电工学原理及应用(Electrical Engineering)经典双语详解讲义 (4)

How to get initial value
S t=0 C
Exa 4.2
R2 L
+ U -
R1
(a)
Knowns：before circuit changing，C、L have no energy; Find：the currents and voltages of all elements。
utransient C
E
New steady state
Old steady state
t
Why the transient response happens? Resistive circuit
K + E _ R t=0 I
I
No transient
•Resistor is a energy-consumption element, current is proportional to voltage, no transient response will happen even if changing source
+ u (0+) –C t=0＋
uC (0 ) 8 iR (0 ) 1m A R2 8
R 8k 2

How to get initial value
•Exercise 2: Given by R1=4Ω, R2=6Ω, R3=3Ω, C=0.1µ F, L=1mH, US=36V, switch S is closed for a long time. Open the switch S when t=0, how to get the initial values of all elements?
•Homework
•P4.2 •P4.6 •P4.8
4.3 First-order RC Circuits

First-order circuit Only one (equivalent) capacitor or inductor is included in a linear circuit. Equivalent circuit of First-order circuit Two parts: one (equivalent) capacitor or inductor; a two terminal network with resistance and sources. L
Answer: vx =5V, ix = 1A
t>>0
Exercise 4.3 Find steady-state values of labeled currents and voltages for t>>0.
Answer: va =50V, ia = 2A
i1 = 2A, i2=1A, i3=1A
Solu (1)Brofore switch
uC (0 ), i L (0 )
uC (0 ) uC (0 ) 0
We know that uC (0 ) 0, i L (0 ) 0 By the law of circuit changing
L (0 ) L (0 ) 0
Example 4.2
S C R2
iC (0+ ) uC (0+) u2(0+_ )
+
+ U -
t=0 R1 (a) circuit L U
+
-
i1(0+ )
R1
+ + u _ 1(0+) _ uL(0+)
R2
iL(0+ )
(b) t = 0+ circiut
(2) By the circuit at t=0+, Find other unknowns
Energy
can not change instantly because of accumulating or decaying period.
1 2 Electric field energy（Wc Cu ） C 2 uC Change WC Charging or discharging
When t=0-, capacitor is considered as open circuit, we get equivalent circuit.
R 1 4k
8 uC (0 ) 12 8 V 48
12V
8k
u (0 ) C – t=0-

How to get initial value
Transients
•The time-varying currents and voltages resulting from the sudden application of sources, usually due to switching. •By writing circuit equations, we obtain integrodifferential equations. These equations can be converted to pure differential equations by differentiating w.r.t time. •The study of transients require us to solve differential equations .

N
or
N
C
4.3 First-order RC Circuits

According to Thevenin Law
N
L
or
N
C
iL + R U + uL L R U uC
iC C
-
-
பைடு நூலகம்

Differential equation of first-order RC circuit
iL + U R uL L + U R uC C
4.2 Initial state and steady state
t=0t=0
t=0+
t
Assume changing circuit when t=0, then t=0– is end point of old steady state; t=0+ is the start point of transient state.
R 1 4k 12V
R1 4k 12V 8k u (0–) C
K
t=0
i R
8k R 2 2mF
u C
8 uC (0 ) 12 8 V 48
u C ( 0 ) u C ( 0 ) 8V
substituting voltage source for uC(0+)
i (0+) R
gradually
K + _E R
uC
E C
uC
t
Energy
can not change instantly because of accumulating or decaying period.
1 2 Magnetic field energy （W LiL ） L 2 WL i Change
iC 、uL alter suddenly

How to get initial value
R 1 4k 12 V
K t=0 i R 8k R 2 2m F u C
Exercise 1: Assuming old circuit is in DC steady state before switch K is closed. how to get uC(0+),iR(0+)? Solution:
Answer: Uc(0)=12V, iL(0)=4A,iR(0)=2A
Conclusions
1. At the moment of changing circuit，uC、 iL can not alter suddenly, but ic, uL can change suddenly . 2. Before circuits changed, if energy-storage elements have no energy, just after circuits changed (t=0 +) the capacitance and inductance are viewed as short circuits and open circuit, respectively.
2. Replace inductances with short circuits.
3. Solve the remaining circuit using methods in chapter 2.
Example 4.1 Find steady-state values of vx and ix in this circuit for t>>0.
WL (0 ) WL (0 ) WC (0 ) WC (0 )
i L ( 0 ) i L ( 0 ) uC (0 ) uC (0 )
The law of changing circuit (换路定则)
From t=0–to t=0+,iL 、 uC can not change suddenly.
Electrical Engineering and Electronics II
Chapter 4 Transients
4 Course Hours
4.Transients