6.1 Sinusoids 6.2 Phasors 6.4 Impedance and Admittance 6.3 Phasor Relationships for Circuit Elements Chapter 6 Sinusoids and Phasors 正弦量和相量

6.1 Sinusoids 正弦量 The sinusoidal voltage Where V m the amplitude ( 幅值 ) of the sinusoid  the angular frequency ( 角频率 ) in radians/s  t the argument ( 相位角 / 相角 ) of the sinusoid  the phase( 相位 ) Or T (s) is the period 周期 f (Hz) is the cyclic frequency 频率

v 1 leads v 2 or v 2 lags v 1 领先 滞后 v 1 and v 2 are out of phase 不同相 If v 1 and v 2 are in phase 同相 v 1 and v 2 are reciprocal phase 反相 v 1 and v 2 are phase quadrature 正交 v1(t)v1(t) v2(t)v2(t) 0

6.2 Phasors 相量 Imaginary axis Real axis Rectangular form 直角坐标形式 A complex number z can be written in three ways: Ⅰ. Complex Number 复数 Polar form 极坐标形式 Exponential form 指数形式

x and y relate to r and  : z may be written as

Ⅱ. The complex number operations 复数运算 Addition: 加法 Subtraction: 减法 Multiplication: 乘法 Division: 除法 Reciprocal: 倒数 Complex conjugate: 共轭复数 The complex numbers:

Ⅲ. Phasor and sinusoid Time domain 时域 Phasor domain 频域 A phasor is a complex number that represents the amplitude and phase of a sinusoid. Maximum value phasor 最大值相量 rms value phasor 有效值相量

The differences between v(t) and should be emphasized: 2. v(t) is time dependent,while is not. 1. v(t) is the instantaneous or time-domain representation,while is the frequency or phasor- domain representation. 3. v(t) is always real with no complex term, while is generally complex.

Example 6.1 Transform these sinusoids to phasors: (a)(a) (b)(b) Solution: (a)(a) (b)(b) Example 6.2 Transform the phasor to sinusoid: f=50Hz Solution:

6.3 Phasor Relationships for Circuit Elements Ⅰ. The resistor

Ⅱ. The inductor

Ⅲ. The capacitor

ElementTime domainFrequency domain R L C Summary of voltage-current relationship

Ⅳ Kirchhoff‘s Laws in the Frequency Domain 频域中的基尔霍夫定律 Around a closed loop KVL ： At a closed surface or a node KCL ：

6.4 Impedance and Admittance 阻抗和导纳 Impedance 阻抗 Impedance angle 阻抗角 Ⅰ. Impedance

The impedance is inductive 感性 The impedance is capacitive 容性 R =Re Z is resistance 电阻 X=Im Z is reactance 电抗 |Z| R X zz impedance triangle 阻抗三角形

For the passive elements:

The admittance ( 导纳 ) Y is the reciprocal of impedance, measured in siemens (S). G =Re Y is conductance 电导 B=Im Y is susceptance 电纳 Ⅱ. admittance

X<0, capacitive Example 6.3 Determine the equivalent impedance seen from the terminals a and b. Solution: Capacitive or inductive?

Ⅲ. Impedance Combinations 阻抗的等效变换 1. series impedances 阻抗的串联 voltage division （分压）

2. parallel impedances 阻抗的并联 current division （分流）

3. Y-  conversion: Y-  转换 A delta or wye circuit is said to be balanced （对称） if it has equal impedance in all three branches. Then:

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！