1. Introduction To Resonant
Circuits
GITAM University
EEE Department

2. Resonance In Electric Circuits
Any passive electric circuit will resonate if it has an inductor
and capacitor.  
Resonance is characterized by the input voltage and current
being in phase. The driving point impedance (or admittance)
is completely real when this condition exists.
In this presentation we will consider (a) series resonance, and
(b) parallel resonance. 

3. Series Resonance Consider the series RLC circuit shown below.
V = VM 0 R L + C V _ I
The input impedance is given by:
The magnitude of the circuit current is;

4. Series Resonance Resonance occurs when,
At resonance we designate w as wo and write;
This is an important equation to remember. It applies to both series
And parallel resonant circuits.

5. Series Resonance
The magnitude of the current response for the series resonance circuit
is as shown below.
|I|
Half power point w1 wo w2 w
Bandwidth:
BW = wBW = w2 – w1

6. Series Resonance The peak power delivered to the circuit is; .
The so-called half-power is given when .
We find the frequencies, w1 and w2, at which this half-power
occurs by using;

7. Series Resonance
After some insightful algebra one will find two frequencies at which
the previous equation is satisfied, they are:
and
The two half-power frequencies are related to the resonant frequency by

8. Series Resonance
The bandwidth of the series resonant circuit is given by;
We define the Q (quality factor) of the circuit as;
Using Q, we can write the bandwidth as;
These are all important relationships.

9. Series Resonance An Observation:
If Q > 10, one can safely use the approximation;
These are useful approximations.

10. Series Resonance An Observation:
By using Q = woL/R in the equations for w1and w2 we have;
and

11. BANDWIDTH OF SERIES RESONANCE
The width of the response is measured by the BANDWIDTH.
BANDWIDTH is the difference between the half-power frequencies.
Resonance frequency can be obtained from the half-power frequencies.
The SHARPNESS of the resonance is measured by the QUALITY FACTOR.
QUALITY FACTOR is the ratio of the resonance frequency to the bandwidth. The higher the Q the smaller is the bandwidth.

12. QUALITY FACTOR OF SERIES RESONANCE

13. BANDWIDTH of SERIES RESONANCE
Current versus frequency for the series resonant circuit.
Half Power Frequencies
Dissipated power is half of the maximum value.
The half-power frequencies 1 and 2 can be obtained by setting,

15. Selectivity
The frequencies corresponding to of the maximum current are called the band frequencies, cutoff frequencies, or half-power frequencies (ƒ1, ƒ2).
Half-power frequencies are those frequencies at which the power delivered is one-half that delivered at resonant frequency.
The range of frequencies between the two are referred to as bandwidth (abbreviated BW) of the resonant circuit.
Since the resonant circuit is adjusted to select a band of frequencies it is called a selectivity curve.
The shape of the curve depends on each element of the series R-L-C circuit.
If resistance is made smaller with a fixed inductance and capacitance, the bandwidth decreases and the selectivity increases.
If the ratio L/C increases with fixed resistance, the bandwidth again decreases with an increase in selectivity.

16. PARALLEL RESONANCE
Resonance is a condition in an RLC circuit in which the capacitive and inductive reactances are equal in magnitude, resulting in a purely resistive impedance.
Parallel resonance circuit behaves similarly but in opposite fashion compared to series resonant circuit.
The admitance is minimum at resonance or impedance is maximum.
Parallel resonant circuit.

17. PARALLEL RESONANCE At Resonance frequency:
1) Admitance is purely resistive.
2) The voltage and current are in phase.
3) The transfer function H()= Y() is Minimum.
4) Inductor and capacitor currents can be much more than the source current.

18. PARALLEL RESONANCE
Voltage versus frequency for the parallel resonant circuit.
The half-power frequencies can be obtained as:

19. Summary of series and parallel resonance circuits:
Characteristic
Series circuit
Parallel circuit ωo Q B
ω1, ω2
Q ≥ 10, ω1, ω2

21. Parallel Resonance
Parallel Resonance
Series Resonance

22. Resonance
Example 1:
Determine the resonant frequency for the circuit below.
At resonance, the phase angle of Z must be equal to zero.

23. Resonance
Analysis
For zero phase;
This gives; or

24. Parallel Resonance Example 2:
A series RLC resonant circuit has a resonant frequency admittance of
2x10-2 S(mohs). The Q of the circuit is 50, and the resonant frequency is
10,000 rad/sec. Calculate the values of R, L, and C. Find the half-power
frequencies and the bandwidth.
A parallel RLC resonant circuit has a resonant frequency admittance of
2x10-2 S(mohs). The Q of the circuit is 50, and the resonant frequency is
10,000 rad/sec. Calculate the values of R, L, and C. Find the half-power
frequencies and the bandwidth.
A parallel RLC resonant circuit has a resonant frequency admittance of
2x10-2 S(mohs). The Q of the circuit is 50, and the resonant frequency is
10,000 rad/sec. Calculate the values of R, L, and C. Find the half-power
frequencies and the bandwidth.
First, R = 1/G = 1/(0.02) = 50 ohms.
Second, from
, we solve
for L, knowing Q, R, and wo to
find L = 0.25 H.
Third, we can use

25. Parallel Resonance Example 2: (continued) Fourth: We can use and
Fifth: Use the approximations;
w1 = wo wBW = 10,000 – 100 = 9,900 rad/sec
w2 = wo wBW = 10, = 10,100 rad/sec