Chapter 21

Resonance

2

Series Resonance• Simple series resonant circuit

– Has an ac source, an inductor, a capacitor, and possibly a resistor

• ZT = R + jXL – jXC = R + j(XL – XC)– Resonance occurs when XL = XC

– At resonance, ZT = R

3

Series Resonance• Response curves for a series resonant circuit

4

Series Resonance

5

Series Resonance• Since XL = L = 2fL and XC = 1/C =

1/2fC for resonance set XL = XC

– Solve for the series resonant frequency fs

(Hz)

(rad/sec)

LCf

LC

s

s

21

1

=

=

6

Series Resonance• At resonance

– Impedance of a series resonant circuit is small and the current is large

• I = E/ZT = E/R

7

Series Resonance• At resonance

VR = IRVL = IXL

VC = IXC

8

Series Resonance• At resonance, average power is P = I2R• Reactive powers dissipated by inductor

and capacitor are I2X• Reactive powers are equal and opposite at

resonance

9

The Quality Factor,Q• Q = reactive power/average power

– Q may be expressed in terms of inductor or capacitor

• For an inductor, Qcoil= XL/Rcoil

RL

RX

RIXIQ LL

s

=== 2

2

10

The Quality Factor,Q• Q is often greater than 1

– Voltages across inductors and capacitors can be larger than source voltage

EV

IRIXQs ==

11

The Quality Factor,Q• This is true even though the sum of the

two voltages algebraically is zero

12

Impedance of a Series Resonant Circuit

• Impedance of a series resonant circuit varies with frequency

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−=

−+=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −+=

−+=

⎟⎠⎞⎜⎝

⎛

RCLC

RCLC

R

CLCjR

CjLjR

ωωθ

ωω

ωω

ωω

121tan

12 22

T

12

T

T

Z

Z

Z

13

Bandwidth• Bandwidth of a circuit

– Difference between frequencies at which circuit delivers half of the maximum power

• Frequencies, f1 and f2

– Half-power frequencies or the cutoff frequencies

14

Bandwidth• A circuit with a narrow bandwidth

– High selectivity• If the bandwidth is wide

– Low selectivity

15

Bandwidth• Cutoff frequencies

– Found by evaluating frequencies at which the power dissipated by the circuit is half of the maximum power

16

Bandwidth

LCLR

LR

LCLR

LR

II

f

f1

422

142

2

2

2

2

22

2

2

11

maxhpf

++==

++−

==

=

17

Bandwidth• From BW = f2 - f1

• BW = R/L• When expression is multiplied by on top

and bottom– BW = s/Q (rad/sec) or BW = fs/Q (Hz)

18

Series-to-Parallel Conversion• For analysis of parallel resonant circuits

– Necessary to convert a series inductor and its resistance to a parallel equivalent circuit

LP

P

S

LS

LS

LSSLP

S

LSSP

XR

RX

Q

XXR

X

RXR

R

==

+=

+=

22

22

19

Series-to-Parallel Conversion• If Q of a circuit is greater than or equal to 10

– Approximations may be made• Resistance of parallel network is

approximately Q2 larger than resistance of series network– RP Q2RS

– XLP XLS

20

Parallel Resonance• Parallel resonant circuit

– Has XC and equivalents of inductive reactance and its series resistor, XLP and RS

• At resonance– XC = XLP

21

Parallel Resonance• Two reactances cancel each other at

resonance– Cause an open circuit for that portion

• ZT = RP at resonance

22

Parallel Resonance

• Response curves for a parallel resonant circuit

23

Parallel Resonance• From XC = XLP

– Resonant frequency is found to be

LCR

LCf

2

121

−=

24

Parallel Resonance• If (L/C) >> R

– Term under the radical is approximately equal to 1

• If (L/C) 100R– Resonant frequency becomes

LCf

21

=

25

Parallel Resonance• Because reactances cancel

– Voltage is V = IR• Impedance is maximum at resonance

– Q = R/XC

• If resistance of coil is the only resistance present– Circuit Q will be that of the inductor

26

Parallel Resonance• Circuit currents are

IQX

IQX

P

CC

P

LL

R

==

==

=

VI

VI

RV

I

27

Parallel Resonance• Magnitudes of currents through the

inductor and capacitor – May be much larger than the current source

28

Bandwidth• Cutoff frequencies are

LCCRRC

LCCRRC1

41

21

141

21

222

221

++=

+−=

29

Bandwidth• BW = 2 - 1 = 1/RC• If Q 10

– Selectivity curve becomes symmetrical around P

30

Bandwidth• Equation of bandwidth becomes

• Same for both series and parallel circuits

P

P

PC

Q

RX

=

=

BW

BW