Resonance – Quantmasters

A.C Circuits made up of resistors, inductors and capacitors are said to be resonant circuits when the current drawn from the supply is in phase with the impressed sinusoidal voltage.
Then:
- the resultant reactance or susceptance is zero.
- the circuit behaves as a resistive circuit.
- the power factor is unity.

Resonance is the phenomenon which finds its applications in communication circuits:
- The ability of a radio or Television receiver
  - (1) to select a particular frequency or a narrow band of frequencies transmitted by broad casting stations or
  - (2) to suppress a band of frequencies from other broad casting stations, is based on resonance.
Thus resonance is desired in tuned circuits, design of filters, signal processing and control engineering.
But it is to be avoided in other circuits. It is to be noted that if R= 0 in a series circuit, the circuits acts as a short circuit at resonance
if R=infinity in parallel circuit, the circuit acts as an open circuit at resonance.

Series Resonance

In a series resonant circuit Resonance can be achieved by:

RLC series resonance circuit

Phasor Diagram of Resonance Condition

The variation of current with frequency

It is observed that there are two frequencies, one above and the other below the resonant frequency, ω₀ at which current is same.
These 2 frequencies are known as cutoff frequencies for which the current becomes 0.707 times of Highest or Resonant current.
ω₁ is the cut off frequency which is less than ω₀ & it is called as lower cut off frequency.
ω₂ is the cut off frequency which is higher than ω₀ & it is called as higher cut off frequency.
ω₁ & ω₂ are also called as half power frequencies because in these frequencies the power becomes half of highest power or resonance Power.
These are called as 3dB frequencies also as decrease to 0.707 is known as 3dB decrease of gain. in dB=20log(Gain)
the resultant reactance is, X = X_L-X_C .
The frequency range between half – power frequencies is ω₁ – ω₂ , and it is referred to as passband or band width.
BW = ω₁ – ω₂ =delta ω
The sharpness of tuning depends on the ratio R/L, a small ratio indicating a high degree of selectivity.
The quality factor of a circuit can be expressed in terms of and of the inductor.
Quality factor =ω₀L/R

In many ways a parallel resonance circuit is exactly the same as the series resonance circuit we looked at in the previous section.
Both are 3-element networks that contain two reactive components making them a second-order circuit, both are influenced by variations in the supply frequency and both have a frequency point where their two reactive components cancel each other out influencing the characteristics of the circuit.
Both circuits have a resonant frequency point.
The difference this time however, is that a parallel resonance circuit is influenced by the currents flowing through each parallel branch within the parallel LC tank circuit.
A tank circuit is a parallel combination of L and C that is used in filter networks to either select or reject AC frequencies.
Consider the parallel RLC circuit below.

A parallel circuit containing a resistance, R, an inductance, L and a capacitance, C will produce a parallel resonance (also called anti-resonance) circuit when the resultant current through the parallel combination is in phase with the supply voltage.
At resonance there will be a large circulating current between the inductor and the capacitor due to the energy of the oscillations, then parallel circuits produce current resonance.
A parallel resonant circuit stores the circuit energy in the magnetic field of the inductor and the electric field of the capacitor.
This energy is constantly being transferred back and forth between the inductor and the capacitor which results in zero current and energy being drawn from the supply.
In the solution of AC parallel resonance circuits we know that the supply voltage is common for all branches, so this can be taken as our reference vector.
Each parallel branch must be treated separately as with series circuits so that the total supply current taken by the parallel circuit is the vector addition of the individual branch currents.

At resonance the parallel LC tank circuit acts like an open circuit with the circuit current being determined by the resistor, R only.
So the total impedance of a parallel resonance circuit at resonance becomes just the value of the resistance in the circuit and Z = R as shown.

Thus at resonance, the impedance of the parallel circuit is at its maximum value and equal to the resistance of the circuit creating a circuit condition of high resistance and low current.
Also at resonance, as the impedance of the circuit is now that of resistance only, the total circuit current, I will be “in-phase” with the supply voltage, V_S.

Note that if the parallel circuits impedance is at its maximum at resonance then consequently, the circuits admittance must be at its minimum and one of the characteristics of a parallel resonance circuit is that admittance is very low limiting the circuits current.
At the resonant frequency, ƒ_r the admittance of the circuit is at its minimum and is equal to the conductance, G given by 1/R because in a parallel resonance circuit the imaginary part of admittance, i.e. the susceptance, B is zero because B_L = B_C as shown.
At resonance the current flowing through the circuit must also be at its minimum as the inductive and capacitive branch currents are equal ( I_L = I_C ) and are 180^o out of phase.
the total current flowing in a parallel RLC circuit is equal to the vector sum of the individual branch currents and for a given frequency is calculated as:

At resonance, currents I_L and I_C are equal and cancelling giving a net reactive current equal to zero.
Then at resonance the above equation becomes.

Parallel Circuit Current at Resonance