Question

A series combination of circuit elements X and Y is connected across an ac source. It is found that the voltage is ahead of current in phase by `pi/4` radian. When element Y is replaced by element Z, the current leads the voltage by `pi/4` radian.

1. Identify the elements X, Y and Z.
2. What will the phase angle and power factor for the circuit be if X, Y and Z were connected in series across the same ac source? What can you say about the current that flows in the circuit in this case?

Answer 1

(a) 1. Voltage ahead of current by `π/4` radians (X and Y combination):

The phase angle of `π/4` indicates that the impedance has both resistive and inductive components, as the voltage leads the current in an RL (Resistor-Inductor) circuit.

The phase angle θ for an RL circuit is given by:

tan θ = `X_L/R`

X_L​ is the inductive reactance, and R is the resistance.

For θ = `π/4`, we have:

`tan (π/4) = 1`

`(X_L)/R = 1`

X_L = R

∴ The element X is a resistor and element Y is an inductor.

2. Current leads voltage by `π/4` radians (Y replaced by Z):

When element Y is replaced by Z and the current leads the voltage by `π/4` radians, this is characteristic of a Resistor-Capacitor (RC) circuit, where the current leads the voltage.

The phase angle θ for an RC circuit is given by:

`tan ⁡θ = X_C/R`

where X_C​ is the capacitive reactance, and R is the resistance.

For θ = `−π/4`

`tan(−π/4)` = −1

`(X_C)/R = -1`

X_C = R

Element Z is a capacitor.

(b) Now, we will consider the case when X, Y, and Z are connected in series across the same AC source:

Element X = Resistor (R)

Element Y = Inductor (L)

Element Z = Capacitor (C)

Impedance of the Series Combination: The total impedance Z of the series combination of a resistor, inductor, and capacitor is given by:

Z = R + j(X_L − X_C)

Since X_L = R and X_C = R from the previous analysis, we get:

Z = R + j(R − R)

= R + j(0)

= R

Thus, the total impedance is purely resistive.

Phase Angle for a purely resistive impedance, the phase angle between the voltage and current is:

Phase Angle = 0

This means that the current and voltage are in phase.

The power factor is the cosine of the phase angle between the voltage and current:

Power Factor = cos ϕ

= cos⁡(0) 

= 1

Thus, the circuit’s power factor is 1, indicating maximum power transfer.

Current in the Circuit in a purely resistive circuit, the current and voltage are in phase, and the current reaches its maximum value, which is determined by Ohm’s law:

I = `V/R`

Where V is the applied voltage of the AC source, and R is the resistance.