course no. 9 technical university of cluj-napoca author: prof. radu ciupa the theory of electric...

Course no. 9

Technical University of Cluj-Napoca

Author: Prof. Radu Ciupa

The Theory of Electric Circuits

About the Course

• Aims– To provide students with an introduction to electric circuits

• Objectives

– To introduce the concept of resonance in electric circuits

• Resonance in electric circuits:

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

Electrostatics

Topics of the course

Chapter 4

RESONANCE IN ELECTRIC CIRCUITS

Friday, April 21, 2023 4

Chapter 4. Resonance in electric circuits.


- 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



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

RX

arctg

1

01

CLX

C

L

1

LC

10



The vector diagram (phasorial representation):Remarks:

a)R does not influence the resonance.

b) U = UR = RI

is minimum (X = 0),

the current is maximum:

c) UL = UC : voltage resonance!

R = X+R = Z 22

maxIR

UI

d) if → overvoltages.

, because →

the characteristic impedance of a series resonant circuit

UUU CL

> or 1 >

=

= 000 RLR

L

RI

LI

U

UL LC

1 0 R

C

LL

LC > =

1

= =

1 =

00 C

L

CL



Q = R

= RI

I =

UU =

UU CL

- the quality factor (or Q-factor)

It shows how many times the voltage across the inductor or across the capacitance of a series resonant circuit (at resonance) is greater than the applied voltage.

- the damping factorR

Qd =

1 =

RC

1 - L

=

C

1 - L = X - X = X

C1

- L + R

U = I

CL

arctan

2

2

- for the R, L, C series circuit:



C1

- L + R

U

C = I

C = U C

1

12

2

0 = U C

2

-2 2

0

d = c

C1

-L+R

UL = I = U L

2

2

0 = U L

d = L 20

-2

2

R

U = I 0 L 0c

Note: Ucmax = ULmax



4.2 RESONANCE IN A PARALLEL CIRCUIT.

-The resonance condition:

Q = 0 => 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 = I CLR

C-L

j-R

U = I

1

1

jB)-(GU = I



The vector diagram (phasorial representation):

Remarks:

a)I = IR = GU = U/R

is minimum (B

= 0),

the current is minimum:

b) IL = IC : current resonance!

minIGUI

c) if → overcurrents.

, because

the characteristic admittance of a parallel resonant circuit

III CL

LC

1 0

G=B+G=Y 22

R < C

= L R

> L

= C

1or

1

10 0

= L

C =

L = C

0 0

1



R = G

=Q

G

= Q

= d1

- the quality factor

- the damping factor

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

CLR22

1

1



4.3 RESONANCE IN REAL CIRCUITS.

Cj+R = Z Lj+R = Z

1 and 2211

Cj

-R = Y

Lj+R = Y

1and

1

2

21

1

C+R

C-L+R

Lj -

L+R

R + L+R

R = Y+Y = Y e

2222

2221

2222

2

2221

121 1

1

1

0 = 1

1

2222

2221

C+R

C-L+R

L = Be

In a parallel circuit, the resonance

occurs when Be = 0



R-R-

LC =

R-CL

R-CL

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

Cj

-RL)j+(R =

Z+Z

ZZ = Z e



4.4 RESONANCE IN INDUCTIVELY COUPLED CIRCUITS.

122

22

211

111

+

1 - + = 0

+

1 - + =

IMjIC

LjR

IMjIC

LjRU

It is possible to obtain resonance in the primary circuit, in the secondary circuit or simultaneously in both circuits.

+ + = 0

= =

1

222

1

211e1

1

1

MjI

IjXR

I

IMjjXRZ

I

U

C-L = X ,

C

1-L = X

222

111

1

where

jX+R

Mj- =

I

I

221

2



X+R

XM-Xj +X+R

MR+R = jX+R

M+jX+R = Z e 22

22

222

122

22

222

1

22

22

111

It results:

The resonance occur when Xe = 0 :

Remarks: a)the resistance R1 does not influence the resonance, while R2 influences the resonance;

b)the realization of the resonance of inductively coupled circuits is important especially in radio frequencies circuits, we can approximate

X+R

XM = X 22

22

222

1

22 XR



M = XX 2221

M = C

L C

-L 22

22

11

1

1

0 1 0201202

201

224 = +)+(-)k-(

where: is the coefficient of coupling,

is the resonance frequency of the primary circuit

is the resonance frequency of the secondary circuit

LL

M = k

21

CL =

11

01

1

CL =

22

02

1



)k-(

)k-(-)+(+ =

2

202

201

22202

201

202

201

00 12

14 '' ,'

d d-d = k rc 4

4

2 (critical coupling)

course no. 9 technical university of cluj-napoca author: prof. radu ciupa the theory of electric...

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