Question

In a single-phase AC circuit, the power consumed by load resistance $R_L$ for an excitation $V_S$ is measured by a wattmeter. The same wattmeter is connected in two different topologies, Topology-A and Topology-B, as shown in the figure. Different branch currents and voltage drops are also marked in the figure. Among the following options, the condition that ensures low error in the wattmeter reading for both the topologies is

 

• A. \( V_L \gg V_C \) for Topology-A and \( I_L \gg I_P \) for Topology-B
• B. \( V_L \gg V_C \) for Topology-A and \( I_L \ll I_P \) for Topology-B
• C. \( V_L \ll V_C \) for Topology-A and \( I_L \ll I_P \) for Topology-B
• D. \( V_L \ll V_C \) for Topology-A and \( I_L \gg I_P \) for Topology-B

Hint:

- In Topology-A (pressure coil across the load), the error is mainly due to the power consumed by the current coil. This error is small if the voltage drop across the current coil ($V_C$) is much smaller than the voltage across the load ($V_L$). - In Topology-B (pressure coil across the source), the error is mainly due to the current in the pressure coil ($I_P$) flowing through the current coil. This error is small if the pressure coil current ($I_P$) is much smaller than the load current ($I_L$).

Answer 1

The Correct Option is A

Step 1: Analyze the error in Topology-A.
In Topology-A, the pressure coil is connected across the load, measuring \(V_L\). The current coil carries \(I_L\). The power measured is approximately \(P_W \approx V_L I_L \cos \phi_L + I_L^2 R_C\). The error is due to the power loss in the current coil, \(I_L^2 R_C\). To minimize this error, \(I_L^2 R_C \ll V_L I_L \cos \phi_L\), which implies \(I_L R_C \ll V_L \cos \phi_L\), or \(V_C \ll V_L\) (assuming \(\cos \phi_L\) is not very small).
Step 2: Analyze the error in Topology-B.
In Topology-B, the pressure coil is connected across the source, measuring \(V_S = V_L + V_C\). The current coil carries \(I_L\). The power measured is approximately \(P_W \approx V_S I_L \cos \phi'\). The error arises because the pressure coil measures \(V_S\) instead of \(V_L\). Another source of error is the current through the pressure coil (\(I_P\)) flowing through the current coil. The total current in the current coil is \(I_L + I_P\). To minimize the error due to \(I_P\), we need \(I_P \ll I_L\). The current \(I_P = V_S / Z_P \approx V_S / R_P\). Thus, we need \(V_S / R_P \ll I_L\), or \(I_L \gg I_P\).
Combining the conditions for low error in both topologies, we need \(V_L \gg V_C\) for Topology-A and \(I_L \gg I_P\) for Topology-B.