Experiment # 3
First Order RC and RL Filters
Objective
The objective of this experiment is to study the behavior of first-order filters, as discussed
in Chapter 6 of your textbook. The experiment will examine the basic filtering effect of
capacitors and inductors as well as the significance of capacitor/inductor placement.
Remember to pay close attention to generator frequency and resistors values used in the
RC Vs the RL networks.
Discussion of Concepts
Capacitors
Review: The impedance of a capacitor is given by ZC = 1/jωC = -j/ωC = jXC, where j
indicates that the impedance of a capacitor is a purely imaginary number, XC = -1/ωC
represents the imaginary portion of ZC and is called the reactance of ZC, and ω = 2πf
where f is the frequency measured in Hz. In this experiment we are interested in the
magnitude of the impedance of this capacitor which is given by │ZC│ = 1/ωC. The
source voltage (produced by the function generator, see Figure1) can be represented by
he following expression.
V(ω,t) = VMcos(2πft) = VMcos(ωt)
The voltage magnitude seen across the capacitor in an RC series network, Figures 1 and
2, is given by the following equation which is computed using the concept of voltage
division.
VC=│V(ZC/(R+ZC))│=VM (│ZC│/│R+ZC│)=VM(-XC/│R+jωC│)=VM(-XC/√(R2+XC2))
In a similar fashion it can be shown that the voltage across the resistor is given by
VR = VM(R/√(R2+XC2))
The source is a function of time which implies that the capacitor voltage is also a function
of time. More importantly, the equation for VC shows that the voltage VC is a function of
frequency.
As you recall, this is the basis for the transfer function equations discussed in class.
Namely, that for a first-order low pass filter, the transfer function, H(f) is given by:
H( f ) =
1

1 + j  f

f
B 

and for a first-order high pass filter, the transfer function, H(f) is given by:

