Joseph Walsh
Modern Instrumentation
750:327:02
February 7, 2004
Lab 2 – AC Intro and Voltage Divider
Introduction: The purpose of this lab was to learn about the AC voltage divider and differentiator, and to gain experience working with the oscilloscope. In general, the AC voltage divider is a lot like the DC case, with a few changes. First of all, there is an AC source rather than a DC source, and the two passive elements need not both be resistors, but rather anything with impedance. Impedance is a complex quantity represented by Z. The impedance of a capacitor is:
Z = -j / (ωC) where ω is the frequency of the input voltage, as measured in radians per second, and j
2
= -1.
The impedance of a resistor is just the same as the resistance:
Z = R
The impedance for any element is measured in ohms, and is just the generalized form of resistance. Therefore, most of the old formulae that dealt with resistance can be applied using impedance instead. In particular, this applies to the output voltage formula for a voltage divider. where the voltage divider is set up as according to the diagram.
The type of AC voltage divider we will look at, the differentiator, consists of an AC (sinusoidal) voltage source, a capacitor, and a resistor. The load is placed in parallel with the resistor. It is called a differentiator because, if the input voltage is varied (over a long time comparable to the time constant), the graph of the output voltage looks like the derivative of the input.
Theoretically, in an RC (resistor-capacitor) circuit with a constant direct current voltage source, the current through the circuit follows the form: i = (V/R)*e -t/τ where τ = RC is the time constant.
The oscilloscope is a useful tool for analyzing the shapes of waves. It can pick up a varying electronic impulse and uses a light to draw the corresponding wave form on a screen.

The first part of this lab was to familiarize ourselves with the scope, so we would know what we were doing in the actual experiment. We hooked the function generator to an input of the oscilloscope and adjusted the controls in order to see the effect of the wave on the screen. When we felt comfortable with the controls, we moved on to the experiment, testing an AC voltage divider.
Part A:
Methods: We chose the capacitor and resistor that we were to use in the system and measured their actual capacitance and resistance as compared to the theoretical values. We set up the AC voltage divider on the circuit board, using the oscilloscope to measure the output voltage in parallel with the resistor. The function generator was set to produce a sinusoidal wave, and we varied the frequency to see the effect it had on the output voltage waveform as compared to the input. The magnitude of the input voltage we used was 4.8 V. The function generator displayed the frequency, f, of the sinusoidal input wave in Hertz, or cycles per second.
In order to convert this to radians, we used the basic relation:
ω = 2πf
We measured the output voltage at frequencies of 100 Hz, 300 Hz, 1 kHz, and so on; with this spacing, we would get roughly even spacing on a logarithmic scale. To record the output voltage, we recorded from the oscilloscope graph both the amplitude of the output wave and the phase difference between the output and input waves. The theoretical amplitude and phase difference were calculated using the theoretical output voltage and the formulae:
A = |V out
| = √ (Re(V out
)) 2 + (Im(V out
)) 2 and 3)
δ = arctan( Im(V out
)/Re(V out
) )
Lastly we plotted the data in a variety of ways to make it easier to examine. (Figures 1, 2,
Results:
V in
= 4.8 V
Element
Capacitor
Theoretical Value
1000 pF
Experimental Value
1222 pF
Resistor 19980 Ω
Frequency (Hz) Impedance in R
20000 Ω
Impedance in C
100
300
1000
3000
10000
30000
100000
300000
1000000
19980 Ω
19980 Ω
19980 Ω
19980 Ω
19980 Ω
19980 Ω
19980 Ω
19980 Ω
19980 Ω
-1.30e6*j Ω
-434000*j Ω
-130000*j Ω
-43400*j Ω
-13000*j Ω
-4340*j Ω
-1300*j Ω
-434*j Ω
-130*j Ω
Theoretical Output voltage (V)
.001 + .074*j
.010 + .220*j
.110 + .719*j
.839 + 1.823*j
3.369 + 2.196*j
4.584 + .996*j
4.780 + .312*j
4.798 + .104*j
4.799 + .031*j

