Name: Power in AC Circuit
Uploaded: 2019-07-19
Duration: 4 min 36 s
Description: The video defines the expression for the instantaneous power in an ac circuit. It also derives the expression for the average power consumed by an AC circuit.

Question

An alternating voltage of angular frequency

ω

, is applied across a series combination of a resistance

R

, an inductance

L

and a capacitance

C

. Write down the formula for the impedance of the circuit.

Accepted Answer

Resonant circuit: In L-C-R circuit, when the value of impedance is least, then the current becomes maximum this phenomenon is called resonance. In this state the frequency of the applied E.M.F. is equal to the natural frequency of the L-C circuit called resonant frequency and this circuit is called resonant electric circuit.

In the figure an inductance $C$ , a capacitance $C$ and a resistance $R$ are connected in series with an alternating voltage. At any instant alternating voltage is given as

${V=V}_{\circ} sinωt . . . . . . . . . (1)$

Where $V_{\circ}$ is the peak value of alternating voltage at any instant, if the current in the circuit be $I,$ then. The potential difference across the inductance $L$ is. $V_{L} = X_{L} I . . . . . . . . (2)$

Similarly, potential difference across the capacitance $C$ is $V_{C} = X_{C} I . . . . . . . . . . . (3)$

And the potential difference across the resistance $L$ is $C$

Now $V_{R}$ and $I$ are in the same phase while $V_{L}$ Leading ahead $I$ by $90 °$ and $V_{C}$ Logging behind $I$ by $90 °$ Hence, the angle between $V_{L}$ and $V_{C}$ is $180 °$ i.e., $V_{L}$ and $V_{C}$ are in opposite phase. Now the resultant of $V_{L}$ and $V_{C}$ is $V_{L} - V_{C}$ Then the angle between $V_{L} - V_{C}$ and $V_{R}$ is $90 °$

If the resultant of $(V_{L} - V_{C})$ and $V_{R}$ is $V .$

Then according to the figure $(2)$ by Pythagoras theorem. $V^{2} = {(V_{L} - V_{C})}^{2} + V_{R}^{2} . . . . . . . (5)$

from equation $(2), (3), (4)$ and $(5)$

$V^{2} = {(X_{L} I - X_{C} I)}^{2} + (RI)^{2}$

$V^{2} = I^{2} {(X_{L} - X_{C})}^{2} + I^{2} R$

$\frac{V^{2}}{I^{2}} = {(X_{L} - X_{C})}^{2} + R^{2}$

$\frac{V}{I} = {(X_{L} - X_{C})}^{2} + R^{2} . . . . . . . . . . (6)$

On comparing equation $(6)$ with ohms law, the right-hand side of equation $(6)$ shows that the effective resistance of this circuit which is called impedance of this circuit and denoted by $Z_{L - C - R }$ then, $Z_{L - C - R} = \sqrt{{(X_{L} - X_{C})}^{2} + R^{2}} . . . . . . . . (7)$

but, $X_{L} = ωL$ and $X_{C} = \frac{1}{ωC}$

$Z_{L - C - R} = \sqrt{{(ωL - \frac{1}{ωc})}^{2} + R^{2}} . . . . . . . . . . . (8)$

Equation $(7)$ and $(8)$ are the required expression for the impedance of the circuit.

An alternating voltage of angular frequency $ω$ , is applied across a series combination of a resistance $R$ , an inductance $L$ and a capacitance $C$ . Write down the formula for the impedance of the circuit.

Important Questions on Alternating Current

Learn and Prepare for any exam you want !

An alternating voltage of angular frequency ω, is applied across a series combination of a resistance R, an inductance L and a capacitance C. Write down the formula for the impedance of the circuit.

Important Questions on Alternating Current

Learn and Prepare for any exam you want !

An alternating voltage of angular frequency $ω$ , is applied across a series combination of a resistance $R$ , an inductance $L$ and a capacitance $C$ . Write down the formula for the impedance of the circuit.