Understand the resonance phenomenon in RLC circuits and learn to calculate the resonance frequency on series and parallel circuits.
To understand this post’s subject, it’s necessary to know the concepts shown in older posts, which links are below.
Resistance, Capacitance, Inductance, Impedance and ReactanceClick here
Basic concepts of alternate current (AC)Click here
Resonance frequency in RLC circuits
It’s the frequency at which capacitive reactance (X_{C}) is equal to inductive reactance (X_{L}). The graph below shows that inductive reactance is directly proportional to frequency, while capacitive reactance describes a hyperbolic curve.
Making replacements to find formula of resonance frequency f:
\omega is in radians per second (rad/s) and f is in Hertz.
Resonant circuit in series
The circuit’s total impedance X_{T} is:
On resonance frequency, reactances are equal, therefore, they cancel each other. Consequently, total impedance Z_{T} on this circuit is equal to the circuit’s total resistance R. The current on this frequency is:
On figure below, phasor diagram on the right shows that voltages on inductor L and capacitor C are out of phase by 180^{\circ} and they cancel each other.
Therefore, on resonance frequency, the voltage on the resistor is equal to the voltage source and power factor is 1.
Resonant circuit in parallel
In this circuit, total admittance Y_{T}, the inverse of total impedance Z_{T}, is:
On practice, inductor has a resistance, although small. Therefore, the RLC circuit below is a more realistic approximation.
In this circuit, the total impedance is relatively high on resonance frequency. The equation of Rp above shows that the resistance of parallel RLC circuit depends on frequency. For power factor becomes 1 in this circuit, inductive and capacitive reactances must be equal, therefore:
Obtaining resonance frequency:
Multiplying the numerator and the denominator inside the square by the relation C/L, we have:
Where f_{s} is the resonance frequency when inductive and capacitive reactances are equal and f_{p} is the resonance frequency to obtain power factor 1 and depends on coil’s resistance Rl. Below is the resonance frequency to obtain maximum impedance (f_{m}).
Order of magnitude of the three frequencies.
Practical applications
The main applications of this phenomenon in AC circuits are filters, oscillators, antenna coupling and frequency tuners on radio and TV receivers.