Frequency
100
300
1000
3000
10000
30000
100000
300000
Amplitude (V) Phase difference (radians)
Theoretical Experimental Theoretical Experimental
.074
.221
.728
2.007
4.021
4.691
4.790
4.799
.070
.22
.7
2
3.8
4.4
4.6
4.6
1.55
1.52
1.42
1.14
.58
.21
.07
.02
1.57
1.48
1.26
1.11
.60
.18
0
0
1000000 4.799 4.6 .01
Analysis: In all the graphs, the scatterplot represents the observed data, while the
0 superimposed curve represents the theoretical plot. Figure 1 plots the gain of the circuit, or the ratio of the amplitudes of the output to the input voltage. The gain is plotted against the frequency on a logarithmic scale. There does not seem to be much discrepancy between the experimental and theoretical data. As the frequency increases, the experimental data drops off slightly below the theoretical curve. This could be due to impedance in the oscilloscope, or inaccuracies in the function generator. Finally, it is obvious that this circuit is a high-pass filter, since almost all voltage at high frequencies gets through, while little voltage passes through at low frequencies.
Figure 2 plots the phase difference against the logarithmic scale of frequency. This was the part of the lab with the most human error. It was difficult estimating the phase differences in the oscilloscope’s scale if little tick marks. For the highest three frequencies, we could not discern a difference due to the actual small phase difference and the heavy weight of the line used to graph the wave forms on the oscilloscope. I cannot conclude much from my results; the theoretical curve may or may not be a good fit. Further experimentation is needed to either validate or refute this claim.
Figure 3 plots the gain against the phase difference. For all but the last point, the theoretical curve lies above and to the right of the data. This leads me to conclude that either the measured gain was too small or the phase difference too small or a combination of the two.
Looking at the closeness of fit of the gain versus the frequency leads me to believe that much of the error, especially for the low gains (less than .2), lies in the measured phase difference, which
I already commented on.
Part B:
Methods: It is important in this section to remember that a longer pulse is the same as a lower frequency, but since we will only be looking at one pulse of the wave, I will not mention frequency. The purpose of this section was to examine the circuit acting as a differentiator. In order to do this, we first set the pulse of the input wave to last relatively long to the theoretical time constant. (Recall the time constant is the resistance multiplied by the capacitance in this case.) We recorded what the input and output graph on the oscilloscope looked like in the cases that the input wave had either a square or triangular form.
Next, for the square wave input, we switched to a pulse length of about twice the theoretical time constant in an effort to empirically measure the time constant. Recall that for a

constant voltage source, the time constant is the time it takes for the current, or equivalently the voltage through the resistor, to drop to 1/e of its initial value. Also, a square wave pulse is the same as a constant direct current during the pulse. Therefore, to estimate the time constant, we measured from the oscilloscope the maximum value of the output voltage and also where the output took on 1/e times that value. The difference in time between those two events is the experimental time constant. We then took numerous readings of voltage versus time from the scope and plotted them along with the theoretical curve given by the theoretical time constant.
Results:
V in
= 4.8 V
R = 19980 Ω input voltage resistance
C = 1222 pF
τ = RC = 24.4 μs
T = 28 μs capacitance theoretical time constant experimental time constant
Analysis: As you can see from the drawings, when the time constant is very short in relation to the length of the pulse, the output voltage (shown in blue) is closely related to the derivative of the input wave function. For instance, in the case of the square wave function, the output voltage is very nearly zero except at the discontinuities, where the output voltage jumps in the direction the input voltage is changing. For the triangular wave, the output voltage shows a nearly constant positive value when the input voltage in increasing uniformly, and a constant negative value when the input is decreasing uniformly. In part A, we saw that for low frequencies of a sinusoidal input, the output voltage was π/2 ahead of the input voltage, just as the cosine function
(the derivative of sine) is π/2 ahead of the sine function.
Our experimental time constant was a little greater than our theoretical value. Figure 4, which graphs the data taken from the oscilloscope and the theoretical decay curve backs up this claim. It shows that as time passes, the experimental data is increasingly higher than the theoretical curve, that it is taking longer to decay the same amount. I can attribute some of this discrepancy to the other resistance in the circuit, such as that in the oscilloscope and function generator.

Square Wave Input
Triangular Wave Input