1. 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

2. 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

3. 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

4. 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

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

6. 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

7. 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

8. 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.

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

10. 6.3 Phasor Relationships for Circuit Elements
Ⅰ. The resistor

11. Ⅱ. The inductor

12. Ⅲ. The capacitor

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

14. 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：

15. Example 6. 3 The voltage is applied to a 0. 1H inductor
Example 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,

16. 6.4 Impedance and Admittance 阻抗和导纳
Impedance angle
阻抗角

17. 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

18. For the passive elements:

19. 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

20. 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,

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

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

23. 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:

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

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

26. 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！