Alternating Current Class 12 Formulas

• AC Voltage:

$v(t) = V_{\text{max}} \sin(\omega t)$

• Peak Value of AC: 

$i(t) = \frac{v(t)}{R} = \frac{V_{\text{max}}}{R} \sin(\omega t)$

• Average Value of AC: 

$V_{\text{avg}} = \frac{2 V_{\text{max}}}{\pi}, \quad I_{\text{avg}} = \frac{2 I_{\text{max}}}{\pi}$

• RMS Value of AC:

$V_{\text{rms}} = \frac{V_{\text{max}}}{\sqrt{2}}, \quad I_{\text{rms}} = \frac{I_{\text{max}}}{\sqrt{2}}$​​

• Kirchhoff’s Loop Rule for an AC circuit with a Resistor

$\mathrm{<br>}{V}_{\text{max}}\sin (\omega t)\; -i(t)R\; =\; 0$

• Current in the AC Circuit with a Resistor

$i(t) = \frac{V_{\text{max}}}{R} \sin(\omega t)$

• Amplitude (Peak Value) of Current

$I_{\text{max}} = \frac{V_{\text{max}}}{R}$

• Instantaneous Power in a Resistive Circuit

$p(t) = v(t) \cdot i(t) = V_{\text{max}} I_{\text{max}} \sin^2(\omega t)$

• Phase Difference Between V and I_r

It means both will reach the respective peak values at the same time.

$\phi = 0^\circ$. 

• Kirchhoff’s Loop Rule for an AC circuit with a Capacitor

$$v_m\sin (\omega t)=\frac{q}{C}$$

• Current in the AC Circuit with a Capacitor

$i(t) = C \cdot V_{\text{max}} \omega \cos(\omega t)$

• Amplitude (Peak Value) of Current

$I_{\text{max}} = V_{\text{max}} \cdot \omega C$

• Capacitive Reactance

$X_C = \frac{1}{\omega C}$

• Instantaneous Power in a Capacitive Circuit

$$p(t)=V_{\text{max}}{I}_{\text{max}}{\sin }^2(\omega t)$$

• Average Power in a Capacitive Circuit

In a full cycle, energy is stored in the first cycle and the same energy is returned in the next half cycle. So

$$P_{\text{avg}}=0$$

• Phase Difference Between V and I_c

Current lags behind the voltage by 90° in a capacitive circuit:

$\varphi =-{90}^{\circ }$

• Kirchhoff’s Loop Rule for an AC circuit with an Inductor

$$v_L=L\frac{di}{dt}$$

• Current in the AC Circuit with An Inductor

$$i(t)=-\frac{V_m}{\omega L}\cos (\omega t)$$

• Amplitude (Peak Value) of Current

$$I_m=\frac{V_m}{\omega L}$$

• $$\text{Inductive Reactance}$$

$$X_L=\omega L\text{<br>}$$

• Instantaneous Power in Circuit

$$p(t)=-\frac{V_m{I}_m}{2}\cos (2\omega t-{90}^{\circ })$$

• Average Power in Circuit

For a full cycle, the average power is zero.

$$P_{\text{avg}}=0$$

• Phase Difference Between V and I_L

The current phase is ahead of the voltage phase.

$$\varphi =+{90}^{\circ }$$

• Current in LCR Circuit

$$I_m=\frac{V_m}{Z}<br>$$

• Impedance

$$Z = \sqrt{R^2 + (X_L - X_C)^2}$$

• If there is no resistance

$$Z=\sqrt{(X_L-X_C{)}^2}$$

• If there is no Inductive Reactance

$$Z=\sqrt{R^2+<br>{\mathrm{(\; X}}_C{)}^2}$$

• If there is no capacitive reactance

$$Z=\sqrt{R^2+(X_L{\mathrm{<br>}}_{}{)}^2}$$

• If inductive and capacitive reactance are equal, It behaves like a circuit with a resistor only. It is called resonance.

$$(X_L - X_C) = 0$$

• Resonant Frequency

$$X_L = X_C \quad \Rightarrow \quad \omega_0 = \frac{1}{\sqrt{LC}}$$

• Phase Angle

$$\tan \varphi =\frac{X_L-X_C}{R}$$

• Quality Factor 

$$Q=\frac{{\omega }_{0}L}{R}=\frac{1}{{\omega }_{0}CR}$$