Transfer Functions and Linear Active Networks Using Operational Amplifiers Sachin Mehta Reno Nevada

1. Transfer Functions and Linear Active Networks Using
Operational Amplifiers
Written By: Sachin Mehta
University of Nevada, Reno

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Abstract: The application of filters in the world of electronics and most everything around us in
the world is an important area that needs to be studied and improved, if possible. Different
types of filters can give the user specific responses and outputs that can be modified quite
simply and with ease, but only if the transfer function is known. The analysis of the transfer
function gives an algebraic representation of a linear, time-invariant filter—and is especially
useful in the frequency domain. The analysis of the circuit depicted in Figure 1 was completed
in various ways and the outputs were then studied and compared. For example, the circuit
shown in Figure 1 was physically put together on the Jameco breadboard, with components
provided in the Capstone Laboratory room. For example, three Bode Plots of the circuit were
assembled: one using Matlab, another from MultiSim, and the last one was generated using a
function generator and the spectrum analyzer in the lab room. Having these different plots
helped illustrate the significance of percent error that arise with varying engineering
techniques. Figure 4 displays the data obtained from the physical circuit compared to the ideal
Bode Plot of the circuit’s transfer function. It is apparent that there is some percent
discrepancy among the obtained data and how the circuit should act in theory.
Procedure: By inspection, this laboratory experiment involved a great deal of circuit theory and
principles however will not be discussed in great detail in the remainder of this report. Using
these circuit principles, equations were derived from the circuit schematic shown in Figure 1. It
was important to recognize that the network in question was implemented using an integrator,
an adder, and an inverting amplifier.
Figure 1: Circuit Schematic of Active Network with Adder, Integrator, & Inverting Amplifier

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Part 1 of the lab experiment had various parts—the first of which involved circuit analysis.
Node equations were developed and manipulated in order to find the voltage transfer function
of the circuit. They can be summarized as is done below, which eventually gave the voltage
transfer function of the electronic circuit—called H(s).
𝑉1 = −𝑉𝑖𝑛 − 𝑉3
𝑉𝑜𝑢𝑡 =
−𝑉1
𝑗 ∗ 𝑅 ∗ 𝐶 ∗ 𝜔
−𝑉𝑜𝑢𝑡 = 𝑉3
𝑉𝑖𝑛 = −𝑉1 − 𝑉3
𝑉𝑜𝑢𝑡 = (
−1
𝑗 ∗ 𝑅 ∗ 𝐶 ∗ 𝜔
) × (−𝑉1 − 𝑉3)
𝑉𝑜𝑢𝑡 = (
−1
𝑗 ∗ 𝑅 ∗ 𝐶 ∗ 𝜔
) × (−𝑉𝑖𝑛 + 𝑉𝑜𝑢𝑡)
𝑉𝑜𝑢𝑡 = (
𝑉𝑖𝑛
𝑗 ∗ 𝑅 ∗ 𝐶 ∗ 𝜔
) − (
𝑉𝑜𝑢𝑡
𝑗 ∗ 𝑅 ∗ 𝐶 ∗ 𝜔
)
𝑉𝑜𝑢𝑡
𝑉𝑖𝑛
× (1 +
1
𝑗 ∗ 𝑅 ∗ 𝐶 ∗ 𝜔
) = (
1
𝑗 ∗ 𝑅 ∗ 𝐶 ∗ 𝜔
)
𝑉𝑜𝑢𝑡
𝑉𝑖𝑛
= (
1
𝑗𝜔𝑅𝐶 + 1
)
𝑉𝑜𝑢𝑡
𝑉𝑖𝑛
( 𝑠) = (
1
𝑅𝐶𝑠+1
) = 𝐻(𝑠) (1)
With Eq. (1)—the transfer function—now known, various responses and transient responses
were able to be analyzed for the circuit. For example, using the Laplace transform theory of
convolution, we were able to determine the unit impulse of the linear active network. Taking
the inverse transform of out transfer function provided us with the impulse response. I
developed a code for Matlab which gave the inverse Laplace transform of our transfer function
above (Eq. (1)).

