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

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.

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.

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.

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.

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

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

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

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.