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

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

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。
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 .
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

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
-
-

Differential equation of first-order RC circuit
iL + U R uL L + U R uC C
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.
• For inductors with dc source, inductance behaves as short circuits.

DC Steady State Response
•The steps in determining the steady state response for RLC circuits with dc sources are: 1. Replace capacitances with open circuits.
uC (0 ) 0, changing moment，capacitance short L (0 ) 0, changing moment，inductance open
uL (0 ) u1 (0 ) U
U C (0 ) 1 (0 ) R
(C (0 ) 0) (uL (0 ) 0) u2 (0 ) 0
+ 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?
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:
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
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.
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
L gradually
K + t=0 E R iL t
iL
E/R
_பைடு நூலகம்
The
causes of transients
1. Energy storage elements
-inductors and capacitors
uC , i L
change gradually；
2.Changing circuit, such as switching source.
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
Electrical Engineering and Electronics II
Chapter 4 Transients
4 Course Hours
4.Transients
4.1 Introduction 4.2 Initial state and steady state 4.3 First-order RC Circuits 4.4 First-order RL Circuits
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.
•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
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
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
3. Before circuits changed, if uC(0-)0, the capacitance can be replaced by an ideal voltage source with uc(0+) at t=0+; if iL(0-)0, the inductance can be replaced by an ideal current source with iL(0+) at t=0+.
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.

DC Steady State Response
•The transient terms for currents/voltages decay to zero with time. •Under Steady State :
• For capacitors with dc source, capacitance behaves as open circuits.
4.1 Introduction
Conception of steady state and transient state R
K R
+ + _
E
uC
Us _
C
New steady state
uC
Old steady state
When t=∞, uc(∞)=Us
When t=0，uc(0)=0