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By inputting the correct resistance and capacitance values, the calculated unit impulse for the
network in Fig. 1 was found to be:
𝑒
−𝑡
𝑅𝐶⁄
𝑅𝐶
=
𝑒
−𝑡
10000 ∗.01 ×10−6⁄
10000∗(0.01×10−6)
= 10000 𝑒
−𝑡
10000∗.01×10−6⁄
= 10000 𝒆−𝟏𝟎𝟎𝟎𝟎𝒕
As in the impulse response, the parameters you find in a second order system determine
aspects of the step response. The damping ratio, ϛ, will determine how much the system
oscillates as the response decays toward steady state. The undamped natural frequency, 𝜔n,
determines how fast the system oscillates during any transient response. And finally, the DC
gain simply determines the size of the steady state response when the input settles out to a
constant value. One can think of the step response simply as the integral of the impulse
response.
Note that the impulse response of a filter, in the time domain, is proportional to the bandwidth
of the filter in the frequency domain. This means that the narrower the impulse—the wider the
bandwidth of the filter. So using Matlab we were able to determine this response and the plot
is shown in Fig. 2. This plot displays the unit impulse response of our active network. It is trivial
to compare this response to the unit impulse function found above and examine that they are
in fact the same.

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Figure 2: Ideal Impulse Response of Transfer Function using Matlab
On the other hand, the step response of the lab’s circuit was found to be the plot shown in Fig.
3. Taking the integral of the impulse response was performed using Matlab giving the step
response of our circuit to be:
𝒔( 𝒕) = ∫ 𝟏𝟎𝟎𝟎𝟎 𝒆−𝟏𝟎𝟎𝟎𝟎𝒕
= −𝒆−𝟏𝟎𝟎𝟎𝟎𝒕
Figure 3: Matlab Output of Step Response of Network

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One of the most important outputs that were found in this lab experiment was the Bode Plot of
the circuit over a specific frequency range. Both the magnitude and phase plots can be viewed
in Fig. 4—and it is important to note that this output shows both the Transfer Functions Bode
Plot, as well as the experimental Bode Plot superimposed on the same plot. It is shown clearly
and specified on the plot which magnitude (dB) data is which.
Figure 4: Bode Plot of Matlab Output and Experimental Data Bode Plot
Part 2 of this lab experiment was the modeling of our circuit in state space representation.
These sets of simultaneous differential equations allows for x to be a state variable vector and
the coefficents to be arranged in matrices. We assumed that x(0) = 0 and then applied the
Laplace transform to obtain the following:

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That equation can be written in terms of the output voltage in our circuit. Using Matlab, the
state space representation of our circuit was found from the following code:
Using Matlab, we again determined models of the impulse response, as well as the unit step
response and developed plots of these. However, for this section of the lab experiment we
plotted the step and impulse of the state-space model of the circuit. The state differential
equations are simply just another way to describe a dynamic system. The state variables shown
above in the clip of Matlab code simply provide and describe the future response of the active
network in Fig. 1. Using Matlab, we were also able to develop the impulse response of the
state-space model (Fig. 5) and the step response of the state-space model (Fig. 6). The state-
space model of our circuit from Fig. 1 was found to be as follows:
𝑥̇ = −10000𝑥 + 128𝑉𝑖𝑛
𝑉𝑜𝑢𝑡 = 78.13𝑥 + 0𝑉𝑖𝑛

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Figure 5: Unit Impulse Response of State-Space Model of Circuit
Figure 6: Unit Step Response of State-Space Model of Circuit
Comparing Figures 2 and 5 with each other show that the state-space model of the circuit is
truly just another way to represent the dynamic system shown in Fig 1. Along the same lines,
one can examine Figures 3 and 6 with each other and can determine that the outputs are
indeed the same. This was double-checked by using the tracing cursor on both of the step

