Network Theory

RESONANCE
A two-terminal network, in general, offers a complex impedance consisting of resistive and
reactive components. If a sinusoidal voltage is applied to such a network, the current is then
out of phase with the applied voltage.

Under special circumstances, however, the impedance offered by the network is purely
resistive and the current (I) and the applied voltage (V) are then in phase. This phenomenon
is called resonance. The frequency of applied signal at which resonance occurs is called the
frequency of resonance.

Definition : Resonance is the condition in a circuit when the Impedance (Z) of the network
is Purely Resistive or when the supply voltage and the supply current are in phase.

At resonance, the circuit Z and Y are real quantities (resistive or conductive). Then the supply
voltage and the supply current are in phase.

Illustration : If V → applied voltage, I → supply current

V | V |  | V | ( − )
I= or | I |  = or | I |  =
Z | Z |  |Z|

 = – Thus, in general the current is out of phase with the applied voltage.

If  = 0, Z = |Z|00 → Resistive (= R)
| V | 
and | I |  = or I=V/R
R

 = Hence, the current (I) and the applied voltage (V) are in phase.

Similarly, I = VY or |I| = |V| |Y|

 = + Thus, in general the current is out of phase with the applied voltage.

If  = 0, Y = |Y|00 → Conductive (= G)
and |I| = G |V| or I = VG

 = The current (I) and the applied voltage (V) are in phase.

At resonance, I = I0 = V / Z = V / Z0 = V / R

V & I are in phase.

If V is complex, then I is complex with the same phase angle.

Class Note by Santanu Das 1

 Network Theory

The nature of resonance depends upon whether the inductance and the capacitance are in series
or in parallel. Accordingly, we classify the resonant circuits into the following two categories:

(i) Series resonant circuit, (ii) Parallel resonant circuit.

Series Resonance
A sinusoidal voltage V sends a current I through a series RLC circuit. The circuit is said to be
resonant when the resultant reactance is zero, i.e., the circuit is purely resistive or V and I are
in phase.

1 V V
Z = R + j L + I= =
j C Z  1 
R + j  L − 
 C 

At resonance, the impedance Z becomes purely resistive, i.e. Z = R and current I = V / R

1 1
Thus, at resonance, Im (Z) = X = 0 i.e., 0 L − =0 or 0 L =
0C 0C

i.e., inductive reactance = capacitive reactance

where 0 is the frequency of resonance in radians/second,

1 1
 0 = or f0 =
LC 2 LC

(1) Series resonance at any desired frequency f0 may be obtained by varying either L, or C or
both.
(2) For fixed values of L and C, series resonance may be achieved by varying the frequency of
applied signal.

It is convenient to obtain resonance at the desired frequency by varying the capacitance

(variable inductor is difficult to obtain)

Class Note by Santanu Das 2

 Network Theory

• Reactance and Impedance Curves of a series RLC Circuit

(Variation of different quantities with frequency)

Z = R + j(L – 1/C)
• Below resonance, the circuit is capacitive and acts as an RC circuit.
• Above resonance, the circuit is inductive and acts as an RL circuit.
• At resonance, the circuit acts like a resistor.
• If R = 0, at resonance the circuit acts like a short circuit.

At f0, Z = R since XL =XC. This is the minimum value of Z. Current |I| is maximum.

At resonance, I = I0 = V / Z = V / Z0 = V / R

V & I are in phase. If V is complex, then I is complex with the same phase angle.

The voltage across L is VL0 = jL I0 & the voltage across C is VC0 = ( –j / C) I0
The voltage across L is equal in magnitude & opposite in phase (1800) to the voltage across C.

The supply voltage V = VR = voltage across R

(if V is complex, VR is also complex with same phase angle).

Class Note by Santanu Das 3

 Network Theory

• Bandwidth
A series RLC circuit gives unequal current response to different frequencies of the voltage.

At the frequency of resonance, the impedance is minimum and the current is maximum.
As the frequency of the applied voltage is either reduced or increased from this
resonance frequency, the impedance increases and the current fall. Thus, a series RLC circuit
possesses frequency selectivity.

The frequencies f1 and f2 at which current I falls to 1/2 (or 0.707) of its maximum value I0 (=
V / R) are called cut-off frequencies or half-power frequencies or 3-dB frequencies.

I0
I1 = I 2 =
2

The difference between the cut-off frequencies (f2 – f1) is called the bandwidth or half-power
bandwidth or 3-dB bandwidth of the circuit.

The ratio of the current at cut-off frequency to the current at resonance in decibels is given by

20 log10 (I1 / I0) = 20 log10 (1/2) = – 3 dB

Thus, I1 is 3 dB lower than I0. Similarly, I2 is 3 dB lower than I0. Hence, f1 and f2 are called 3
dB frequencies.

