Question

You are given three circuit elements X, Y and Z. When the element X is connected across an a.c. source of a given voltage, the current and the voltage are in the same phase. When the element Y is connected in series with X across the source, voltage is ahead of the current in phase by π/4 . But the current is ahead of the voltage in phase by π/4when Z is connected in series with X across the source. Identify the circuit elements X, Y and Z. When all the three elements are connected in series across the same source, determine the impedance of the circuit.Draw a plot of the current versus the frequency of applied source and mention the significance of this plot.

Answer 1

When the element X is connected across an a.c. source of a given voltage, the current is in phase with the applied voltage. This implies that the circuit element X is a resistance R.

When the circuit element Y is connected in series with X across an a.c. source, the voltage is found to lead the current by a phase angle π/4. This indicates that the circuit
element Y is an inductance L. The circuit element Z is a capacitance C as it is found that the current leads the voltage by a phase angle π/4.

When the three circuit elements—resistance, inductance and capacitance—are connected in series with the a.c. source, the impedance of the circuit is given by

`Z=sqrt(R^2+(X_L+X_C)^2`

where R is the resistance, XL is the inductive reactance and XC is the capacitive reactance

The variation of current with the changing frequency of applied voltage in a circuit in which X, Y and Z are connected in series is as shown below:

Such an LCR circuit connected in series admits maximum current corresponding to a
given frequency of a.c. This frequency is known as series resonance frequency.
At resonant frequency ω₀, the impedance of the LCR circuit is minimum, and hence, the
current becomes maximum.

For frequencies greater than or less than ω₀, the values of current are less than the maximum current value I₀. As a series resonance circuit admits maximum peak current through it at resonance, it is called an acceptor circuit. Such circuits find application in radios, TV receiver sets etc.