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

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.

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

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

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

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 + + - -

3. Series Connection Series-aiding connection 顺接 equivalent inductance M 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

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

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

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

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

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

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

Method 2: We can get decoupling equivalent T circuit thus

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

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

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.

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

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

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！