10.1 Mutual Inductance

10.2 Resonance

Chapter 10 Magnetically Coupled Circuits and Resonance磁耦合电路及谐振

When two loops with or without contacts between them affect each other through the magnetic field generated by one of them, they are said to be magnetically coupled.

i1N2

N1

+ –v121 1′ 2 2′+ –v21+ –v22

+ –v11

i2

1111 iL

dt

diL

dt

dv 1

111

11

12121 iM

dt

diM

dt

dv 1

2121

21

dt

diL

dt

dv 2

222

22

dt

diM

dt

dv 2

1212

12

2222 iL21212 iM

MMM 2112

10.1 Mutual Inductance 互感

12

11

2221

Self-induced voltage 自感电压

Magnetic linkage 磁链

Mutual inductance 互感

Mutual voltage 互感电压

Self- inductance 自感

1. inductance (L) and mutual inductance (M)

+ –v2

i1N2

N1

1 1′ 2 2′+ –v1

+ –v12

+ –v21+ –v22

+ –v11

i2●●

dt

diL

dt

dv 1

111

11

dt

diM

dt

dv 121

21

dt

diL

dt

dv 2

222

22

dt

diM

dt

dv 212

12

dt

diM

dt

diLvvv 12

221222 dt

diM

dt

diLvvv 21

112111

Mutual Inductance is the ability of one inductor to induce a voltage across a neighboring inductor, measured in henrys (H).

12

11

22

21

Dotted terminals 同名端

1 1′ 2 2′● ●

1 and 2′are Dotted terminals

i1 i2

2′

M

L1 L2

1 2

1′

If a current enters the dotted terminal of one coil, the reference polarity of the mutual voltage in the second coil is positive at the dotted terminal of the second coil.

i1

dt

diMv 1

2

2. dot convention

+

-2′

M

L1 L2

1 2

1′

i1

+

-dt

diMv 1

2

2′

M

L1 L2

1 2

1′

MLLL 221

MLLL 221

3. Series connection

• Series-opposing connection 反接

• Series-aiding connection 顺接 ML1 L2i

v+ -

dt

diMLL

dt

diLv )2( 21

dt

diMLL

dt

diLv )2( 21

equivalent inductance

equivalent inductance

ML1 L2i

v+ -

L

v+ -

i

L

v+ -

i

The phase form

dt

diMLL

dt

diLv )2( 21

IMjLjLjV )2( 21

MjLjLjLj 221

ILjV

The equivalent impedance

then

Example 10.1 Calculate the phasor currents and in the circuit.1I 2I

4j

120125j 6j

1I2I

3jSolution:

For loop 1: KVL gives

123)54( 21 IjIjj

For loop 2: KVL gives

03)126( 12 IjIj

So A39.49131 I

A04.1491.22 I

1021

kLL

Mk

The coupling coefficient k is a measure of the magnetic coupling between two coils.

4. The coefficient of coupling

When k<0.5, loosely coupled 疏耦合 k>0.5, tightly coupled 紧耦合If k=1, perfectly coupled 全耦合

10.2 Resonance 谐振

CjLjR

I

VS

1

Z

)1

(ZC

LjR

When 01

]ZIm[ C

L

Resonance is a condition in an RLC circuit in which the capacitive and inductive reactance are equal in magnitude, thereby resulting in a purely resistive impedance.

1. Series Resonance 串联谐振 I

-+

R

rmsS VV

Lj

Cj1

-

-++RV

CVLV

2. The resonant frequency 谐振频率)/(

1srad

LCo

or

)(2

1Hz

LCfo

RZ

R

VI S

I

-+

R

rmsS VV

Lj

Cj1

-

-++RV

CVLV

CLRS VVVV KVL gives:

RV I

LV

CV

SVRS VV so

（ 3 ） The magnitude of Z is minimum and Irms is maximum

CRR

LQ

0

0 1

CL VC

ILIV 0

0

1

I

-+

R

rmsS VV

Lj

Cj1

-

-++RV

CVLV

Note

（ 1 ） The equivalent impedance is purely resistive

（ 4 ） The inductor voltage VL and capacitor voltage VC can be much more than the source voltage VS

（ 2 ） The voltage and current are in phaseSV I

The quality factor 品质因数

LjCj

RV

IY

11

)1

(1

LCj

R

01

L

C

Resonant occurs when

)/(1

sradLC

o or

)(2

1Hz

LCfo

3. Parallel Resonance 并联谐振

RI

V LjCj

1R Lj

Cj1

R SS II

LICI

The resonant frequency

SR II

RY

1So purely resistive

部分电路图和内容参考了： 电路基础（第 3 版），清华大学出版社 电路（第 5 版），高等教育出版社 特此感谢！