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 $\dfrac{\pi}{4}$ radian. When element $Y$ is replaced by element $Z$, the current leads the voltage by $\dfrac{\pi}{4}$ radian.

(a) Identify the elements $X$, $Y$ and $Z$.

(b) 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?

Hint:

Voltage ahead means a resistor plus an inductor, current ahead means a resistor plus a capacitor. Then X_L = X_C = R.

Answer 1

Step 1: Understanding the Concept:
In a series circuit with a resistor \(R\), the phase angle \(\phi\) between voltage and current follows \(\tan\phi = \dfrac{X_L - X_C}{R}\). Voltage leads current for an inductor, and current leads voltage for a capacitor.

Step 2: Part (a) - First combination X and Y:
The voltage leads the current by \(\dfrac{\pi}{4}\). So the circuit is partly inductive, and the phase angle is less than \(\dfrac{\pi}{2}\). A pure resistor alone gives zero phase angle, and a pure inductor alone gives \(\dfrac{\pi}{2}\). A value of \(\dfrac{\pi}{4}\) needs both a resistor and an inductor.

Step 3: Part (a) - Second combination X and Z:
After replacing Y with Z, the current leads the voltage by \(\dfrac{\pi}{4}\). This needs a capacitor together with a resistor. Since Y gave a lead of voltage and Z gives a lead of current, Y and Z are different kinds of elements.
Both X and Y, and X and Z, give the same angle \(\dfrac{\pi}{4}\), so \(\tan\phi = 1\). This means: \(X_L = R\) and \(X_C = R\).
So: X is a resistor (R), Y is an inductor (L), and Z is a capacitor (C).

Step 4: Part (b) - All three in series:
Now \(X_L = X_C = R\), so \(X_L - X_C = 0\).
\[ \tan\phi = \dfrac{X_L - X_C}{R} = 0 \quad \Rightarrow \quad \phi = 0 \]
\[ \text{Power factor} = \cos\phi = 1 \]

Step 5: Part (b) - Current in the circuit:
The impedance is \(Z = \sqrt{R^2 + (X_L - X_C)^2} = R\), the minimum possible value. This is the condition of series resonance. The current is in phase with the voltage and has its maximum value \(I = \dfrac{V}{R}\).

Final Answer:
X is a resistor, Y is an inductor and Z is a capacitor. In the series combination, \(\phi = 0\), the power factor is 1, and the current is maximum and in phase with the voltage. \[ \boxed{\phi = 0,\quad \cos\phi = 1} \]