Question

You have a resistor of resistance 230 Ω , an inductor of inductance 0.410 H , a capacitor of capacitance 6.50 μF and a voltage source that has a voltage amplitude of 35.0 V and an angular frequency of 220 rad/s . The resistor, inductor, capacitor, and voltage source are connected to form an L-R-C series circuit.

1.What is the impedance of the circuit?

2.What is the current amplitude?

3.What is the phase angle of the source voltage with respect to the current?

4.What is the voltage amplitude across the resistor?

5.What is the voltage amplitude across the inductor?

6.What is the voltage amplitudes across the capacitor?

7.Explain how it is possible for the voltage amplitude across the capacitor to be greater than the voltage amplitude across the source.

Answer 1

Sol:

Given

Resistance = 230 Ω  

Capacitor = 6.50 μF ==> 6.50 *10^-6 F

Inductance = 0.410 H

Voltage = 35.0 V

Angular frequency = 220 rad/s

Circuit - (1) :

1.) Impedance of the circuit :

Impedance of the circuit is

• 

• 

now we know X_L and X_C values so substitute in equation-(1)

2.)  Current amplitude:

Current amplitude is

3. ) Phase angle of the source voltage with respect to the current:

The source voltage of the phase with respect to current (I) leads voltage (V) , then the phase angle is

4.)  Voltage amplitude across the resistor:

Voltage amplitude across the resistor is

5.) Voltage amplitude across the Inductor:

Voltage amplitude across the inductor is

6.) Voltage amplitudes across the capacitor:

Voltage amplitudes across the capacitor is

7.Explain how it is possible for the voltage amplitude across the capacitor to be greater than the voltage amplitude across the source:

Given voltage source is 35.0 V and voltage amplitudes across the capacitor is 37.06 V  

It is a phenomenon referred to as “series resonance”, where the inductive and capacitive properties of the circuit cancels each other out, and basically only the resistive part remains (in theory).

A similar phenomenon is occurs if you have an inductor(L) and capacitor(C) in parallel. At a certain frequency This is nothing but it is “parallel resonance”.
This can create what is called a “tank circuit”, where the impedance seen from the outside is large, but the current inside the circuit is large (due to the capacitor and inductor alternately supplying each other.When the circuit is at or near resonance, the voltages across the capacitor and inductor can be larger than the input voltage. The inductor voltage leads the resistor's voltage by 90 while the capacitor's voltage lags the resistor's voltage by 90 degrees, The inductor voltage is 180 degrees out of phase with the capacitor's voltage. At resonance they cancel each other out.