电路原理课件讲义英文版 Chapter_7

Conclusion qc (0+) = qc (0－) vC (0+) = vC (0－)
L (0+)= L (0 )
－
iL(0+)= iL(0－)
4. Determination of initial condition 例1
求 iC(0+) i 10k 40k 10V k iC (1) 由0－电路求 uC(0－)或iL(0－)
iC(0+)
例2
1 K
t = 0时闭合开关k , 求 uL(0+) 4 1
解
4
先求
i L (0 )
电 感 短 路
L
iL
+
uL
10V
0+电路
-
10V
1
4
10 i L (0 ) 2A 1 4

+
10V 2A
uL
uL (0 ) 0 uL (0 ) 0
由换路定律:
w p t
2. Derivation of differential Equations
RC circuit By KVL, there is vs(t) (t >0) R +
i
vC
–
C
dvc Ri vc vS (t ) i C dt dvc RC vc vS (t ) dt
Taking voltage as circuit variable:
(t >0) R + vs
i
vL
–
L
R vL dt vL vS (t ) L dvL dvS (t ) R vL L dt dt
Firstorder circuit
Ri vL vc vS (t )
dvc iC dt
2
(t >0) ＋ vS(t) － － R +
i
di vL L dt
vC
C
vL
＋ –
L
d vc dvc LC 2 RC vc vS (t ) dt dt
Taking current as circuit variable:
di 1 Ri L idt vS (t ) dt C
High-order circuit
dnx d n 1 x dx an n an1 n1 a1 a0 x e( t ) t 0 dt dt dt
3. Initial condition
(1) Concepts of t = 0＋ and t = 0－ A circuit is switched at t=0 0- : the moment just before switching 0+ : the moment just after switching
dvS (t ) di d i 1 R L 2 i dt dt C dt
2
Second -order circuit
First-order circuit
dx a1 a0 x e ( t ) t 0 dt
Second-order circuit
d2x dx a2 2 a1 a0 x e( t ) t 0 dt dt
1. The circuit comprises energy storage elements of capacitors or inductors.
2. Energy stored will change when the circuit changes, and it needs time to store or release energy
f (0 ) lim f ( t )
t 0 t 0

f (0 ) f (0 )
f(t)
f (0 ) lim f ( t )
t 0 t 0
f (0 ) f (0 )
t
0－0 0＋
Initial condition: values of v, i and their derivative at t = 0＋
Taking current as circuit variable:
Firstorder circuit
1 Ri idt vS (t ) C
di i dvS (t ) R dt C dt
RL circuit
di vL L Ri vL vS (t ) dt di Ri L vS (t ) dt
7.1 Introduction
RC circuits: a circuit comprising a resistor and capacitor RL circuits: a circuit comprising a resistor and an inductor
A first-order circuit: is characterized by a first-order

i() is finite, so
In addition,

0

0
i ( )d 0 vC (0+) = vC (0－)
q (0+) = q (0－)
(3) Initial condition of RL circuit
iL
+
v
L
1 iL (t ) L
1 L
0
t
v( )d
t 0

-
1 v( )d L

v( ))d
When t = 0+
1 t iL (0 ) v( )d L 0 1 0 iL (0 ) iL (0 ) v( )d L 0
0

v() is finite, so
v( )d 0 0 iL(0＋)= iL(0－) LiL L (0＋)= L (0－)
i
t
Former steady stateLeabharlann Transient Process
New steady state Transient term
Circuit switching
The construction or state of a circuit is changed
Why does transient process exit?
1. Analysis of transient process
RL circuit (t = 0) Vs
K
i
R +
The circuit is in steady state (稳定 状态)before switch K is closed.
i = 0 , vL = 0
L
vL
–
The circuit will reach new steady state after K has been closed for a long.

uL(0+)= - RIS
例3
求K闭合瞬间各支路电流和电感电压 由0－电路得： 2 解
+
48V
+
K
L
iL
uL 2
3 C
+
2 iL
3
-
-
48V
-
2 + uC －
由0+电路得：
i L (0 ) i L (0 ) 48 / 4 12 A
uC (0 ) uC (0 ) 2 12 24V
Recall
dvC Capacitor: iC C dt 1 t vC iC dt C diL vL L dt 1 t iL vL dt L 1 t vC iC dt vC (t0 ) C t0
Inductor:
1 t iL vL dt iL (t0 ) L t0
differential equation (一阶微分方程).
A second-order circuits: is characterized by a secondorder differential equation (二阶微分方程). It consists of resistors and the equivalent of two energy storage elements.
---Lord Kelvin
Chapter 7 First-order Circuits
7.1 Introduction 7.2 The Source-free RC Circuit 7.3 The Source-free RL Circuit 7.4 Singularity Functions 7.5 Step Response of an RC Circuit 7.6 Step Response of an RL Circuit
(t →) R + Vs
i
vL= 0, i=Vs /R
L
i
VS
vL
–
VS/R
?
VL
0 Transient process t1 New steady statet Former steady state (过渡过程) Transient term
RC circuit
(t = 0) Vs
K
i
R
The circuit is in steady state before switch K is closed.
(2) Initial condition of RC circuit
i
+ vc -