Formula Sheet: AC Basics

Comprehensive formula sheet extracted from FAD1022 Tutorial 5 — AC Basics

1. Voltage Divider & Bridge Circuits

1.1 Voltage Divider Rule

$$V_{out} = V_{in} \frac{R_2}{R_1 + R_2}$$

Variable Description SI Unit
$V_{out}$ Output voltage $\text{V}$
$V_{in}$ Input voltage $\text{V}$
$R_1, R_2$ Series resistors $\Omega$

1.2 Wheatstone Bridge (Balanced Condition)

At equilibrium (balanced bridge, galvanometer current $I_G = 0$):

$$\frac{R_1}{R_2} = \frac{R_3}{R_4}$$

Or solving for unknown resistance:

$$R_{unknown} = R_3 \frac{R_2}{R_1}$$

Variable Description SI Unit
$R_{unknown}$ Unknown resistance $\Omega$
$R_1, R_2, R_3$ Known resistances $\Omega$

2. Sinusoidal AC Quantities

2.1 Instantaneous Current

$$I(t) = I_0 \sin(\omega t + \phi_I)$$

2.2 Instantaneous Voltage

$$V(t) = V_0 \sin(\omega t + \phi_V)$$

2.3 General Form

$$x(t) = X_0 \sin(\omega t + \phi)$$

Variable Description SI Unit
$I(t), V(t)$ Instantaneous current / voltage $\text{A}$ / $\text{V}$
$I_0, V_0$ Peak / maximum values $\text{A}$ / $\text{V}$
$\omega$ Angular frequency $\text{rad s}^{-1}$
$f$ Frequency $\text{Hz}$
$T$ Period $\text{s}$
$\phi$ Phase constant / initial phase rad
$t$ Time $\text{s}$

3. Frequency Relationships

3.1 Angular Frequency

$$\omega = 2\pi f$$

3.2 Period

$$T = \frac{1}{f} = \frac{2\pi}{\omega}$$

3.3 Frequency from Period

$$f = \frac{1}{T}$$

Variable Description SI Unit
$\omega$ Angular frequency $\text{rad s}^{-1}$
$f$ Frequency $\text{Hz}$ (Hertz)
$T$ Period $\text{s}$

4. RMS (Root Mean Square) Values

4.1 RMS Current

$$I_{rms} = \frac{I_0}{\sqrt{2}} \approx 0.707 , I_0$$

4.2 RMS Voltage

$$V_{rms} = \frac{V_0}{\sqrt{2}} \approx 0.707 , V_0$$

4.3 Peak from RMS

$$I_0 = \sqrt{2} , I_{rms} \approx 1.414 , I_{rms}$$

$$V_0 = \sqrt{2} , V_{rms} \approx 1.414 , V_{rms}$$

4.4 General Definition (for any periodic function)

$$X_{rms} = \sqrt{\frac{1}{T} \int_0^T x^2(t) , dt}$$

For sinusoidal signals, this reduces to $X_{rms} = X_0/\sqrt{2}$.

Variable Description SI Unit
$I_{rms}$ RMS current $\text{A}$
$V_{rms}$ RMS voltage $\text{V}$
$I_0$ Peak current $\text{A}$
$V_0$ Peak voltage $\text{V}$

5. Ohm's Law for AC Circuits (Resistive)

5.1 Peak Values

$$V_0 = I_0 R$$

5.2 RMS Values

$$V_{rms} = I_{rms} R$$

5.3 Instantaneous

$$V(t) = I(t) R$$

Variable Description SI Unit
$R$ Resistance $\Omega$

6. Phase Relationships

6.1 Phase Difference

$$\Delta\phi = \phi_V - \phi_I$$

6.2 In-Phase (Purely Resistive Circuit)

$$\Delta\phi = 0 \quad \Rightarrow \quad V \text{ and } I \text{ are in phase}$$

6.3 Voltage Leads Current

$$\Delta\phi > 0 \quad \Rightarrow \quad V \text{ leads } I \text{ by } \Delta\phi$$

6.4 Current Leads Voltage

$$\Delta\phi < 0 \quad \Rightarrow \quad I \text{ leads } V \text{ by } |\Delta\phi|$$

Variable Description SI Unit
$\Delta\phi$ Phase difference rad or degrees
$\phi_V$ Voltage phase angle rad
$\phi_I$ Current phase angle rad

7. Power in AC Circuits (Resistive)

7.1 Instantaneous Power

$$P(t) = I(t) V(t) = I_0 V_0 \sin^2(\omega t)$$

7.2 Average Power

$$P_{avg} = I_{rms} V_{rms} = I_{rms}^2 R = \frac{V_{rms}^2}{R}$$

7.3 Alternative Form Using Peak Values

$$P_{avg} = \frac{1}{2} I_0 V_0 = \frac{I_0^2 R}{2} = \frac{V_0^2}{2R}$$

Variable Description SI Unit
$P(t)$ Instantaneous power $\text{W}$
$P_{avg}$ Average / real power $\text{W}$

8. Summary of Key Relationships

Concept Formula
Voltage divider $V_{out} = V_{in} \dfrac{R_2}{R_1 + R_2}$
Wheatstone bridge (balanced) $\dfrac{R_1}{R_2} = \dfrac{R_3}{R_4}$
Instantaneous current $I(t) = I_0 \sin(\omega t + \phi)$
Instantaneous voltage $V(t) = V_0 \sin(\omega t + \phi)$
Angular frequency $\omega = 2\pi f = \dfrac{2\pi}{T}$
RMS current $I_{rms} = \dfrac{I_0}{\sqrt{2}}$
RMS voltage $V_{rms} = \dfrac{V_0}{\sqrt{2}}$
Ohm's Law (AC, resistor) $V_{rms} = I_{rms} R$
Average power $P_{avg} = I_{rms} V_{rms} = I_{rms}^2 R$
Phase difference $\Delta\phi = \phi_V - \phi_I$

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