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. 

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, 

V_R = I_R = V

V_L = Q. V∠90° where Q = ωL/R

V_C = Q. V∠ − 90° where Q = 1/ωCR

The behavior of series RLC circuit is given by 

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 = f_H − f_L =f_0/Q

Where, Q = ω₀L/R = 1/ω₀CR

Also √f_H. f_L = 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.

 

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.

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−V_R−V_L−V_C = 0
⇒ V−IR−I(jX_L)−I(−jX_C) = 0
⇒ V = IR+I(jX_L)+I(−jX_C)
⇒ V = I[R+j(X_L−X_C)]  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(X_L−X_C)

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 f_r. In series RLC circuit resonance occurs, when the imaginary term of impedance Z is zero, i.e., the value of X_L − X_C should be equal to zero.
⇒ X_L = X_C
Substitute X_L = 2πfL and  in the above equation.



Therefore, the resonant frequency f_r of series RLC circuit is

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(X_L − X_C)
Substitute X_L = X_C in the above equation.
Z = R+j(X_C − X_C)
⇒ 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 X_L − X_C = 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.

(d) Voltage across Resistor

The voltage across resistor is
V_R = IR
Substitute the value of I in the above equation.

⇒ V_R = V
Therefore, the voltage across resistor at resonance is V_R = V.

(e) Voltage across Inductor

The voltage across inductor is
V_L = I(jX_L)
Substitute the value of I in the above equation.


⇒ V_L = jQV
Therefore, the voltage across inductor at resonance is V_L = jQV.
So, the magnitude of voltage across inductor at resonance will be
|V_L| = QV
Where Q is the Quality factor and its value is equal to

(f) Voltage across Capacitor

The voltage across capacitor is
V_C = I(−jX_C)
Substitute the value of I in the above equation.


⇒ V_C = −jQV
Therefore, the voltage across capacitor at resonance is V_C = −jQV.
So, the magnitude of voltage across capacitor at resonance will be
|V_C| = QV
Where Q is the Quality factor and its value is equal to
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 Resonance

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

Solution: 

Given, X_L = X_C = 20

∴ Circuit is in resonance

∴V_S = V_R = 100V

Q − factor = ωL/R = X_L/R = 20/10 = 2

∴ Voltage across inductor V_L = V.Q∠90°

=100.2∠90° = 200j

∴ Magnitude of V_L = 200V

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

Solution: The impedance of above circuit is given by:

For resonance, imaginary part = 0 

∴ 1 – 4ω² = 0, ∴ ω₀ = ½ = 0.5