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

6.1 Sinusoids 正弦量 The sinusoidal voltage Or Where Vm 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) f (Hz) is the frequency 频率 T (s) is the period 周期

v1 and v2 are phase quadrature 正交 If v1(t) v2(t) v1 and v2 are out of phase 不同相 If v1leads v2 or v2 lags v1 领先 滞后 v1 and v2 are in phase If 同相 v1 and v2 are phase quadrature 正交 If v1 and v2 are reciprocal phase 反相 If

6.2 Phasors 相量 Ⅰ. Complex Number 复数 A complex number z can be written in three ways: Imaginary axis Rectangular form 直角坐标形式 Polar form 极坐标形式 Real axis 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 The complex numbers: Addition: 加法 Subtraction: 减法 Multiplication: 乘法 Division: 除法 Reciprocal: 倒数 Complex conjugate: 共轭复数

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

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

Example 6.1 Transform these sinusoids to phasors: (b) Solution: (a) (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

Summary of voltage-current relationship Element Time domain Frequency domain R L C

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

Example 6. 3 The voltage is applied to a 0. 1H inductor Example 6.3 The voltage is applied to a 0.1H inductor. Find the steady-state current through the inductor. Solution: The phasor form of v is For the inductor, Hence, Converting this to the time domain,

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

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

For the passive elements:

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

Example 6.4 Find v(t) and i(t) in the circuit. Solution: The phasor form of vs is The impedance of capacitor is Hence, Converting to the time domain,

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:

Example 6.5 Determine the equivalent impedance seen from the terminals a and b. Solution: Capacitive or inductive? X<0, capacitive

Example 6.5 Find vo(t) in the circuit. Solution: By the current division principle, Hence, Converting to the time domain,

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！