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response plots. At a time of 0.0004053 seconds, the amplitude reached was a mere 0.9826 V.
On both plots (Fig. 3 & 6)—this data point was determined to be the exact same—showing that
the results obtained from the transfer function’s step response and the state-space step
response were equivalent.
Part 3 of this laboratory experiment involved the detailed design of the same circuit from Fig. 1.
Using an online electronics dealer—Mouser—I was able to determine the estimated cost that it
would take in order to construct this filter, using three LF347N operational amplifier, one
capacitor, and six resistors. Of course, the total cost shown in Table 1 ($ 95.01) is estimated
due to the various and abundant electronics parts that are being manufactured today.
Table 1: Part Supply List for Active Network shown in Fig. 1
Component
Type
Description
Mouser Part # Mfr. Part # Manufacturer Voltage
Rating or
Operating
Supply
Voltage(V)
Tolerance Power
Rating
Price
per
Unit($)
Units Total
Price ($)
Operational
Amplifier
595-LF347N LF347N Texas
Instruments
7 to 36 +/- 3.5 V to
+/- 18 V
N/A 0.75 3 2.25
Metal Foil
Resistor (10 kΩ)
71-
S102JT10K000TB
Y078510K0000T9L Vishay
Precision
Group
300 0.01% 600mW 15.25 6 91.50
Capacitor
(0.01µF)
598-
715P10356KD3
715P10356KD3 Cornell
Dubilier
600 5% N/A 1.26 1 1.26
95.01
My detailed design also involved the implementation of the MultiSim program—and electrical
circuit design suite—that is very convenient and helpful in construction and analysis. Fig. 7
depicts the circuit that I designed—again using the same LF347N Wide Bandwidth Quad JFET
Input Operational Amplifier and the same resistors and capacitors from the schematic in Fig. 1.
The Fig. 8 below shows all the pin-pin connections, and the only part not quite realistic is the
use of all three of the amplifiers’ voltage rails. Since in reality, the actual pin connection of the
op amp (LF347N) shown in the parts list has a total of only two of voltage rails (pins) that will
power each of the 4 inverting op amps in the package.

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Figure 7: Linear Active Network with Operational Amplifier Design with MultiSim
Figure 8: Pin Connections for LF347N Operational Amplifier
Part 4 of the laboratory experiment was the practical part of the experiment and involved the
actual construction of a physical circuit on the breadboard. Using six 10 kΩ resistors provided
by the course instructor and one 0.01 µF capacitor, along with a wire kit, we were able to
design the same circuit in Fig. 1. We supplied the circuit with +/- 5 Vp-p and studied the outputs
from both the oscilloscope, as well the dynamic spectrum analyzer (Stanford Research Systems
Model SR785). By using the function generator, we supplied our circuit at a specified frequency

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and obtained the plots shown in Fig. 9—which depicts the Bode Plot of our circuit (magnitude
plot). By examining the plot carefully, we can see the cutoff frequency (or the -3dB point).
Figure 9: Spectrum Analyzer Output of Bode Plot (Magnitude plot)
The output of the phase from the spectrum analyzer, on the other hand, is depicted in Fig. 10
but also clearly shows the cursor at the -3dB point (or the cutoff frequency)—giving the cutoff
frequency of our circuit.
Figure 10: Spectrum Analyzer Output of Bode Plot (Phase plot)
Knowing the theoretical cutoff frequency equation (Eq. 2), we were able to compare the
experimental corner frequency with it. The calculation of the theoretical cutoff frequency was
done as follows:
fc =
1
2 𝜋 𝑅 𝐶
=
1
2 𝜋 10000 ∗(0.01∗10 −6)
= 𝟏𝟓𝟗𝟏. 𝟓𝟓 𝑯𝒛

