Question

In figure below if $Z_L = Z_c$ and reading of ammeter is 1 A. Find value of source voltage $V$.

Answer 1

100 V

Answer 2

The circuit diagram shows a series AC circuit consisting of a resistor ($R$), an inductor ($L$), and a combination of another inductor ($Z_L$) and a capacitor ($Z_C$) connected in series. The source voltage is $V$ at a frequency of $30$ Hz, and the ammeter reading indicates the total current $I = 1$ A.

Given values: Resistance $R = 80 \Omega$ Inductance $L = \frac{1}{\pi}$ H Frequency $f = 30$ Hz Current $I = 1$ A

The problem states "if $Z_L = Z_C$". In the context of the diagram, $Z_L$ refers to the inductive reactance $X_L$ of the inductor inside the dashed box, and $Z_C$ refers to the capacitive reactance $X_C$ of the capacitor inside the dashed box. The condition $Z_L = Z_C$ implies that the magnitudes of their reactances are equal, i.e., $X_L = X_C$.

These two components ($Z_L$ and $Z_C$) are connected in series within the dashed box. The impedance of this series combination is given by: $Z_{box} = jX_L - jX_C$ Since $X_L = X_C$, we have: $Z_{box} = jX_L - jX_L = 0$ This means the portion of the circuit inside the dashed box acts as a short circuit at this frequency.

Therefore, the effective circuit consists only of the resistor $R = 80 \Omega$ and the inductor $L = \frac{1}{\pi}$ H connected in series with the voltage source and ammeter.

First, calculate the angular frequency $\omega$: $\omega = 2 \pi f = 2 \pi \times 30 = 60 \pi \text{ rad/s}$

Next, calculate the inductive reactance of the inductor $L = \frac{1}{\pi}$ H: $X_L = \omega L = (60 \pi) \times \left(\frac{1}{\pi}\right) = 60 \Omega$

Now, calculate the total impedance ($Z_{total}$) of the series R-L circuit: $Z_{total} = \sqrt{R^2 + X_L^2}$ $Z_{total} = \sqrt{(80 \Omega)^2 + (60 \Omega)^2}$ $Z_{total} = \sqrt{6400 + 3600}$ $Z_{total} = \sqrt{10000}$ $Z_{total} = 100 \Omega$

Finally, use Ohm's law for AC circuits to find the source voltage $V$: $V = I \times Z_{total}$ $V = 1 \text{ A} \times 100 \Omega$ $V = 100 \text{ V}$

The value of the source voltage $V$ is 100 V.