Capacitive Circuitsalternating Currents Topic
JEE Advanced
- Capacitor
- Capacitive reactance: $$X_C=\frac{1}{2\pi fC}$$
- Current flowing: $$I=\frac{V}{X_C}$$
$$X_C= \frac{1}{2\pi \times 50 \text{ Hz} \times 100\times 10^{-6} \text{ F}}=318.31\Omega$$
$$I=\frac{200\text{ V}}{318.31\Omega}=\boxed{0.628 \text{ A}}$$
- AC Circuit with Inductor and Capacitor:
- Impedance: $$Z=\sqrt{R^2+(X_L-X_C)^2}$$
- Current flowing: $$I=\frac{V}{Z}$$
$$X_L=2\pi f L=2\pi \times 50\text{ Hz}\times 0.1\text{ H}=31.42 \Omega$$
$$Z=\sqrt{10^2+(31.42-318.31)^2}=\boxed{300.85 \Omega}$$
$$I=\frac{200\text{ V}}{300.85\Omega}=\boxed{0.665 \text{ A}}$$
- Step-up Transformer:
- Voltage across secondary: $$V_s=\frac{N_s}{N_p}V_p$$
$$V_s=\frac{200\ \text{ turns}}{100\text{ turns}}\times100\text{ V}=\boxed{200\text{ V}}$$
CBSE Board Exams
- Capacitor
-
Capacitive reactance: $$X_C=\frac{1}{2\pi fC}$$
-
Current flowing: $$I=\frac{V}{X_C}$$ $$X_C=\frac{1}{2\pi\times 50 \text{ Hz}\times 100\times10^{-6}\text{ F}}=\boxed{318.31 \Omega}$$
$$I=\frac{200\text{ V}}{318.31\Omega}=\boxed{0.628\text{ A}}$$
- AC Circuit with Inductor and Capacitor
-
Impedance: $$Z=\sqrt{R^2+(X_L-X_C)^2}$$
-
Current flowing: $$I=\frac{V}{Z}$$
$$X_L=2\pi fL=2\pi\times50\text{ Hz}\times0.1\text{ H}=31.42\Omega$$
$$Z=\sqrt{10^2+(31.42-318.31)^2}=\boxed{300.85\Omega}$$
$$I=\frac{200\text{ V}}{300.85\Omega}=\boxed{0.665\text{ A}}$$
- Step-up Transformer:
- Voltage across secondary: $$V_s=\frac{N_s}{N_p}V_p$$
$$V_s=\frac{200\text{ turns}}{100\text{ turns}}\times100\text{ V}=\boxed{200\text{ V}}$$
- AC Generator:
-
Peak voltage: $$V_{peak}=V_{rms}\sqrt{2}$$ $$V_{peak}=220 \text{ V}\times \sqrt{2}=\boxed{311\text{ V}}$$
-
Peak current: $$I_{peak}=\frac{V_{peak}}{R}$$ $$I_{peak}=\frac{311 \text{ V}}{100 \Omega}=\boxed{3.11\text{ A}}$$
- Inductor with AC Source:
- $$I_{rms}=\frac{V_{rms}}{Z}, where Z=\sqrt{R^2+X_L^2}$$
$$X_L=2\pi fL=2\pi (60\text{ Hz})(0.1\text{ H})=37.7\Omega$$
$$Z=\sqrt{10^2+37.7^2}=38.6 \Omega$$
- Current amplitude: $$I_{rms}=\frac{120\text{ V}}{38.6 \Omega}=\boxed{3.11 \text{ A}}$$
- Capacitor with AC Source:
With $$X_C=\frac{1}{2\pi fC}=26.52 \Omega$$
- Current amplitude: $$I=V_{rms} \frac{1}{X_C}=\frac{120\text{ V}}{26.52\Omega}=\boxed{4.52 \text{ A}}$$
