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Chapter 4:Resonance Circuits

1

• Resonance in electric circuits:

- Resonance in a series circuit- Resonance in a parallel circuit- Resonance in real circuits

Resonance circuits

The Resonance Effect

Inductive reactance increases as the frequency is increased, but capacitive reactance decreases with higher frequencies.

Because of these opposite characteristics, for any LC combination, there must be a frequency at which the XL equals the XC; one increases while the other decreases.

This case of equal and opposite reactance is called resonance, and the ac circuit is then a resonant circuit.

The frequency at which XL = XC is the resonant frequency.

At the resonant frequency, the inductive reactance and capacitive reactance are equal.

In a series ac circuit, inductive reactance leads by 90°, compared with the zero reference angle of the resistance, and capacitive reactance lags by 90°.

XL and XC are 180° out of phase.

The opposite reactances cancel each other completely when they are equal.

Chapter 4. Resonance in electric circuits.

5

• Resonance it is energy trapping case

- at resonance : Q = 0 (meaning X = 0 or B = 0)

- the phase shift between I and U is 0 (sin φ = 0)

Remarks:

a) X = 0 corresponds to the series resonance,

b) B = 0 corresponds to the parallel resonance

c) at resonance the current has an extreme (in series and ideal parallel resonance)

Series Resonance

LCπ2

1f r

Frequency Response

20 V

f

4 Ω

1 μF1 mH

1 2 3 4 5 6 7 8 9 10Frequency in kHz

5

0

3

4

2

1

Cu

rren

t in

A

LCπ2

1f r =

1

2 π 1× 10−3 × 1× 10−6

= 5.03 kHz

Tuesday, April 3, 2018 7

4.1 RESONANCE IN A SERIES CIRCUIT.

-The resonance condition:

Q = 0=>X = 0

or

- the angular resonant frequency:

)1

(C

LjRZ

22 )1

(C

LR

U

Z

UI

RC

Larctg

R

Xarctg

1

01

C

LX

C

L

1

LC

10

8

The vector diagram (phasorial representation):

Remarks:

a)R does not influence the resonance.

b) U = UR = RI

is minimum (X = 0),

the (in series and ideal parallel resonance)

:

c) UL = UC : voltage resonance

R = X+R = Z 22

maxIR

UI

UUU CL

Circuit behavior before and after resonance

9

Q Magnification Factor of Resonant Circuit

The quality, or figure of merit, of the resonant circuit, in sharpness of resonance, is indicated by the factor Q.

The higher the ratio of the reactance at resonance to the series resistance, the higher the Q and the sharper the resonance effect.

The Q of the resonant circuit can be considered a magnification factor that determines how much the voltage across L or C is increased by the resonant rise of current in a series circuit.

Resonant Rise in VL and VC

4 W

0.25 mF4 mH

Q = 32

5 A20 V

5 kHz

VL = I × XL = 640 V

VC = I × XC = 640 V

32 × 20 V = 640 V

VL = I × XL = 155 V

VC = I × XC = 155 V

7.8 × 20 V = 155 V

R = 4 W

L

20 V

5 kHz

5 A

Q = 7.8

1 mF

1 mH

QVS = VX

4 W20 V

1 mF1 mH

4 W20 V

0.25 mF4 mH

5

1 2 3 4 5 6 7 8 9 10Frequency in kHz

0

3

4

2

1

Cu

rren

t in

A Half-power

point

Q = 7.8 Q = 32

Increasing the L/C Ratio Raises the Q

14

The magnitude of the current response for the series resonance circuitis as shown below.

mV

R

2

mV

R

w

|I|

wow1 w2

Half power point

Bandwidth:

BW = wBW = w2 – w1

15

The peak power (in series circuits) delivered to the circuit is;

2

mVP

R

The so-called half-power is given when2

mVI

R

We find the frequencies, w1 and w2, at which this half-poweroccurs by using;

2 212 ( )R R wL

wC

16

After some insightful algebra one will find two frequencies at whichthe previous equation is satisfied, they are:

2

1

1

2 2

R Rw

L L LC

and

2

2

1

2 2

R Rw

L L LC

The two half-power frequencies are related to the resonant frequency by

1 2ow w w

17

The bandwidth of the series resonant circuit is given by;

2 1b

RBW w w w

L

We define the Q (quality factor) of the circuit as;

1 1o

o

w L LQ

R w RC R C

Using Q, we can write the bandwidth as;

owBW

Q

These are all important relationships.

