phi)$$
3.3 Lead & Lag Relationships
- Leading: Signal reaches peak/zero-crossing earlier than the reference.
- Lagging: Signal reaches peak/zero-crossing later than the reference.
- Phase angle $\phi$ = magnitude of lead or lag.
Example — Voltage leads current by $\frac{\pi}{2}$:
$$V(t) = V_0 \sin\left(\omega t + \frac{\pi}{2}\right)$$
$$I(t) = I_0 \sin(\omega t)$$
Example — Current leads voltage by $\frac{\pi}{2}$:
$$I(t) = I_0 \sin(\omega t)$$
$$V(t) = V_0 \sin\left(\omega t - \frac{\pi}{2}\right)$$
4. Impedance (L16)
4.1 Definition of Impedance
$$Z = \frac{V_{\text{rms}}}{I_{\text{rms}}} = \frac{V_0}{I_0}$$
| Variable | Description | Unit |
|---|---|---|
| $Z$ | Impedance | $\Omega$ (Ohm) |
| $V_{\text{rms}}$ | RMS voltage | V |
| $I_{\text{rms}}$ | RMS current | A |
| $V_0$ | Peak voltage | V |
| $I_0$ | Peak current | A |
[!tip] Impedance is a scalar quantity. In DC circuits, impedance behaves like resistance.
5. Pure Resistive Circuit (PRC) — L16
5.1 Current & Voltage Equations
$$I = I_0 \sin(\omega t)$$
$$V_R = V_0 \sin(\omega t) = V$$
5.2 Phase Relationship
$$\Delta\phi = \omega t - \omega t = 0$$
- Current and voltage are IN PHASE in a pure resistor.
- Phase difference: $\Delta\phi = 0$
5.3 Impedance in Pure Resistor
$$Z = \frac{V_{\text{rms}}}{I_{\text{rms}}} = \frac{V_0}{I_0} = R$$
| Variable | Description | Unit |
|---|---|---|
| $R$ | Resistance | $\Omega$ |
6. Pure Capacitive Circuit (PCC) — L16
6.1 Current & Voltage Equations
$$V_C = V = V_0 \sin(\omega t)$$
$$I = I_0 \sin\left(\omega t + \frac{\pi}{2}\right) \quad \text{or} \quad I = I_0 \cos(\omega t)$$
6.2 Phase Relationship
$$\Delta\phi = \omega t - \left(\omega t + \frac{\pi}{2}\right) = -\frac{\pi}{2}\ \text{rad}$$
- Current LEADS voltage by $\frac{\pi}{2}$ radians ($90^\circ$).
- Voltage LAGS behind current by $\frac{\pi}{2}$ radians ($90^\circ$).
6.3 Capacitive Reactance, $X_C$
$$X_C = \frac{1}{2\pi f C} = \frac{V_{\text{rms}}}{I_{\text{rms}}} = \frac{V_0}{I_0}$$
| Variable | Description | Unit |
|---|---|---|
| $X_C$ | Capacitive reactance | $\Omega$ |
| $f$ | Frequency of AC source | Hz |
| $C$ | Capacitance | F (Farad) |
Frequency dependence:
$$X_C \propto \frac{1}{f}$$
[!note] Capacitive reactance is inversely proportional to frequency. As $f \uparrow$, $X_C \downarrow$.
7. Pure Inductive Circuit (PLC) — L16
7.1 Current & Voltage Equations
$$V = V_0 \cos(\omega t) \quad \text{or} \quad V = V_0 \sin\left(\omega t + \frac{\pi}{2}\right)$$
$$I = I_0 \sin(\omega t)$$
7.2 Phase Relationship
$$\Delta\phi = \left(\omega t + \frac{\pi}{2}\right) - \omega t = \frac{\pi}{2}\ \text{rad}$$
- Voltage LEADS current by $\frac{\pi}{2}$ radians ($90^\circ$).
- Current LAGS behind voltage by $\frac{\pi}{2}$ radians ($90^\circ$).
7.3 Inductive Reactance, $X_L$
$$X_L = 2\pi f L = \frac{V_{\text{rms}}}{I_{\text{rms}}} = \frac{V_0}{I_0}$$
| Variable | Description | Unit |
|---|---|---|
| $X_L$ | Inductive reactance | $\Omega$ |
| $f$ | Frequency of AC source | Hz |
| $L$ | Self-inductance | H (Henry) |
Frequency dependence:
$$X_L \propto f$$
[!note] Inductive reactance is directly proportional to frequency. As $f \uparrow$, $X_L \uparrow$.
8. Memory Aid — CIVIL Mnemonic
A quick way to remember which quantity leads in capacitor and inductor circuits:
| Circuit | Lead/Lag Relationship |
|---|---|
| C (Capacitor) | I leads V |
| L (Inductor) | V leads I (or I lags V) |
In other words:
- In a capacitor: Current leads voltage by $90^\circ$.
- In an inductor: Voltage leads current by $90^\circ$.
9. Phase Summary Table
| Circuit | Current Equation | Voltage Equation | Phase Difference $\Delta\phi$ | Lead/Lag |
|---|---|---|---|---|
| Pure Resistive (PRC) | $I = I_0 \sin(\omega t)$ | $V_R = V_0 \sin(\omega t)$ | $0$ | In-phase |
| Pure Capacitive (PCC) | $I = I_0 \sin(\omega t + \frac{\pi}{2})$ | $V_C = V_0 \sin(\omega t)$ | $-\frac{\pi}{2}$ | I leads V |
| Pure Inductive (PLC) | $I = I_0 \sin(\omega t)$ | $V_L = V_0 \sin(\omega t + \frac{\pi}{2})$ | $+\frac{\pi}{2}$ | V leads I |
10. Reactance & Impedance Summary Table
| Circuit | Impedance / Reactance | Formula | Frequency Dependence |
|---|---|---|---|
| Pure Resistive (PRC) | Resistance $R$ | $Z = R = \frac{V_0}{I_0}$ | Independent of $f$ |
| Pure Capacitive (PCC) | Capacitive Reactance $X_C$ | $X_C = \frac{1}{2\pi f C}$ | $X_C \propto \frac{1}{f}$ |
| Pure Inductive (PLC) | Inductive Reactance $X_L$ | $X_L = 2\pi f L$ | $X_L \propto f$ |
[!tip] For Finals
- RMS values are used for power calculations and everyday electrical ratings (e.g., 230 V RMS home supply → $V_0 = 230\sqrt{2} \approx 325$ V).
- $\omega = 2\pi f$ is essential for converting between time-domain equations and frequency-domain analysis.
- Remember the CIVIL mnemonic for phase relationships.
- Reactances depend on frequency: capacitors block low frequencies, inductors block high frequencies.