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If you look at the magnitude Bode Plot, it is clear that the -3.014 dB point takes place at a
frequency of 1.696 kHz or 1696 Hz. There lies a percent error among this value and the
theoretical value. The percent error is as follows:
|1696 − 1591.55|
1591.55
× 100 = 6.56 %
This amount of error could be the result of percent discrepancies in the resistors that were
used, as well as the capacitor. Ideally, the resistors should have been 10 kΩ, but in reality they
are not. In addition, the calibration of the machinery and equipment that was used could have
affected our results. Moreover, the percent discrepancy that was calculated was done using
the -3.014 dB point from the spectrum analyzer plot—not the exact -3 dB point. This definitely
had an impact on the 6.56% result.
Sending a square wave at a specified voltage to the input of our circuit, using a frequency of 1
kHz allowed us to obtain the step response of our linear active network. Fig. 11 clearly shows
the oscilloscope output screenshot that was obtained during the experiment.
Figure 11: Oscilloscope Output of Physical Circuit
In an ideal amplifier, the input voltage is a unit step function, and the output voltage is also a
unit step function, but the practical amplifiers used in the experiment cannot change their
output instantaneously, if the input changes suddenly. Comparing this output to Fig. 3 & Fig. 6

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shows that the circuit which we built was in fact the linear network which we wanted. The step
responses of both the state-space model and the transfer function both look similar to the unit
step response that is displayed above in Fig. 11.
Part 5 of the laboratory experiment involved using MultiSim to obtain a step response of the
network in Fig. 7. This response could be compared to the ideal Matlab response, as well as the
experimental step response in order to see if the circuit was behaving as it should. The step
response output or transient analysis obtained from MultiSim is shown below in Fig. 12.
Figure 12: Step Response with MultiSim
The output shown above shows a input signal given to the circuit and the output response (the
exponential plot). The output above shows a similar step response to that of Fig. 3 (Step
Response of Transfer Function with Matlab), Fig. 6 (Step Response of State-Space Model with
Matlab) and Fig. 11 (Experimental Oscilloscope Step Response Output).
Part 6 of the laboratory experiment used circuit theorems and the relatively minimal use of
algebra. Knowing the cutoff frequency that was found above from the Eq. (2), we designed
another low pass filter that would meet the same exact specifications of the circuit shown in
Fig. 1.
Rearranging the cutoff frequency equation—and solving for the capacitance gives 𝐶 =
1
2𝜋𝑅 𝑓𝑐
Choosing a 10 kΩ resistor and plugging in 1.696 kHz as the corner frequency gives:
𝐶 =
1
2𝜋10000 ∗(1.696×103 )
= 9.384 × 10−9
𝐹

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Building a schematic of our circuit design using MultiSim (Fig. 13) allowed us to obtain a step
response for the circuit. Inputting a 5 V input voltage resulted in the step response of Fig. 14.
Figure 13: Equivalent Circuit Schematic of Circuit in Fig. 1
Figure 14: MultiSim Step Response Output of Equivalent Circuit

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The filter that is displayed above is a simple first order low-pass active filter. It has a simple RC
filter stage and has a unity gain feedback.
The transfer function of this low pass filter is the same as that shown in Eq. (1). For this circuit
the transfer function is as follows:
(
1
𝑅𝐶𝑠+1
) = (
1
10000 ∗(9.384∗10−9)𝑠+1
) = (
1
9.384∗10−5 𝑠+1
) = 𝐻𝑒𝑞 (𝑠)
We then applied Matlab in order to analyze the step response of H(s) shown above. The code
used for this was as follows:
The step response that we obtained as a Matlab output is depicted in Fig. 15 below. By
comparing this to the response obtained in Fig. (3), as well as Fig. (6), we found that these plots
are in fact similar. This coincides with our goal of finding an equivalent circuit to that of Fig. 1,
by using only one operational amplifier.

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Figure 15: Matlab Output of Step Response of Equivalent Circuit Transfer Function
In addition, we were able to compare the Bode Plots from the plots shown earlier in this report.
Figure 16: Bode Plot of Equivalent Circuit’s Transfer Function