Frequencies f1 and f2 are also called half-power frequencies because the power dissipation in
the circuit at these frequencies is half of the (maximum) power dissipation at the resonant
frequency f0. This may be seen as below :

P0 = Power dissipation at f0 = I02 R

P1 = Power dissipation at f1 = I12 R = I02 R / 2 = P0 / 2
P2 = Power dissipation at f2 = I22 R = I02 R / 2 = P0 / 2

Power ratio (in decibel) = 10 log10 (P1 / P0) = 10 log10 (1/2) = – 3 dB

Thus, at half-power frequencies, power is 3 dB less than that at resonance.

Class Note by Santanu Das 4

 Network Theory

• Calculation of BANDWIDTH in terms of circuit parameters:

V V
I= =
Z R + j ( L − 1/ C )

1
Let 2 be such a frequency that 2 L − =R
2C
V V I
Then at frequency 2, I2 = and Magnitude I2 = = 0
R + jR 2R 2
where
I0 is the maximum current at resonant frequency f0 and I0 = V / R

Thus, 2 radians/sec (or f2 Hertz) gives the upper cut-off frequency.

1
Similarly, let 1 be such a frequency that, 1L − = −R
1C

[negative sign of right side indicates that below resonance the capacitive reactance (1/1C) is
greater than the inductive reactance 1L]

V V I
Then the current at 1, I1 = and Magnitude I1 = = 0
R − jR 2R 2

Thus, 1 radians/sec (or f1 Hertz) forms the lower cut-off frequency.

At 1, (1L – 1/1C) = – R

or 12 LC – 1 = 1 RC or 12 + (R / L) 1 – (1/LC) = 0

2
R  R  1
 1 = −    + → –ve value of 1 is meaningless.
2L  2 L  LC

2
R  R  1
Hence, 1 = − +   +
2L  2 L  LC
2
R  R  1
At 2, (2L – 1/2C) = + R Hence, 2 = + +   +
2L  2 L  LC
R R R R
  = 2 − 1 = + = or f = f 2 − f1 = Hz
2L 2L L 2L

2
 2
1   R 
2
 R  1
Moreover, 12 =    + −  = = 02
  2 L  LC   2 L  LC
 
So, the resonant frequency is the geometric mean of the lower and upper cut-off frequencies.
Class Note by Santanu Das 5
 Network Theory

• Quality Factor or Q Factor

A resonator stores energy in the capacitor (in electric field) and in the inductor (in magnetic
field). The presence of resistance in the circuit results in power loss and causes a decay of the
stored energy.

Quality factor Q of a resonator is used to Determine the loss, Determine the quality and
Measure the bandwidth of the resonator.

Maximum Energy Stored

Definition : Q = 2
Energy Loss (dissipated) per cycle

W W
 Q = 2 = 0 (Dimensionless)
PLT PL

where T = l/f = period of oscillation,

PL = time average Power Loss (dissipated) in resonator,
W = We + Wm = Maximum energy stored in capacitor (We) & inductor (Wm)

0 f resonant frequency
From this, Q= = 0 =
 f bandwidth

0 L
For a Series R-L-C Circuit, since  = R / L , Q=
R

Then Quality factor or Q factor → the ratio of inductive reactance to resistance at resonance.

1 1
Since 0 L = , Q may be expressed as Q=
0C 0CR

• Voltage Magnification

At resonance, the current I0 = V / R

The voltage across the inductor L at resonance : VL = I0 XL = (V / R) (0L) = Q V

The voltage across the capacitor C at resonance : VC = I0 XC = (V / R) (1/0C) = Q V

Voltage magnification = Voltage across L or C at resonance / Supply voltage at resonance

= VL / V = VC / V = Q

At resonance, the voltages across L and C each is equal to Q times the applied voltage V.

Q is considerably greater than the unity.

Thus, the voltage across L (or C) may be greater than applied voltage.

Because of this, the Q-factor is also sometimes referred to as the circuit magnification factor
or simply the magnification factor.

Class Note by Santanu Das 6

 Network Theory

Parallel Resonance
A parallel RLC circuit shown in Fig. is connected to a sinusoidal source.

Analysis of parallel resonance circuit may be done more conveniently in terms of admittances
instead of impedances.
Y (Admittance) = G (Conductance) + j B (Susceptance)

1 1
Y = YR + YL + YC = + + j C
R jL
The admittance “seen” by the source is given by
1  1 
= + j  C − 
R  L 
At resonance, the imaginary part of Y is zero.
That is, at  = 0, Im (Y) = 0

1 1 1
 0C − =0 and 0 = or f0 =
0 L LC 2 LC

Here, I=VY

At resonance, the admittance, Y = 1/R or Z = R.

Y & Z are real.  I & V are in phase.

Susceptance and Admittance Curves of a Parallel RLC Circuit

(Variation of different quantities with frequency)

Class Note by Santanu Das 7

 Network Theory

• Below resonance, the circuit is inductive and acts like an RL circuit.

