• A.C circuit with capacitive load.

• Purely Resistive load: Current is in phase with voltage

• Inductive load (L): Current lags behind voltage

• Capacitive load (C) : Current leads the voltage

• E - voltage

• L - inductor

• I - current

• c - capacitive

• The Ratio of Voltage Maximum to Current Maximum.

• Resistive circuit $I_M = \frac {V_m}{R}$

• Inductive load : Inductive Resistance

• $\Chi_L = \omega L$

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

• Capacitive Load : $\Chi_C = \frac {1}{\omega C}$

• $I_m = \frac{V_m}{\Chi_C} = V_m . \omega C$

• $V(t) = V_m \sin \omega t$

• $I(t) = I_m \sin (\omega t + \phi)$

• $\phi$ - phase by which current leads voltage

• Resistive : $\phi$ = 0

• Inductive : $\phi = \frac {-\pi}{2}$

• $V(t) =V_ m e^{i \omega t} =V_m[\cos \omega t + i \sin \omega t]$

• I(t) =$\operatorname{I_m} e^{i(\omega t+\Phi)}$

• Complex Impedance: $Z=\frac{V(t)}{I(t)}=\frac{V_m}{I_m} \cdot e^{-\iota \varphi}$

• For resistive circuit: $\varphi=0, \quad Z=R$

• For inductive circuit: $\varphi=-\pi / 2$, $\quad Z=\omega L e^{i \pi / 2}=i \omega L$

• For a capacitive: $Z=\frac{1}{\omega c} e^{-i \pi / 2 } = (-i) \frac{1}{\omega c}=\frac{1}{i \omega c}$.

• $Z = R+i\left[\omega_L-\frac{1}{\omega C}\right]$

• $\quad =R+i\left[\Chi _ L - \Chi_ C\right]$

• |Z| =$\sqrt{R^2+\left(\Chi_ L - \Chi_C\right)^2}$

• $\tan \varphi=\frac{\Chi_ C - \Chi_L}{R}$

• Impedance triangle: $|\Chi_ C - \Chi_L|$

• Average for a resistive circuit

• $\frac {I_m R}{2}$ : rms current

• $\frac {I_ m}{\sqrt 2} = I_{rms}$

• $P \equiv I_{rms}^2 R$

• Inductors and Capacitors

• $V(t)-V_R - V_L - V_c = 0$

• $V(t) = I R + L \frac{d I}{d t}+\frac{Q}{C}$

• Current must be the same through the three elements.

• $I = I_m \sin(\omega t + \varphi)$

• Voltage is in series

• $V = V_m \sin{\omega t}$

• $I= I_m \sin (\omega t+\phi)$.

• $|\vec{OA}|= (\Chi_C-\Chi_L) I_{m }$

• $V_m^2 = (V_m^m)^2+(V_C^m-V_L^m)^2$

• $=(I_m R)^2+I_m^2(\Chi_C-\Chi_L)^2$

• $V_m=I_m \sqrt{R^2+(\Chi_C-\Chi_L)^2}=I_m Z$

• $Z = \sqrt{R^2+ (\Chi_C-\Chi_L)^2}$

• $Z \longrightarrow R$

• $Z \longrightarrow \Chi_C , \Chi_L$

• $Z =\sqrt{R^2+\left(\Chi_C-\Chi_L\right)^2}$

• Example: r.m.s current of 2A is passing through the circuit.

• Z =$\sqrt{R^2+\left(\Chi_C-\Chi_L\right)^2}$

• =$\sqrt{80^2+60^2} = 100\Omega$

• $V^{r m s} =I^{r m s} \times Z =200 V-r m s$

• $V_R = I R = 2 \times 80 = 160 V(r m s)$

• $V_C =2 \Chi_C=200 V(r m s)$

• $V_L =2 \Chi_L=80 V(r m s)$

• $V_R$ = 160 V(rms)

• $V_L$ = 80 V

• $V_C$ = 200 V

• Resulting voltage lags by $\varphi$

• $tan \varphi = \frac {120}{160} = \frac{3}{4}$

• $\varphi = 37^0$ (0.64 rods)

• Time Lag Between Current Max and Voltage Max

• $\frac {\varphi}{\omega} = \frac{0.64}{400}$ = 1.6 ms.

• what happens when $\omega$ increases.

• $\varphi \rightarrow 0$ for capacitive circuit.

• $tan \varphi = \frac {\Chi_C - \Chi_L}{R}$ capacitive behaves lika a conductor.

• For inductive circuit

• $\langle P\rangle =I_{r m s}^2 \cdot R$

• $= 4 \times 80$

• $=320$ watts

• Example:- (RC Circuit)

• $R=3 \Omega, C=2.5 \times 10^{-4} \mathrm{F}$

• $\omega=1000 \mathrm{rad} / \mathrm{s}$

• $v_{max }= 5 \mathrm{~V}$

-

• $\Chi_C = \frac{1}{\omega C} = \frac{1}{1000 \times 2.5 \times 10^{-4}}$

• $V=5 \sin \omega t = 4\Omega$

• $I=I_m \sin (\omega t+\varphi)$

• $Z=\sqrt{R^2 + \Chi_C^2} = 5 \Omega, \quad I_m = 1 A$

• $v_n^{\max }=3 \mathrm{V}, \mathrm{v}_C^{\max }=\frac{I}{\omega C}=4 \mathrm{V}$

• $V_c = I X_c$

• $tan \varphi = \frac {4}{3}$

• $\Chi_L = \omega L = 2000\Omega$

• $\Chi_C = \frac {1}{\omega C} = 5000\Omega$

• $Z = \sqrt {R^2 + (\Chi_C -\Chi_L)^2} = 1800\Omega$

• $I_m = \frac {140}{1800} = 0.078 A , I_{rms} = 55 MA$

• $V_{R}^{max} = 78 V$

• $V_{C}^{max} = I^{max} . \Chi_C = 39 V$

• $V_{V}^{max} = 156 V$

• $tan \varphi = \frac {\Chi_C -\Chi_L}{R} = \varphi = - 56^0$