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The Bode Plot above, if examined, is of close comparison to that plot shown in Fig. 4 from this
lab report. There is likely some amount of percent error that could be a result of Matlab using
the theoretical transfer function to plot the magnitude and phase of the equivalent circuit,
while MultiSim did not. These errors are usually always the case in experiments, but the goal is
always to reduce these percentages to as minimal as possible. This can provide proof that the
data provided is very accurate.
Part 7-Questions
a) A low pass filter, like the circuit constructed for this lab, has many applications in
practice. One important thing that all audiophiles must have is a subwoofer—which
takes out all high frequencies and only uses the low frequencies. This results in the
shaking of vehicles and the noise filled movie theatres. Digital algorithms also use these
filters in order to smooth data sets, blur images, and things of that nature. Their
importance in the field of electronics will never be underestimated and will always be
used. Musicians also find the need for filters in order to have the required spectral
information that they want to play and harmonics at different levels.
b) The circuit constructed during this lab can be thought of as stable because the poles of
the closed loop transfer function are found in the left half of the s-plane. The left half of
this plane is equivalent to being negative and real—meaning that if a pole is located
here, then the systemis stable. Another aspect of this stability is the Nyquist Diagram
of this circuit and its transfer function. Shown below, in Fig. 14, is the Nyquist Plot for
this lab experiment’s circuit.
Figure 14: Nyquist Diagram of Transfer Function and Circuit

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c) When analyzing a unit impulse response of a filter circuit, it is easiest to relate it to the
transfer function of the circuit—that is the voltage transfer function or H(s). Their
relationship is simply that the inverse Laplace Transform of the transfer function gives
the impulse response of the circuit. However, one has to make sure that the transfer
function is in terms of the s-domain (or the frequency domain). For our circuit, larger
values for the resistors, and the capacitor would in turn make it take longer for our
circuit to reach its steady state value. This would result in a change of the corner
frequency—making it change to a lower frequency.
Summary: The laboratory experiment conducted and discussed about in this report provided an
in-depth view into the world of filter analysis and its relationship to feedback control systems.
Designing the linear active network (Fig. 1) was completed in order to analyze various parts of
circuit theory and engineering. The Bode Plots, made throughout this laboratory experiment,
were all constructed using different engineering tools that were at our disposal—including
MultiSim, Matlab, and the Spectrum Analyzer from the laboratory room itself. The Bode Plots
in Fig. 4 show an important aspect of this lab—and the use of simple transfer functions.
Obtaining the Bode Plot from the transfer function with Matlab, and comparing it to our Bode
Plot made from actual experimental data reinforced the principle fact that Eq. (1) was indeed
the response of the circuit in Fig. 1. It is important to note that the spectrum analyzer output
(Fig. 9 & Fig. 10) found the cutoff frequency of our circuit to be 1.696 kHz at a phase of 44.43°.
In ideal conditions, this phase should have been 45°--and the percent discrepancy between our
experimental data and the expected value was calculated to be 1.283%. This was considered a
negligible amount of error—meaning that our experimental data pulled from our constructed
breadboarded circuit and the spectrum analyzer was very accurate. This percent error was
most likely the result of using resistors and capacitors that are never 100% the correct values.
In addition, the equipment like the function generator, the power supply, and even the
spectrum analyzer may not have been calibrated properly. One important aspect of this lab
was the modeling of a circuit—whether it is with a transfer function, the state-space model, or
even with the use of specific engineering software. This laboratory experiment took all three of
these methods and applied them to the analysis of the active network seen in Fig. 1. The
results found by applying the transfer function—Eq. (1)—found the step response (Fig. 3) to be
the same as when the state-space transformation was used (Fig. 6). These plots—being the
exact same—clearly displayed that there are in fact various ways to model the active filter from
Fig. 1. In fact, even another completely different circuit (Fig. 13) gave a step response exactly
the same as that of Fig. 3. This result displays the principle of filter design and that different
circuits can be used for the same application with the same cutoff frequency. Moreover, the
application of Laplace Transform theory showed how mathematical equations relate to circuit

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analysis and the impulse response/step response. This laboratory experiment delved into the
fact that the impulse response simply means how the circuit behaves due to a specific input.
This laboratory experiment was indicative of the type of work that might be done at an
electronics firm of some sort, where it is crucial to be familiar with how software (like Matlab
and MultiSim) work –and how they relate and can be used in order to find all sort of data (Bode
Plots, Laplace Transforms, Integrals, Filter Design, AC Analyses, Transient Analyses, etc).