Resonance in Series RLC Circuit

Consider a series RLC circuit where a resistor, inductor and capacitor are connected in series across a voltage supply. This series RLC circuit resonates at a specific frequency known as the resonant frequency.
In this circuit containing inductor and capacitor, the energy is stored in two different ways.

1. When a current flows in an inductor, energy gets stored in magnetic field.
2. When a capacitor is charged, energy gets stored in static electric field.

The magnetic field in the inductor is built by the current, which is provided by the discharging capacitor. Similarly, the capacitor is charged by the current produced by collapsing magnetic field of inductor and this process continues on and on, causing electrical energy to oscillate between the magnetic field and the electric field. In some cases, at certain frequency called resonant frequency, the inductive reactance of the circuit becomes equal to capacitive reactance which causes the electrical energy to oscillate between the electric field of the capacitor and magnetic field of the inductor. This forms a harmonic oscillator for current. In RLC circuit, the presence of resistor causes these oscillation to die out over period of time and is called damping effect of resistor.

Variation in Inductive Reactance and Capacitive Reactance with Frequency

Variation of Inductive Reactance Vs Frequency

We know that inductive reactance X_L = 2πfL means inductive reactance is directly proportional to frequency (X_L and prop ƒ). When the frequency is zero or in case of DC, inductive reactance is also zero, the circuit acts as a short circuit; but when frequency increases; inductive reactance also increases. At infinite frequency, inductive reactance becomes infinity and circuit behaves as open circuit. It means that, when frequency increases inductive reactance also increases and when frequency decreases, inductive reactance also decreases. So, if we plot a graph between inductive reactance and frequency, it is a straight line linear curve passing through origin as shown in the figure above.

Variation of Capacitive Reactance Vs Frequency

The formula for capacitive reactance X_C = 1 / 2πfC shows that frequency and capacitive reactance are inversely proportional. At zero frequency (DC), capacitive reactance is infinite, and the circuit acts as an open circuit. As frequency increases, capacitive reactance decreases and becomes zero at infinite frequency, making the circuit act as a short circuit. The graph of capacitive reactance versus frequency forms a hyperbolic curve.

Inductive Reactance and Capacitive Reactance Vs Frequency

From the above discussion, it can be concluded that the inductive reactance is directly proportional to frequency and capacitive reactance is inversely proportional to frequency, i.e at low frequency X_L is low and X_C is high but there must be a frequency, where the value of inductive reactance becomes equal to capacitive reactance. Now if we plot a single graph of inductive reactance vs frequency and capacitive reactance vs frequency, then there must occur a point where these two graphs cut each other. At that point of intersection, the inductive and capacitive reactance becomes equal and the frequency at which these two reactances become equal, is called resonant frequency, f_r.
At resonant frequency, X_L = X_L


At resonance f = f_r and on solving above equation we get,

Variation of Impedance Vs Frequency

At resonance in a series RLC circuit, the inductive and capacitive reactances cancel each other out. Thus, the only opposition to current flow is due to the resistor. At this point, the total impedance is equal to the resistance (Z = R) with no imaginary part. This impedance at resonant frequency is called dynamic impedance, which is always less than the total impedance of the circuit. Before resonance, f_r capacitive reactance dominates, and after resonance, inductive reactance dominates. At resonance, the circuit behaves purely resistive, allowing maximum current flow.

Resonant Current

In series RLC circuit, the total voltage is the phasor sum of voltage across resistor, inductor and capacitor. At resonance in series RLC circuit, both inductive and capacitive reactance cancel each other and we know that in series circuit, the current flowing through all the elements is same, So the voltage across inductor and capacitor is equal in magnitude and opposite in direction and thereby they cancel each other. So, in a series resonant circuit, voltage across resistor is equal to supply voltage i.e V = V_r.
In series RLC circuit current, I = V / Z but at resonance current I = V / R, therefore the current at resonant frequency is maximum as at resonance in impedance of circuit is resistance only and is minimum.
The above graph shows the plot between circuit current and frequency. At starting, when the frequency increases, the impedance Z_c decreases and hence the circuit current increases. After some time frequency becomes equal to resonant frequency, at that point inductive reactance becomes equal to capacitive reactance and the impedance of circuit reduces and is equal to circuit resistance only. So at this point, the circuit current becomes maximum I = V / R. Now when the frequency is further increased, Z_L increases and with increase in Z_L, the circuit current reduces and then the current drops finally to zero as frequency becomes infinite.

Power Factor at Resonance

At resonance, the inductive reactance is equal to capacitive reactance and hence the voltage across inductor and capacitor cancel each other. The total impedance of circuit is resistance only. So, the circuit behaves like a pure resistive circuit and we know that in pure resistive circuit, voltage and the circuit current are in same phase i.e V_r, V and I are in same phase direction. Therefore, the phase angle between voltage and current is zero and the power factor is unity.

Application of Series RLC Resonant Circuit

Since resonance in series RLC circuit occurs at particular frequency, so it is used for filtering and tuning purpose as it does not allow unwanted oscillations that would otherwise cause signal distortion, noise and damage to circuit to pass through it.
Summary
For a series RLC circuit at certain frequency called resonant frequency, the following points must be remembered. So at resonance:

1. Inductive reactance X_L is equal to capacitive reactance X_C.
2. Total impedance of circuit becomes minimum which is equal to R i.e Z = R.
3. Circuit current becomes maximum as impedance reduces, I = V / R.
4. Voltage across inductor and capacitor cancels each other, so voltage across resistor V_r = V, supply voltage.
5. Since net reactance is zero, circuit becomes purely resistive circuit and hence the voltage and the current are in same phase, so the phase angle between them is zero.
6. Power factor is unity.
7. Frequency at which resonance in series RLC circuit occurs is given by