Tài liệu Circuits & Electronics P15 docx

6.002 Fall 2000 Lecture
1
15
6.002
CIRCUITS
AND
ELECTRONICS
Second-Order Systems
6.002 Fall 2000 Lecture
2
15
Second-Order Systems
C
A
B
5V
+
–
5V
C
GS
large
loop
2KΩ
50Ω
2KΩ
Demo
Our old friend, the inverter, driving another.
The parasitic inductance of the wire and
the gate-to-source capacitance of the
MOSFET are shown

[Review complex algebra appendix for next class]
S
6.002 Fall 2000 Lecture
3
15
Second-Order Systems
C
A
B
5V
+
–
5V
C
GS
large
loop
2KΩ
50Ω
2KΩ
Demo
+
–
5V
C
GS
2KΩ
B
L
Relevant circuit:

S
6.002 Fall 2000 Lecture
4
15
Now, let’s try to speed up our inverter by
closing the switch S to lower the effective
resistance
t
v
A
5
0
v
B
0
t
v
C
0
t
Observed Output
2
kΩ
2kΩ
6.002 Fall 2000 Lecture
5
15
t
v
A

5
0
v
B
0
t
v
C
0
t
Observed Output
~50Ω
50
Ω
Huh!
6.002 Fall 2000 Lecture
6
15
v
,
i
state variables
+
–
C
L
+
–
)(
tv

)(tv
I
)(
ti
Node method:
dt
dv
Cti
=)(
dt
dv
Cdtvv
L
t
I
=−
∫
∞−
)(
1
2
2
)(
1
d
t
vd
Cvv
L
I

=−
I
vv
d
t
vd
LC =+
2
2
time
2
dt
di
Lvv
I
=−
idtvv
L
t
I
=−
∫
∞−
)(
1
Recall
First, let’s analyze the LC network
6.002 Fall 2000 Lecture
7
15

Recall, the method of homogeneous and
particular solutions:
( ) ( )
tvtvv
HP
+=
1
2
3
Find the particular solution.
Find the homogeneous solution.
L
4 steps
The total solution is the sum of the
particular and homogeneous.
Use initial conditions to solve for the
remaining constants.
Solving