AC Voltage applied to a Series LCR Circuit: Definition, Derivation, Application & Class 12 Notes

When alternating current is applied to the series LCR circuit, the behavior of the circuit will depend on the frequency of the applied voltage. To make a series LCR circuit, we need to connect inductor, capacitor and resistor in series. In series LCR circuit, resistor allow the flow of current with no phase shift, inductor causes the current to lag, and capacitor allows the current to lead the voltage. The total impedance(Z) of the LCR circuit is expressed as: $Z=\sqrt{R^2+{(X_L-X_C)}^2}$

Where, 

• Inductive reactance: $X_L=\omega L$
• Capacitive reactance: $X_C=\frac{1}{\omega C}$
• Angular frequency: $\omega =2\pi f$

The circuit may behave like a resistive, inductive, or capacitive circuit based on the frequency. At a resonant frequency (f₀), the inductive and capacitive reactances cancel out, and the circuit behaves like a pure resistor:

$f_0=\frac{1}{2\pi \sqrt{LC}}$

At resonance: 

• Current is maximum

• Impedance is minimum

• Power transfer is most efficient

• Current and voltage are in phase

 LCR circuits are essential in resonant power systems, tuned circuits, and radio receivers.

Also Check:

NCERT Class 11 Notes	CBSE NCERT Class 11 Physics Notes
Class 11 Chemistry NCERT notes	Class 11 Maths NCERT notes

1. Circuit Description
A LCR series circuit consists of:

• Inductor (L)
• Capacitor (C)
• Resistor (R)

LCR connected in series across the AC voltage source:

$v\left(t\right)=V_m\sin \left(\omega t\right)$

Let, 

• Current through the circuit $I(t)$
• Inductive reactance $X_L=\omega L$
• Capacitive reactance: $X_C=\frac{1}{\omega C}$

Read More: NCERT Solutions

2. Impedance of the Circuit
The total opposition to current (impedance) is expressed as: $Z=\sqrt{R^2+{(X_L-X_C)}^2}$

3. Current in the Circuit
Ohm's Law in AC circuits: $I\left(t\right)=\frac{V_m}{Z}\sin (\omega t-\phi )$

Where,

• $\phi$ is the phase angle between current and voltage, and is expressed as 

$\phi ={\tan }^{-1}\left(\frac{X_L-X_C}{R}\right)$

• The circuit is inductive if $X_L>X_C$
• The circuit is capacitive if $X_C>X_L$
• When $X_L=X_C$ the circuit is resonant, and current is in phase with the voltage.

4. Resonant Frequency

At resonance: $X_L=X_C\Rightarrow {\omega }_{0}L=\frac{1}{{\omega }_{0}C}$

${\omega }_0=\frac{1}{\sqrt{LC}}\Rightarrow f_0=\frac{1}{2\pi \sqrt{LC}}$

At this frequency:

• Z=R, Impedance is minimum
• Current is maximum
• Power factor = 1

 

A series LCR circuit is used in electrical and electronic systems. Here are the major applications of the LCR series circuit. 

1. Tuning Circuits in Radio and TVs

• A series LCR circuit is used to select specific frequencies.

• They are critical in tuned receivers, which help the devices to isolate a desired station frequency using resonance. 

2. Band-Pass Filters

• A series LCR circuit passes only a narrow band of frequencies at resonance, acting as a band-pass filter.  

• It is used in audio processing, instrumentation, and communication systems. 

3. Power Supply and AC Signal Control

• It is used to suppress noise, stabilize AC voltage or allow specific frequency signals to pass through in a power circuit. 

4. Resonant Wireless Power Transfer

• A series LCR circuit is used in advanced technologies to tune wireless power systems for maximum efficiency by matching resonant frequencies. 

5. Oscillator Circuits

• These circuits are parts of LC oscillators that generate high-frequency signals for wireless transmitters, radios and function generators.

 

Students can find the link for Class 12 Physics NCERT notes below.

Sl. No	Name of Chapter
1	Chapter 1: Electric Charges and Fields
2	Chapter 2: Electrostatic Potential and Capacitance
3	Chapter 3: Current Electricity
4	Chapter 4: Moving Charges and Magnetism
5	Chapter 5: Magnetism and Matter
6	Chapter 6: Electromagnetic Induction
7	Chapter 7: Alternating Current
8	Chapter 8: Electromagnetic Waves
9	Chapter 9: Ray Optics and Optical Instruments
10	Chapter 10: Wave Optics
11	Chapter 11: Dual Nature of Radiation and Matter
12	Chapter 12: Atoms
13	Chapter 13: Nuclei
14	Chapter 14: Semiconductor Electronics: Materials, Devices and Simple Circuits

To practice for the Class 12 board exam can find the NCERT solutions for the Physics paper.

Sl. No	Name of Chapter
1	Chapter 1 Electric Charges and Fields
2	Chapter 2 Electrostatic Potential and Capacitance
3	Chapter 3 Current Electricity
4	Chapter 4 Moving Charges and Magnetism
5	Chapter 5 Magnetism and Matter
6	Chapter 6 Electromagnetic Induction
7	Chapter 7 Alternating Current
8	Chapter 8 Electromagnetic Waves
9	Chapter 9 Ray Optics and Optical Instruments
10	Chapter 10 Wave Optics
11	Chapter 11 Dual Nature of Radiation and Matter
12	Chapter 12 Atoms
13	Chapter 13 Nuclei
14	Chapter 14 Semiconductor Electronics: Materials, Devices and Simple Circuits