18

If Q > 10, one can safely use the approximation;

1 22 2

o o

BW BWw w and w w

These are useful approximations.

By using Q = woL/R in the equations for w1and w2 we have;

2

2

1 11

2 2ow w

Q Q

2

1

1 11

2 2ow w

Q Q

and

If Q < 10

Bandwidth of Resonant Circuit

Parallel Resonance

When L and C are in parallel and XL equals XC, the reactive branch currents are equal and opposite at resonance.

Then they cancel each other to produce minimum current in the main line.

Since the line current is minimum, the impedance is maximum.

22

Consider the circuits shown below:

I R L C

V

I

R L

CV

jwLjwC

RVI

11

jwCjwLRIV

1

23

jwLjwC

RVI

11

jwCjwLRIV

1

We notice the above equations are the same provided:

VI

RR

1

CL

If we make the inner-change,then one equation becomes the same as the other.

For such case, we say the one circuit is the dual of the other.

If we make the inner-change,then one equation becomes the same as the other.

For such case, we say the one circuit is the dual of the other.

Duality between series and parallel resonance

24

RbereplacedR

1

LbyreplacedC

CbyreplacedL

What this means is that for all the equations we have derived for the parallel resonant circuit, we can use for the series resonant circuit provided we make the substitutions

25

-The resonance condition:

B = 0

- the angular resonant frequency:

Remark: In practice (that is having real L and C), the resonant frequencies

are different for the series and parallel connections.

01

CL

B C

L

1

LC

10

CjU+Lj

U+

R

U = I

I+I+I = ICLR

C-L

j-R

U = I

1

1

jB)-(GU = I

26

The vector diagram (phasor representation):

Remarks:

a) I = IR = GU = U/R

is minimum (B = 0),

the (in series and ideal parallel resonance)

b) IL = IC : current resonance!

minIGUI

c) if → overcurrents.III CL

G=B+G=Y 22

27

I+I+I = I

CU = I

L

U = I

R

U = I

CLR

C

L

R

)I-I(+I = I

C-L

j-R

U = I

CLR

22

1

1

Circuit behavior in ideal parallel resonance

28

Parallel ResonanceSeries Resonance

R

LwQ O

LCw

O

1

LCw

O

1

RCwQo

L

RwwwBW

BW )(

12 RCwBW

BW

1

29

Determine the resonant frequency for the circuit below.

jwRCLCw

jwLLRCw

jwCjwLR

jwCRjwL

ZNI

)1(

)(

1

)1

(

2

2

At resonance, the phase angle of Z must be equal to zero.

Resonance In Practical Circuits.

30

jwRCLCw

jwLLRCw

)1(

)(2

2Analysis

For zero phase;

LCw

wRC

LCRw

wL22 1()(

This gives;

12222 CRwLCw

or

)(

122CRLC

wo

31

RESONANCE IN Practical CIRCUITS.

Cj+R = Z Lj+R = Z

1 and 2211

C

j-R

= Y Lj+R

= Y

1and

1

2

2

1

1

C+R

C-L+R

Lj -

L+R

R +

L+R

R = Y+Y = Y e

22

22

2221

22

22

2

2221

1

21 1

1

1

0 = 1

1

22

22

2221

C+R

C-L+R

L = Be

In a parallel circuit, the resonance

occurs when Be = 0

32

R-

R-

LC =

R-C

L

R-C

L

LC =

22

2

21

2

22

21

0

1

1

Remarks:

a) if there is resonance;

b) if there is no resonance;

c) if , → the same as for a series resonant circuit

d) if , → the resonance can occur at any frequency

in this case:

and or and 2121 <R <R >R >R

R>>R R<<R 2121 or

21 RRLC

= 1

0

21 RR0

00

=

1 2

21

21

c-Lj+R

C

j-RL)j+(R

= Z+Z

ZZ = Z e

Q is often established by coil resistance.

31.61

31.6==

rS

XLQ =

20 V

5.03 kHzC = 1 mF L = 1 mH

rS = 1 W

If the circuits contains resistance other than the coil resistance the Q of the whole circuit differs from the Q of

the coil.