j  f

 fB 
H( f ) =

1 + j  f

f
B 

where
fB =
1
2πRC
Inductors
Review: The impedance of a inductor is given by ZL = jωL = jXL, where j indicates that
the impedance of inductor is a purely imaginary number, XL = ωL represents the
imaginary portion of ZL and is called the reactance of ZL and ω = 2πf where f is the
frequency measured in Hz. In this experiment we are interested in the magnitude of the
impedance which is simply given by │ZL│ = XL = ωL. The voltage magnitude seen
across the inductor in an RL series network, Figures 3 and 4, is given by the following
equation which is found using the concept of voltage division.
VL=│V(ZL/(R+ZL))│=VM (│ZL│/│R+ZL│)=VM(XL/│R+jXL│)=VM(XL/√(R2+XL2))
In a similar fashion it can be shown that the voltage across the resistor is given by
VR = VM(R/√(R2+XL2))
The equation for VL shows that the voltage VL is a function of frequency. As above, this
is the basis for the inductor based filters discussed in class. The transfer functions for the
high and low pass filters employing inductors are the same as those given above, with the
exception that:
fB =
R
2πL
Preliminary Lab Exercise
Capacitor Circuits
For the circuit shown in Figure 1, use a spreadsheet to compute XC, VC, and VR for
frequencies from 1000 Hz to 100,000 Hz (calculate every 1000 Hz). Also compute the
magnitude and phase angle of the transfer function for each of these frequencies,
assuming that the output voltage is measured across the vertical element on the right hand
2
side of the circuit. Repeat these calculations (on a separate sheet) for the circuit shown in
Figure 2. Also calculate fB for each of the circuits.
Inductor Circuits
On a separate sheet for each circuit, compute XL, VL, and VR, |H(f)|, and <H(f) for the
circuits shown in Figures 3 and 4. Do this over the same range of frequencies you used
for the capacitor circuits above. Again, calculate fB for each of these circuits.
Cascade Circuits
On a separate sheet for each circuit, calculate the overall transfer function (at each of the
frequencies analyzed above) for the cascaded circuits shown in Figures 5 and 6 below.
Do this both by multiplying the individual transfer functions (of the two original circuits)
and by analyzing the cascaded circuit and calculating the expected output voltage
(Again, across the rightmost element, the Resistor R2 in Figure 5 and the capacitor in
Figure 6). If your results differ, try to explain why.
Procedure
Capacitor Circuits
1. Construct the circuit of Figure 1.
R
1k
V(w,t)
C
0.1u F
Figure 1
2. With VM set to 5 V, set the frequency of the function generator to various values in
order to verify the responses you calculated using your spreadsheet. You do not need
to test this at all 100 values, but collect enough data to verify your calculations. This
obviously requires the collection of more data in regions of the curve where things
seem to be changing rapidly, but you can get away with collecting less at other
3
frequencies. At each frequency, compare V(ω,t) and VC by measuring the peak
voltage of VC and the phase shift of VC with respect to the generator voltage V(t).
3. Modify the circuit shown in Figure 1 to resemble that of the circuit shown in Figure
2. For this experiment we wish to look at the phase shift of the voltage across the
resistor compared to the source voltage. The current through the resistor is I = VR/R
and it will have the same phase as VR with respect to the source voltage. This is the
key point to this portion of the experiment, for the current through the resistor is also
the current through the capacitor. In addition, this is the same current as in the RC
circuit of Figure 1. For the same frequencies used above, take measurements of the
source and resistor voltages.
C
0.1 uF
R
V(w,t)
1k
Figure 2:
Inductor Circuits
1. Construct the circuit of Figure 3.
R1
10k
1
L1
10mH
V(w,t)
2
Figure 3
2. With VM set to 5V, set the frequency f of the function generator to the appropriate
values, as determined by a procedure similar to that you used in step 2 for the circuit
4
of Figure 1. Note that although the procedure is similar, the frequencies of interest
will not be. At each frequency, compare V(ω,t) and by measuring the peak voltage of
VL and the phase shift of VL with respect to the generator voltage V(ω,t).
3. Modify the circuit shown in Figure 3 to resemble that of the circuit shown in Figure
4. For this experiment we wish to look at the phase shift of the voltage across the
resistor compared to the source voltage. The current through the resistor is I = VR/R
and it will have the same phase as VR with respect to the source voltage. This is the
key point to this portion of the experiment, for the current through the resistor is also
the current through the inductor. In addition, this is the same current as in the RL
circuit of figure 3. For the same frequencies used above, take measurements of the
source and resistor voltages.
L1
1
2
10mH
R1
V(w,t)
10k
Figure 4
Cascade Circuits
1. Construct the circuit shown in Figure 5 (which replaces the input voltage of Figure 4
with the output of Figure 2). One might assume that this will produce a filter whose
transfer function is the product of the transfer functions of the two original circuits.
C
L1
1
0.1uF
2
10mH
R1
v(w,t)
10k
R2
1k
Figure #5: Cascade Filter (Cap HPF with Inductor LPF)
5
2. With VM set to 5V, set the frequency f of the function generator to the appropriate
values, as determined by a procedure similar to that you used in step 2 for the circuit of
Figure 1. Note that although the procedure is similar, the frequencies of interest will not
be. At each frequency, compare V(ω,t) and by measuring the peak voltage of VR and the
phase shift of VR with respect to the generator voltage V(ω,t).
3. Construct the circuit shown in Figure 5 (which replaces the input voltage of Figure
with the output of Figure 3). Again, one might assume that this will produce a filter
whose transfer function is the product of the transfer functions of the two original
circuits.
R1
R2
1
10k
1k
L1
v(w,t)
10mH
C
0.1uF
2
Figure #6: Cascade Filter (Inductor HPF with Cap LPF)
4. With VM set to 5V, set the frequency f of the function generator to appropriate values,
as with the earlier portions of the experiment. At each frequency, compare V(ω,t) and by
measuring the peak voltage of VC and the phase shift of VC with respect to the generator
voltage V(ω,t).
Analysis
Capacitor Circuits
For the analysis portion of this experiment use the graphing capabilities of your
spreadsheet software to plot both VC and VR as a function of frequency (this should be
done both for the measured and calculated values). Compare the phase shift of the
capacitor voltage with the phase shift of the resistor voltage. Draw conclusion
6
concerning the current through and voltage across a capacitor for the phase shift data.
Which RC circuit, Figure 1 or 2, looks like a low pass filter and which looks like a high
pass filter? Compare the magnitude of the appropriate transfer function with your
measured results and explain any differences.
Inductor Circuits
For the analysis portion of this experiment plot (use spreadsheet software) both VL and
VR as a function of frequency (this should be done both for the measured and calculated
values). Compare the phase shift of the inductor voltage with the phase shift of the
resistor voltage. Draw conclusion concerning the current through and voltage across the
inductor for the phase shift data. Which RL circuit, Figure 3 or 4, looks like a low pass
filter and which looks like a high pass filter? Compare the magnitude of the appropriate
transfer function with your measured results and explain any differences.
Cascade Circuits
Cause your spreadsheet program to plot your estimated output based on multiplying the
transfer functions, the estimate based on circuit analysis of the entire circuit, and the
observed results on the same axis. Which set of predictions do the observed values match
more closely? Why are the two predictions so different (Hint, re-read your class notes
and the sections of Chapter 6 where we derived the transfer function equations and
review all assumptions made).
7