Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE) PDF Download

Introduction

Resonance occurs in electric circuits due to the presence of energy storing elements like inductor and capacitor. It is the fundamental concept based on which, the radio and TV receivers are designed in such a way that they should be able to select only the desired station frequency.
There are two types of resonances, namely series resonance and parallel resonance. These are classified based on the network elements that are connected in series or parallel. 

In this document, let us discuss about series resonance.

What is Series Resonance? 

Resonance in electric circuits is because of the presence of energy storing elements called capacitor and inductor. At a fixed frequency f0, the elements L and C will exchange their energy freely as a function of time which results in sinusoidal oscillations either across inductor (or) capacitor.

Consider a series RLC circuit at resonance. 

Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Z = R + jωL +1/jωC

= R + j (ωL − 1/ωC)

At resonance impedance is purely real i. e. ω0L =1/ω0C

∴ ω0 = 1/√LC

Z(jω0 ) = R

At ω = ω0

VR = IR = V

VL = Q. V∠90° where Q = ωL/R

VC = Q. V∠ − 90° where Q = 1/ωCR

The behavior of series RLC circuit is given by 

Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Phasor Diagram:

Notes: 

i. The LC combination in a series RLC circuit acts like a short circuit at resonance.

ii. At resonance the AC circuit behaves like dc circuit.

iii. Generally, the series RLC circuit frequency response is similar to band pass filter 

response.

Frequency Response

Bandwidth = fH − fL =f0/Q

Where, Q = ω0L/R = 1/ω0CR

Also √fH. fL = f0

Note:

i. At resonance, Z is minimum ⇒ I = maximum ⇒ so it is called acceptor circuit. 

ii. Since voltage across L and C elements are magnified by ‘Q’ times, hence series RLC 

circuit at resonance is also called as voltage magnification circuit.

 Q =ω0L/R = 1/ω0CR = (1/R)x (√(L/C) = ω0  × (Maximum energy stored in L and C at resonance / Average power dissipated at resonance)

iv. As Q is more ⇒ circuit is said to be more selective and oscillations produced are of high 

quality.

v. For the physical existence of the circuit a minimum R is to be maintained in the circuit

which is known as critical resistance of the circuit, where damping ratio is 1.

 Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE)

For finding resonant frequency always equal the imaginary part of impedance (or) admittance to be zero. 

Series Resonance Circuit Diagram

If the resonance occurs in series RLC circuit, then it is called as Series Resonance. Consider the following series RLC circuit, which is represented in phasor domain.


Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE)
Here, the passive elements such as resistor, inductor and capacitor are connected in series. This entire combination is in series with the input sinusoidal voltage source.


Apply KVL around the loop:

V−VR−VL−V= 0
⇒ V−IR−I(jXL)−I(−jXC) = 0
⇒ V = IR+I(jXL)+I(−jXC)
⇒ V = I[R+j(XL−XC)]  Equation 1


The above equation is in the form of V = IZ.
Therefore, the impedance Z of series RLC circuit will be:
Z = R + j(XL−XC)

Parameters & Electrical Quantities at Resonance

Now, let us derive the values of parameters and electrical quantities at resonance of series RLC circuit one by one.

(a) Resonant Frequency

The frequency at which resonance occurs is called as resonant frequency fr. In series RLC circuit resonance occurs, when the imaginary term of impedance Z is zero, i.e., the value of XL − XC should be equal to zero.
⇒ XL = XC
Substitute XL = 2πfL and Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE) in the above equation.
Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE)
Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE)
Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE)
Therefore, the resonant frequency fr of series RLC circuit is
Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE)
Where, L is the inductance of an inductor and C is the capacitance of a capacitor.
The resonant frequency fr of series RLC circuit depends only on the inductance L and capacitance C. But, it is independent of resistance R.

(b) Impedance

We got the impedance Z of series RLC circuit as
Z = R+j(XL − XC)
Substitute XL = XC in the above equation.
Z = R+j(XC − XC)
⇒ Z = R+j(0)
⇒ Z = R
At resonance, the impedance Z of series RLC circuit is equal to the value of resistance R, i.e., Z = R.

