Adnan Menderes University
Elec. & Electronics Eng. Dept.
EE209 Circuit Laboratory I
2015-2016
EXPERIMENT 9
SECOND ORDER CIRCUITS
OBJECTIVES
 The square wave responses of RLC circuits are studied,
 Resistor value will be observed experimentally for the critically damped response,
 Voltages across the components (R,L,C) are obtained using DSO for underdamped,
overdamped and critically damped responses.
EQUIPMENT LIST
1. Digital Storage Oscilloscope (GW Instek GDS-1072A-U)
2. Function/Arbitrary Waveform Generator (AATech AWG 1020A)
3. Breadboard
4. Resistor: 1k (POT)
5. Capacitors: 100nF
6. Inductor : 1 mH
PRELIMINARY WORK
1) Consider the series RLC circuit of Fig. 9.1.
C = 100 nF
L = 1 mH
Vin(t) = Vpeak.u(t) volts
Fig. 9.1
a. Show that the differential equation for VC(t) is given as,
d2
R d
1
1
Vc(t ) 
Vc(t ) 
Vc(t ) 
Vin(t )
2
dt
L dt
LC
LC
The characteristic equation for this differential equation is
s 2  2s   20  0
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 of the characteristic equation are repeated the response is called critically
damped. Determine the value of the resistor R = Ro 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 > Ro (overdamped response):
i.
Solve the differential equation and show that the variables VC(t) and iL(t) are given
below.
Vpeak
Vc(t )  Vpeak 
( s2 e s1t  s1e s2 t ),
s1  s2
1 Vpeak s1t
(e  e s2 t ),
L s1  s2
where s1 and s2 are the real distinct roots of the characteristic equation.
i L (t ) 
ii.
Determine VL(t), VR(t), iC(t) and 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 = 3Ro.
e. For R = Ro (critically damped response):
i.
Show that the variables VC(t) and iL(t) are given as,
Vc(t )  Vpeak  Vpeak ( s0 t  1)e s0t ,
Vpeak s0t
te ,
L
where s0 is the only (repeated) root of the characteristic equation.
i L (t ) 
ii.
Repeat Part 1.d.ii. for R = Ro.
iii.
Repeat Part 1.d.iii. for R = Ro.
f. For R < Ro (underdamped response):
i.
Show that the variables VC(t) and iL(t) are given as,
Vc(t )  Vpeak  Vpeak
i L (t ) 
 0 t

e sin( d t  arctan d ),
d

Vpeak t
e sin( d t )
d L
where s1    j d and s2    j d are the complex conjugate roots of the
characteristic equation and  0 
ii.
iii.
1
.
LC
Repeat Part 1.d.ii.
Repeat Part 1.d.iii for R = Ro/4.
EXPERIMENTAL WORK
1. Set up the circuit of Fig. 9.1. Adjust the square wave output of the signal generator so that
Vin(t) is a 2 Vp-p square wave with 500Hz frequency.
a) Determine experimentally R = Ro 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.
ii.
iii.
R0 (min. position of POT)
R=R0
R1k (max. position of POT)