chpt8_正弦

Addition and subtraction of phasors of the same frequency can be performed by phasor diagram, which plays an important role in sinusoidal steady-state analysis.
u1 ( t ) u2 (t ) 2 U 1 sin( w t + Ψ 1 ) Im( 2 U 2 sin( w t + Ψ 2 ) Im(

2U1e

jw t
) ) e
jw t
2U
2
e
jw t

u ( t ) u 1 ( t ) + u 2 ( t ) Im( Im(
differentiation:
di dt
di dt

d dt

2 I sin( w t + i )

)
id t

2 I sin( w t + y i )d t 2 I cos( w t + y i ) ω 2I sin( w t + y i ) ω 2
2 I cos( w t + y i ) w 2 w I sin( w t + y i +

2U1e

jw t
) + Im(
2U

2
)
jw t
2U1e
jw t
+
2U
2
e
jw t
) Im(
2 (U 1 + U 2 ) e

)
then： i1 i2 = i3
I1 I2 I3
U U1 + U 2
U
a b= c lga + lgb=lgc
The course is essentially a kind of transformation.
Part 2 AC Circuit
Chapter 8 Sinusoids Steady-State Analysis
Emphasis：
phase difference phasors Impedance and Admittance phasor diagram Sinusoidal steady-state analysis using phasors
I 1 T

def
I
1 T

T 0
i ( t )d t
2

T 0
I m sin ( w t + Ψ ) d t
2 2

T 0
sin ( w t + Ψ ) d t
2

T 0
1 cos 2 ( w t + Ψ ) 2
dt
1 2
t
T 0

1 2
T

I Im
1 T
Im
2T 2u ( t Nhomakorabea) Ri ( t ) + L di ( t ) dt
Find i ( t ).
First-order linear differential equation with constant coefficients
8.4 Phasor relationships for circuit elements
j
A | A | e
| A |
relationships： A=a+jb A=|A|ej =|A|
| A | a 2 + b 2 b θ arctg a
Im Rectangular form b Polar form O
A
|A|

a Re
or
a | A | cos θ b | A | sin θ
(1) amplitude (maximum)Im：the range in which a sinusoid changes
(2) angular frequency(w )：the radians that a sinusoid changes per second w 2 f 2 unit： rad/s ，弧度 / 秒 T (3) initial phase angle(y )：the starting point of a sinusoid 3. phase difference between two sinusoids of the same frequency Let u(t)=Umsin(w t+y u), i(t)=Imsin(w t+y i) then phase difference between them is： j = (w t+y u)- (w t+y i)= y u-y i
power analysis of sinusoidal circuit
8. 1 Fundamental Concepts
1. sinusoid：~ is a signal that has the form of the sine or cosine function. i + u i waveform： _ T O t Expression for instantaneous value： i(t)=Imsin(w t+y)
y/w
Period T and frequency f : T ：the time of one complete cycle f ：the number of cycles per second unit：s，秒
f =1/T
unit：Hz(Hertz)，赫(兹)
2. three elements of a sinusoid：
| A1 | | A2 | θ1 θ2
| A1 | θ 1 | A2 | θ 2
| A1 | e | A2 | e

| A1 | | A2 |
e
j( θ 1 θ 2 )
~ are better performed in polar form. (3) Rotating factor： ej =cos +jsin =1∠ ej/2 =j , e-j/2 = -j, ej= –1 , so +j, –j, -1 can be considered as rotating factor.
I
U
R
O Instantaneous power：
pR uRi
2U
R
wt
u=i
2 I sin (ω t + Ψ i )
2
U R I [1 cos 2 (ω t + Ψ i )]
II. phasor representation 1. Phasor representation of the sinusoid
magnitude of the phasor represents the rms value of the sinusoid angle of the phasor represents the initial phase of the sinusoid
equal to the initial phase difference
j >0， u leads i by j，or i lags u by j (u reaches its maximum prior to i )；
j <0， i leads u by j ，or u lags i by j ( i reaches the maximum prior to u). u, i u u and i are out of phase. i Particularly：
Note: the symbols of instantaneous value、maximum and rms value of ac voltage and current:
i , Im , I
2. rms values of sinusoidal voltages and currents Let i(t)=Imsin(w t+ )
2 jw I
2
w I y i +

idt
I w
yi
2

I jw
For the sake of expression, we definite: Phasor Phasor dI jw I differentiation: integration:
dt

j =0， in phase：
j = (180o ) ，
opposite in phase： Set： | j | (180°)
yu yi j
O
wt
8. 2 Effective value of periodic voltage or current
1. effective value
U
Im
U2 U1
U
Im
U1
U2
60

41 . 9
60

41 . 9
30


30

Re
Re
(2) . Differentiation and integration of sinusoids
i
2 I sin( w t + y i ) I I y i
integration:
(2) Multiplication and division