Name: AC Voltage Applied to a Resistor
Uploaded: 2019-07-19
Duration: 7 min 45 s
Description: A purely resistive circuit has very little inductance at its typical frequency. This video explains various aspects of it and also shows the plotting of curr...

Question

Derive an expression for the impedance of a series LCR circuit, when an AC voltage is applied to it.

Accepted Answer

Reactance: The effective opposition offered by the component of the A.C. circuit to the flow of current in the circuit is referred to be Reactance.

Impedance: The effective opposition offered by combination of resistors, inductors and capacitors to the flow of current is referred to be Impedance.

Instantaneous current in $L - C - R$ circuit:

Suppose, $I$ be the root-mean-square value of current flowing through all the circuit elements.

Potential difference across Inductor $(L)$ : $V_{L} = I X_{L}$ ............................ $(1)$

Potential difference across Capacitor $(C)$ : $V_{C} = I X_{C}$ ............................ $(2)$

Potential difference across Resistor $(R)$ : $V_{R} = I R$ ............................ $(3)$

Since, $V_{R}$ and $I$ are in phase. So, $V_{R}$ is represented by $O A$ in the direction of $I$ .

The current lags behind the $V_{L}$ , by angles of $\frac{π}{2}$ . So, $V_{L}$ is represented by $O B$ perpendicular to the direction of $I$ .

The current leads $V_{C}$ by angle of $\frac{π}{2}$ . So, $V_{C}$ is represented by $O F$ perpendicular to the direction of $I$ .

Suppose, $V_{L} > V_{C}$ . Then, resultant of $V_{R}$ and $(V_{L} - V_{C})$ is given by $O H$

$O H = \sqrt{O A^{2} + O D^{2}} \Rightarrow V = \sqrt{V_{R}^{2} + {(V_{L} - V_{C})}^{2}}$ ............................. $(4)$

Substituting equation $(1)$ , $(2)$ and $(3)$ in equation $(4)$ we get,

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

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

Here, $Z = \sqrt{R^{2} + {(X_{L} - X_{C})}^{2}}$ is the impedance.

Now, $\tan ϕ = \frac{V_{L} - V_{C}}{V_{R}} = \frac{I (X_{L} - X_{C})}{I R} = \frac{X_{L} - X_{C}}{R} \Rightarrow ϕ = \tan^{- 1} \frac{X_{L} - X_{C}}{R}$

Hence the expression for impedance in series $L - C - R$ has been established.

Derive an expression for the impedance of a series LCR circuit, when an AC voltage is applied to it.

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