• Above resonance, the circuit is capacitive and acts like an RC circuit.
• At resonance, the circuit acts like a resistor.
• If the conductance G = 0 (R = ), at resonance the circuit behaves like an open circuit.

At resonance, Y is minimum and conductive, i.e. Z is maximum and resistive.

 I is minimum and I = I0 = V / R or magnitude I0 = V / R

The variation of | I | with  :

• Bandwidth
A parallel RLC circuit gives unequal response to different frequencies of the voltage.

At the frequency of resonance, the impedance is maximum and the current is minimum.
As the frequency of the applied voltage is either reduced or increased from this
resonance frequency, the impedance decreases and the current increases. Thus, a parallel RLC
circuit possesses frequency selectivity.

Fig. shows the variation of current I with frequency.

The frequencies f1 and f2 at which current I rises to 2 (or 1.414) times the minimum current
Imin (= V / R) are called cut-off frequencies or half-power frequencies or 3-dB frequencies.

The difference between the cut-off frequencies (f2 – f1) is called the bandwidth or half-power
bandwidth or 3-dB bandwidth of the circuit.

Class Note by Santanu Das 8

 Network Theory

• Calculation of BANDWIDTH in terms of circuit parameters :

1  1 
I = VY = V  + j  C − 
R  L  

1 1
Let 2 be such a frequency that 2C − =
2 L R

1 1 V
Then at frequency 2, I2 = V  + j  and Magnitude I2 = 2 = 2I0
R R R
where
I0 is the minimum current at resonant frequency f0 and I0 = V / R

Thus, 2 radians/sec (or f2 Hertz) gives the upper cut-off frequency.

1 1
Similarly, let 1 be such a frequency that, 1C − =−
1L R

[negative sign of right side indicates that below resonance the capacitive susceptance (1C) is
less than the inductive susceptance (1/ 1L)]

1 1 V
Then at 1, I1 = V  − j  and Magnitude I1 = 2 = 2I0
R R R

Thus, 1 radians/sec (or f1 Hertz) forms the lower cut-off frequency.

2
1  1  1
At 1, (1C– 1/1L) = – 1 / R  1 = − +   +
2RC  2RC  LC

2
1  1  1
At 2, (2C – 1/2L) = + 1 / R  2 = + +   +
2 RC  2 RC  LC

1 1 1
  = 2 − 1 = + =
2 RC 2 RC RC

2
 2  2
 1  1  −  1  = 1 = 02
Moreover, 12 =    +
  2 RC  LC   2 RC  LC
 

So, the resonant frequency is the geometric mean of the lower and upper cut-off frequencies.

Class Note by Santanu Das 9

 Network Theory

• Q of a Parallel R-L-C Circuit

0 f resonant frequency
Q= = 0 =
 f bandwidth

Since  = 1 / RC Q = 0RC

1 R
Since 0 L = , Q may be expressed as Q=
0C 0 L

• Current Magnification

At resonance, the current drawn from the supply source is I0 = V / R

The current through the capacitor C at resonance : IC = V / XC = V0C

The current through the inductor L at resonance : IL = V / XL = V / 0L

Current magnification = Current through C or L at resonance / Supply current at resonance

= IC / I0 = IL / I0 = Q

Thus, at resonance, the current through C and L each is equal to Q times the source current I.

Class Note by Santanu Das 10

 Network Theory

Resonance in Practical Parallel Circuit

A practical parallel resonant or anti-resonant circuit consists of an inductor L with associated
small series resistance in parallel with a capacitor C. Such a parallel tuned circuit is of great
practical importance because it is the basis of tuned circuits (tuned amplifiers, oscillators etc.)

1 r − jL
The admittance of the inductive branch : Y1 = = 2
r + jL r + 2 L2

Admittance of capacitor C is given by YC = jC

r  L 
Total admittance Y = Y1 + YC = + j C − 2
r + L
22 2
 r + 2 L2 

At resonance, Im [Y] = 0

0 L L
 0C − =0 or r 2 + 02 L2 =
r + 02 L2
2
C

1 r2 1 1 r2
Hence, 0 = − or f0 = −
LC L2 2 LC L2

• Impedance of Parallel Resonant Circuit

r
At resonance : Ya =
r + 02 L2
2

L rC
But r 2 + 02 L2 = Hence, Ya =
C L

L
Hence, Impedance at resonance is given by, Z a = Ra =
rC

This is a pure resistance and is often called the equivalent or dynamic resistance at resonance
of the parallel tuned circuit.

Class Note by Santanu Das 11

 Network Theory

• Current Magnification

V V
The current drawn from supply source at resonance : I0 = =
Ra L / Cr
The current in the capacitor branch : IC = V / XC = V0C

Current magnification = Current through C or L at resonance / Supply current at resonance

= IC / I0 = 0L / r = Q

Thus, at resonance, the current through C and L each is equal to Q times the source current I.

Class Note by Santanu Das 12