Question

The impedance of an AC circuit is $Z = 100\angle 30^\circ$, the resistance of the circuit is?

Answer 1

Recall the expression for impedance of series LCR circuit in terms of phase angle. Equate the given impedance of the circuit with the original expression for impedance. After equating the equation, find the value of the term ${X_L} - {X_C}$ and substitute it into the equated quantity.

Formula used:
$Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \angle {\tan ^{ - 1}}\left( {\dfrac{{{X_L} - {X_C}}}{R}} \right)$
Here, R is the resistance, ${X_L}$ is the inductive reactance and ${X_C}$is the capacitive reactance.

Complete step by step answer:
We have the expression for impedance of AC circuit is,
$Z = \sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} \angle {\tan ^{ - 1}}\left( {\dfrac{{{X_L} - {X_C}}}{R}} \right)$ …… (1)
Here, R is the resistance, ${X_L}$ is the inductive reactance and ${X_C}$is the capacitive reactance.

We have given the impedance of AC circuit is,
$Z = 100\angle 30^\circ$ …… (2)
Therefore, comparing equation (1) and (2), we get,
$\sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} = 100$ …… (3)
And,
${\tan ^{ - 1}}\left( {\dfrac{{{X_L} - {X_C}}}{R}} \right) = 30^\circ$
$\Rightarrow \dfrac{{{X_L} - {X_C}}}{R} = \tan 30^\circ$
$\Rightarrow {X_L} - {X_C} = \dfrac{1}{{\sqrt 3 }}R$ …… (4)
Substituting equation (2) in equation (1), we get,
$\sqrt {{R^2} + {{\left( {\dfrac{R}{{\sqrt 3 }}} \right)}^2}} = 100$
$\Rightarrow \sqrt {\dfrac{{3{R^2} + {R^2}}}{3}} = 100$

Taking the square of the above equation, we get,
$\dfrac{{3{R^2} + {R^2}}}{3} = 10000$
$\Rightarrow 4{R^2} = 30000$
$\Rightarrow {R^2} = \dfrac{{30000}}{4}$
$\Rightarrow {R^2} = 7500$
$\Rightarrow R = \sqrt {7500}$
$\therefore R = 86.6\,\Omega$

Therefore, the resistance of the circuit is $86.6\,\Omega$.

Additional information:
When the resistor, capacitor and inductor are connected in series with each other, the circuit is known as series LCR circuit. All the three components provide resistance in the circuit. The resistance offered by the capacitor is known as capacitive reactance and it is denoted as ${X_C}$ and the resistance offered by the inductor is known as inductive reactance.

Note: The factor $\angle {\tan ^{ - 1}}\left( {\dfrac{{{X_L} - {X_C}}}{R}} \right)$ is known as phase angle. The phase angle may be positive or negative depending on whether the voltage provided by the source leads or lags the current in the circuit. When the inductive reactance equals capacitive reactance, the phase angle becomes $45^\circ$. This condition is known as resonance.