Download Second Order Circuits

Second Order Circuits
ES-3
Download:
http://www.ece.tufts.edu/~hopwood/downloads/es3/SecondOrderCircuits.ppt
Second Order Circuits
• Contain two independent reactive
components
• This results in a second order differential
equation containing d2i/dt2 or d2v/dt2
Example: Series RLC circuit
L
VS(t)
i(t)
+
C
KVL:
VL + VR + VC = Vs
R
vL (t )  vR (t )  vC (t )  vS (t )
t
di
1
L  iR   i (t )dt  vC (0)  vS (t )
dt
C0
dvS
d 2i
di i (t )
L 2 R 
0 
dt
dt C
dt
d 2i R di i (t ) 1 dvS



2
dt
L dt LC L dt
d 2i
di
1 dvS
2
 2  0 i (t ) 
2
dt
dt
L dt
R
1
...where  
and 0 
.
2L
LC
Find the natural response : vS  0
d 2i
di
2

2



0 i (t )  0
2
dt
dt
di
d 2i
2



2

0 i (t )  0
2
dt
dt
try a solution in the form :
i (t )  kest
substituting :
s 2 kest  2skest  02 kest  0
giving :
s 2  2s  02  0
Use the quadratic equation :
 2  (2 ) 2  4(1)02
    2  02
s
2(1)
Three Cases
• Case 1 - Overdamped: >o  large R:
R
>
2L
1
LC
s1     2  02  0
s2        0
2
2
0
The natural response is two decaying exponentia ls :
in (t )  K1e s1t  K 2 e s2t
Three Cases
• Case 2 - Underdamped: o  small R:
R

2L
1
LC
s1     2  02     1 02   2    j 02   2
s2     2  02    j 02   2
The natural response is two decaying phasors
in (t )  K1e s1t  K 2 e s2t
t
 K1e e
j ( 02  2 ) t
2
2
t  j ( 0  ) t
 K 2e e
 Ke t cos(nt   )...where n  02   2
Three Cases
• Case 3 – Critically damped: o
R

2L
1
LC
s1          0
2
2
0
s2        0    0
2
2
0
The natural response is
in (t )  K1e t  K 2tet
Examples
L
VS(t)
L = 10uH
i(t)
+
C
C = 10nF
R
o = 3.16x106 rad/sec (504 kHz)
R = 10, 63.2, and 1000 ohms
 = 5x105, 3.16x106, and 5x107 s-1
vs
t
Underdamped
Critically Damped
Overdamped
Overdamped: R=1000W
10V
5V
0V
0s
0.2us
0.4us
0.6us
0.8us
1.0us
V(R1:2)
Time
1.2us
1.4us
1.6us
1.8us
2.0us
Underdamped: R=10W
10V
5V
0V
-5V
-10V
1us
0s
2us
3us
4us
5us
V(R1:2)
Time
6us
7us
8us
9us
10us
Critically Damped: R=63.2W
10V
5V
0V
-5V
1us
0s
2us
3us
4us
5us
V(R1:2)
Time
6us
7us
8us
9us
10us
Mechanical Analogy
Wikipedia.com
Automobile:
Mass
 inductor
Suspension:
Spring  capacitor
Shock Absorber: Damper  resistor
Force  voltage
Velocity  current