Download Preliminary Work

MECE 232
Lab0
Second Order Circuits
Preliminary Work:
1) Consider the series RLC circuit of Fig.1.
C = 4.7 nF
L = 0.1 H
Vin(t) =Vpeak. u(t) volts
Fig. 1
a. Show that the differential equation for Vc(t) is given as,
d2
R d
1
1
V (t ) 
VC (t ) 
VC (t ) 
Vin (t )
2 C
L dt
LC
LC
dt
The characteristic equation for this differential equation is,
The roots of the characteristic equation are denoted by s1 and s2 when they are distinct
and by s0 when the roots are repeated.
b. When the roots if the characteristic equation are repeated the response is called
critically damped. Determine the value of the resistor R=R0 for the critically damped
response.
c. Determine the values of the currents and voltages at t  0  , t  0  and t   for all
circuit elements.
d. For R>R0 (overdamped response):
i. Solve the differential equation and show that the variables Vc(t) and iL(t) are given
below.
VC (t )  V peak 
i L (t ) 
V peak
s1  s 2
( s 2 e s1t  s1e s2t ) ,
1 V peak
(e s1t  e s2t )
L s1  s 2
where s1 and s2 are the real distinct roots of the characteristic equation.
ii. Determine VL(t), VR(t), ic(t), iR(t) using the above results.
iii. Determine and sketch VL(t), Vc(t), VR(t), ic(t), iL(t) and iR(t) for R=3R0.
e. For R=R0 (critically damped response):
i. Show that the variables Vc(t) and iL(t) are given as,
VC (t ) V peak V peak (s0 t 1) e s0t ,
i L (t ) 
V peak
t e s0 t
L
where s0 is the only (repeated) root of the characteristic equation.
ii. Repeat Part 1.d.ii for R=R0.
iii. Repeat Part 1.d.iii for R=R0.
f. For R<R0 (underdamped response):
i. Show that the variables Vc(t) and iL(t) are given as,
VC (t )  V peak  V peak
i L (t ) 
V peak
dL
 0 t
d
e sin ( d t  arctan
),
d

e t sin ( d t )
where s1     j d and s 2     j d are the complex conjugate roots of the
characteristic equation and  0 
ii. Repeat Part 1.d.ii.
iii. Repeat Part 1.d.iii for R=R0/4.
1
.
LC
2) Consider the parallel RLC circuit of Fig. 2.
R1 = 10 KΩ
C = 0.1 μF
L = 0.1 H
Vin(t) = Vpeak. u(t) volts
Fig. 2
a. Obtain the differential equation for Vc(t).
b. Determine the value of R=R0 for the critically damped response.
c. Determine the value of Vc(t) at t  0  , t  0  and t   .
EXPERIMENTAL WORK
1) Set up the circuit of Fig. 1. Adjust the square wave output of the signal generator so
that Vin(t) is a 2 Vp-p square wave with 150Hz frequency.
a. Determine experimentally R=R0 for the critically damped response.
b. Observe and plot (Ac components only) Vin(t), VL(t), VR(t) and VC(t), also note the
DC levels for the following cases:
i. R=3R0 ,
ii. R=R0 ,
iii. R=R0/4
2) Set up the circuit of Fig.2. and repeat Part 1.
3) Set up the following circuit for f = 10kHz. Observe and plot V0(t) for the following
cases:
a. S1 and S2 open. b. S1 closed and S2 open. c. S1 and S2 closed.
R1 = 10 KΩ
R2 = 10 KΩ
C = 0.1 μF
L = 0.1 H