电工学原理及应用(Electrical Engineering)经典双语详解讲义 (5)

sinz cosz 90


•Root-Mean-Square Values or Effective Values
Vrms
1 T
T

0
v t dt
2
1 2 I rm s i t dt T0
T
Pavg
V R
2 rms
Pavg I
Average Power
10 30 V V2
•Adding Sinusoids Using Phasors
Vs V1 V2 20 45 10 30 14.14 j14.14 8.660 j5 23.06 j19.14 29.97 39.7 V
A | A |
A a b
2 2
Im
b |A|
A

0 a
Re
a | A | cos
b arctan a

b | A | sin
Conversion between Rectangular and polar
•Conversion
A | A |
A A cos j A sin A cos jsin
b A a b arctan a
2 2
A a jb
•Exponential form
Euler’s
Identity
Im b |A|
A
e
j
cos j sin

0
a
Re
A A cos j sin Ae
j
•Three forms
Imaginary part
a Re[A]
b Im[ A] 2 2 2 2 2 Magnitude a b A A a b b Angle (相角 ) arctan a 2 Conjugate A a j b
A A AA A
•Polar form
•Phase Relationships
•If V1 leads V2 by φ and V1 =V2=V, then j1
V1 Ve Ve e V2 e j90 V jV •If φ =90°, then V1 2 2
j
j( 2 )
•Phase Relationships
•Then when standing at a fixed point, if V1 arrives first followed by V2 after a rotation of θ , we say that V1 leads V2 by θ . •Alternatively, we could say that V2 lags V1 by θ . (Usually, we take θ≤180o as the smaller angle between the two phasors.)
v(t ) 100 cos( t ) 100
A voltage given by v (t ) 100 cos(100t )is applied to a 50Ω resistance. Find the rms value of the voltage and the average power delivered to the resistance.
•The importance of steady-state sinusoidal analysis

Electric power transmission and distribution by sinusoidal currents and voltages

Sinusoidal signals in radio communication


Rectangular form
Polar form

Exponential form

Arithmetic operations of complex numbers
•Rectangular form
A a jb

Im
b
A
|A|
Imaginary unit
j
1
0

a
Re


Real part

All periodic signals are composed of sinusoidal components according to Fourier analysis
5.1 Sinusoidal Currents and Voltages
•Parameters of Sinusoidal Currents and Voltages
A B
and subtracting
C A B e1 je 2
e1 a c e2 b d
•Arithmetic Operations
Given
Product
A a jb
B c jd
j A
C A B A e Ce
j C
Be
j B
θ
•The phasor for a sinusoid is a ‘snapshot’ of the corresponding rotating vector at t = 0.
•Phase Relationships
•To determine phase relationships from a phasor diagram, consider the phasors to rotate counterclockwise.
•Adding Sinusoids Using Phasors
•Example: v1 t 20 cos t 45 V
v2 t 10sin t 60 V
Find vs v1 v2 ?
Solution:
V1 20 45 V
Given
A a jb
5.2 PHASORS
•Phasor Definition
Time function: v1 t V1 cosωt θ1
Sinusoidal steady-state analysis is greatly facilitated if currents and voltages are represented as vectors or Phasors. peak Phasor: V1 V11
Vrms
Vm 2
70.71V
V 2 rms 70.712 Pavg 100 W R 50 v 2 (t ) p (t ) 200 cos 2 (100 t ) W R
5.2 PHASORS：Preparation
Complex Numbers

3 forms of comg Sinusoids Using Phasors
Step 1: Determine the phasor for each term Step 2: Add the phasors using complex arithmetic. Step 3: Convert the sum to polar form. Step 4: Write the result as a time function.
•Vm is the peak value, unit is volt •ω is the angular frequency, unit is radians per second •f is the frequency，unit is Hertz (Hz) or inverse second. •θ is the phase angle, unit is radian or degree.
Angular velocity
正弦量可以表示为在复 平面complex plane上按 逆时针方向旋转的相量 的实部real part。
•Sinusoids can be visualized as the real-axis projection of vectors rotating in the complex plane.
•Parameters of Sinusoidal Currents and Voltages
1 Frequency f T Angular frequency 2 T 2f
To uniformity, This textbook expresses sinusoidal functions by using cosine function rather than the sine function.
•rms Phasor or effective phasor: Vrms1 Vrms11 V1 2
•In this book, if phasors are not labeled as rms, then they are peak phasors.
•Phasors as Rotating Vector
Rectangular form
A a jb
A | A |
A | A | e
j
Polar form Exponential form
•Arithmetic Operations
Given
Identity Adding
A a jb
B c jd a c, b d
vs t 29.97 cos t 39.7 V

•Adding Sinusoids Using Phasors