(c) Current flowing through the Circuit

Substitute XL − XC = 0 in Equation 1
V = I[R+j(0)]
⇒ V = IR
⇒ I = V/R
Therefore, current flowing through series RLC circuit at resonance is I = V/R
At resonance, the impedance of series RLC circuit reaches to minimum value. Hence, the maximum current flows through this circuit at resonance.

Question for Series Resonance
Try yourself:What happens to a series RLC circuit at resonance?
 
View Solution

(d) Voltage across Resistor

The voltage across resistor is
VR = IR
Substitute the value of I in the above equation.
Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE)
⇒ VR = V
Therefore, the voltage across resistor at resonance is VR = V.

(e) Voltage across Inductor

The voltage across inductor is
VL = I(jXL)
Substitute the value of I in the above equation.
Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE)
Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE)
⇒ VL = jQV
Therefore, the voltage across inductor at resonance is VL = jQV.
So, the magnitude of voltage across inductor at resonance will be
|VL| = QV
Where Q is the Quality factor and its value is equal to Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE)

(f) Voltage across Capacitor

The voltage across capacitor is
VC = I(−jXC)
Substitute the value of I in the above equation.
Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE)
Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE)
⇒ VC = −jQV
Therefore, the voltage across capacitor at resonance is VC = −jQV.
So, the magnitude of voltage across capacitor at resonance will be
|VC| = QV
Where Q is the Quality factor and its value is equal toSeries Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE)
Note: Series resonance RLC circuit is called as voltage magnification circuit, because the magnitude of voltage across the inductor and the capacitor is equal to Q times the input sinusoidal voltage V.

RLC ResonanceRLC Resonance

Example 1: In a series RLC circuit, R = 10 Ω, XL = 20 and XC = 20. Then determine the voltage across the inductor if VS = 100 V.

Solution: 

Given, XL = XC = 20

∴ Circuit is in resonance

∴VS = VR = 100V

Q − factor = ωL/R = XL/R = 20/10 = 2

∴ Voltage across inductor V= V.Q∠90°

=100.2∠90° = 200j

∴ Magnitude of VL = 200V

Example 2: Determine the resonant frequency of the circuit given below:

Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Solution: The impedance of above circuit is given by:

Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE)

Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE)

For resonance, imaginary part = 0 

∴ 1 – 4ω2 = 0, ∴ ω0 = ½ = 0.5

Question for Series Resonance
Try yourself: In a series RLC circuit, R = 10 Ω, XL = 30 Ω, and XC =30 Ω. If the source voltage, VS, is 200V, what is the magnitude of the voltage across the inductor at resonance?
 
View Solution

The document Series Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE) is a part of the Electrical Engineering (EE) Course Network Theory (Electric Circuits).
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FAQs on Series Resonance - Network Theory (Electric Circuits) - Electrical Engineering (EE)

1. What is Series Resonance?
Ans. Series resonance is a phenomenon that occurs in an electrical circuit when the inductive and capacitive reactances cancel each other out at a specific frequency, resulting in a purely resistive impedance.
2. How is Series Resonance depicted in a Phasor Diagram?
Ans. In a phasor diagram, series resonance is represented by the voltage across the resistor being in phase with the current flowing through the circuit, while the voltages across the inductor and capacitor are equal in magnitude but opposite in phase.
3. What is the Frequency Response of a Series Resonance circuit?
Ans. The frequency response of a series resonance circuit shows a sharp peak in voltage amplitude at the resonant frequency, with the impedance being at its minimum value, indicating maximum current flow through the circuit.
4. Can you provide a Circuit Diagram of a Series Resonance Circuit?
Ans. A series resonance circuit consists of a resistor, inductor, and capacitor connected in series, with the voltage source driving the circuit. The resistor represents the resistance in the circuit, while the inductor and capacitor provide the reactive components.
5. What are the Parameters and Electrical Quantities at Resonance in a Series Resonance Circuit?
Ans. At resonance in a series resonance circuit, the voltage across the resistor is maximum, the current flowing through the circuit is maximum, and the total impedance of the circuit is at its minimum value. The power factor is unity at resonance.
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