The switch in the following circuit closes at time t- 0. The voltage source is a ramp input of slope one (t) t). The inductor has a value of 1 H and the capacitance c is 0.25 (v( a. Find the transfer function relating the input voltage to the current in the circuit. b. Design the resistor such that the system is underdamped with a damping ratio of c. sketch the location of the poles (label with X) and zeros (label with of the transfer function. d. Sketch the time response for the current. Label the steady state current, maximum percent overshoot, peak time, and settling time. CCOO
Solution
Transfer function V(t)/I(t) = Z(t) = R + jwL + 1/jwC
or R + Ls + 1/Cs
(RCs + LCs 2 + 1 )/Cs
(s 2 + Rs/L + 1/LC)/s/L
L =1 and C= 0.25
s 2 + Rs + 4 = 0
Here w n 2 = 4 so w n = 2
2(zeta)w n = R
2(1.414) = R
R = 2.828 ohm
Poles are P 1 = 0 and Zeroes Z 1 = -1.414 +1.414 i and Z 2 = -1.414 -1.414 i
For t <0 switch is open i.e. I L (0 - ) = 0 =Â Â I L (0 + )
And Vc(0 - ) = 0 = Vc(0 + )
Peak overshoot = e -(zeta)(pi)/sq root(1-(zeta) 2
= 4.33 %
.
APM Welcome, APM North West Network Conference, Synergies Across Sectors
The switch in the following circuit closes at time t- 0- The voltage s.docx
1. The switch in the following circuit closes at time t- 0. The voltage source is a ramp input of slope
one (t) t). The inductor has a value of 1 H and the capacitance c is 0.25 (v( a. Find the transfer
function relating the input voltage to the current in the circuit. b. Design the resistor such that the
system is underdamped with a damping ratio of c. sketch the location of the poles (label with X)
and zeros (label with of the transfer function. d. Sketch the time response for the current. Label
the steady state current, maximum percent overshoot, peak time, and settling time. CCOO
Solution
Transfer function V(t)/I(t) = Z(t) = R + jwL + 1/jwC
or R + Ls + 1/Cs
(RCs + LCs 2
+ 1 )/Cs
(s 2
+ Rs/L + 1/LC)/s/L
L =1 and C= 0.25
s 2
+ Rs + 4 = 0
Here w n
2
= 4 so w n = 2
2(zeta)w n = R
2(1.414) = R
R = 2.828 ohm
Poles are P 1 = 0 and Zeroes Z 1 = -1.414 +1.414 i and Z 2 = -1.414 -1.414 i
For t <0 switch is open i.e. I L (0 -
) = 0 =Â Â I L (0 +
)
And Vc(0 -
) = 0 = Vc(0 +
)
Peak overshoot = e -(zeta)(pi)/sq root(1-(zeta) 2
