Figure 1
Objective:
To investigate the step response and frequency characteristics of RLC circuits. To investigate relation between the characteristics of the time and frequency response of an RLC circuit.
Content:
Figure 1
Consider
the RLC circuit given in Figure 1 with R = 50 ohm,
C = 0.1 uF and L = 1mH. You are asked to investigate:
1)
1) The step response. That is: the outputs: vRout(t),
vCout(t) and vLout(t) if
our input vin(t)
= u(t).
2) 2) The frequency characteristics. If the input vin(t) is a sinusoidal signal with a certain frequency, the three outputs are sinusoidal waves of the same frequency. However, with the magnitude of the input signal unchanged, the magnitude (and the phase) of the output changes with the frequency of the input signal. The circuit selects different frequencies to pass through the filter.
Pre-lab:
a)
a)
Predict how the output signals
vRout(t), vCout(t)
and vLout(t) would vary, without analysis.
b)
b)
Try to predict how the frequency response of VRout(f),
VCout(f) and VLout(f) would
vary from f
=
0
Hz to f
=
infinity
without analysis. (Textbook P
718).
c)
c) For the RLC circuit, Find the transfer function of VRout(s)/Vin(s),
VCout(s)/Vin(s), VLout(s)/Vin(s),
respectively.
d)
d)
Use the Matlab function step(num,
den) to get the
step response for the three outputs. Check your prediction in a) and explain the
result.
e)
e) Use the
Matlab function freqs(num,
den) to get the frequency
response of the three outputs. Check
your prediction in b) and explain your result.
f)
f) Let R = 400 ohm, with C and L unchanged, repeat c) d) e). Compare the
time and frequency response curves with what you got in d) and e). Try to
explain the differences you observe.
g)
g) Let R = 50 ohm, C = 0.1 uF and L = 33 mH, repeat c) d) e).
Compare the time and frequency response curves with what you got in f). Try to
explain the differences you observe.
h) h) For the values : R = 50 ohm, C = 0.1 uF and L = 1 mH, the input is a square wave (using the Matlab function square( ) ), whose magnitude is 1 V and period T should not be less than 0.001s. Using y = lsim(num,den,x,t) to get the three responses vRout(t), vCout(t) and vLout(t). The second half cycle of square wave leads to the natural response of the RLC circuit when vC(0+) = 1 V. Try to explain the result. (Textbook P377-380)
Experimental:
1)
You should construct the circuit with proper components and input square
wave from the function generator, measure and record input signals and the three
output waveforms by oscilloscope.
2) Input sinusoidal signals from function generator, observe and record the magnitude of three outputs as the frequency varies. Plot frequency—magnitude relation curves of the circuit.