1. Chapter 10 Magnetically Coupled Circuits and Resonance
磁耦合电路及谐振
10.1 Mutual Inductance
10.2 Resonance

2. 10.1 Mutual Inductance 互感
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.

3. 1. Self-Inductance (L) and Mutual Inductance (M)
Magnetic linkage 磁链 N1 N2 i1 i2 + –
v11 + –
v21 + –
v12 2 2′ + –
v22 1 1′
Self- inductance 自感
Mutual inductance 互感
Self-induced voltage
自感电压
Mutual voltage
互感电压

4. Mutual Inductance is the ability of one inductor to induce a voltage across a neighboring inductor, measured in henrys (H). N1 N2 i1 i2 + –
v12
v21
v22
v11 ● ● 2 2′ 1 1′ + – v1 + – v2 2′ M L1 L2 1 2 1′
Dotted terminals 同名端

5. 1 and 2′are dotted terminals● ● 1 1′ 2 2′
1 and 2′are dotted terminals 2′ M L1 L2 1 2 1′

6. 2. Dot Convention
同名端惯例
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. 2′ M L1 L2 1 2 1′ 2′ M L1 L2 1 2 1′ i1 i1 + + - -

8. 3. Series Connection Series-aiding connection 顺接 equivalent inductanceM L1 L2 i v + -
equivalent inductance v + - i
decoupling equivalent circuit
去耦等效电路
Series-opposing connection 反接 M L1 L2 i v + - v + - i
equivalent inductance

9. The phase form for series-aiding connection
The equivalent impedance
then

10. 4. Parallel Connection Same-side parallel 同侧并联
decoupling equivalent circuit
different-side parallel
异侧并联
decoupling equivalent circuit

11. Linear transformer
Decoupling equivalent T circuit
线性变压器
Decoupling equivalent T circuit

12. 5. The Coefficient of Coupling
耦合系数
The coupling coefficient k is a measure of the magnetic coupling between two coils.
When k<0.5, loosely coupled
疏耦合
k>0.5, tightly coupled
紧耦合
k=1, perfectly coupled
全耦合

13. Example 10.1 Calculate the phasor currents and in the circuit.
Solution:
For loop 1: KVL gives
For loop 2: KVL gives So

14. Example 10.2 Calculate the input impedance in the circuit.
Solution:
Method 1:
We can apply a voltage source
and calculate the current
For loop 1: KVL gives
For loop 2: KVL gives
thus

15. Method 2:
We can get decoupling equivalent T circuit
thus

16. 10.2 Resonance 谐振 1. Series Resonance 串联谐振
Resonance is a condition in an RLC circuit in which the capacitive and inductive reactances are equal in magnitude, thereby resulting in a purely resistive impedance.
1. Series Resonance
串联谐振
When
resonance results

17. 2. The Resonant Frequency 谐振频率or
KVL gives: so

18. The quality factor
品质因数
Note that at resonance:
（1）The equivalent impedance is purely resistive, thus, Z=R.
（2）The voltage and current are in phase.
（3）The magnitude of Z is minimum and Irms is maximum.
（4）The inductor voltage VL and capacitor voltage VC can be much more than the source voltage VS.

19. 3. Parallel Resonance 并联谐振 Resonance occurs when or
The resonant frequency So
purely resistive

20. Example 10.3 In the circuit, M=0.05 H，find the resonant frequency f0.
Solution:
Series-aiding connection
the equivalent inductance
so,

21. 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！