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Frequency Response
Of an Active Low-Pass Filter
A. Pinkham, J. Discolo
Abstract— This experiment comprised of building an inverting
low pass filter circuit using the LM358 Operational amplifier.
Low pass filters are circuits that only allow certain low
frequencies, usually determined by the design of the circuit, from
an input signal. Data acquisition came from using MATLAB and
Simulink to model the and display the output frequencies the
circuit allowed to pass. Theoretical and experimental calculations
were computed and compared.
I. INTRODUCTION
Inverting low-pass filters are meant to only allow lower
frequencies to pass through a circuit, which is built during the
experiment. What is tested in this experiment is how real
world data and results compare to the theoretical, ideal models
developed. More specially, there are four factors measured.
The peak to peak input and output voltages, the period of input
signal, and the time delay between the input and output
signals. This data is used to calculate experimental magnitude
ratio and phase lag, which will be discussed in more detail
further in the report.
II. METHODS
First agenda of the experiment involves the construction of
the inverting low pass filter to be used for data acquisition:
A. Analysis
This circuit is considered a first order system, and thus the
magnitude ratio is defined by
(1)
where ω is the frequency, τ is the time constant. Equivalently
B is the out output voltage, A is the input voltage and K is the
static gain, defined by
K= R2/R1
(2)
R2 and R1 values for this circuit will be 1MΩ. Although the
resistors are not quite exactly 1MΩ, the real static gain was
calculated using the real resistor values found by a voltmeter.
For typical low pass filters, a -3dB cutoff frequency is used,
and the formulation for that is as follows
(3)
Where R2 and C2 are the resistor and capacitor respectively.
B. Experimental Program
With a properly constructed inverted low-pass filter with the
correct Simulink modeling tools, the data was collected.
Figure 1. Inverting Low-Pass Filter using the LM358 Op-Amp
This circuit utilizes the inverting input of the op-amp. Two
resistors are used in series around the inverting input to the
voltage out. One added feature of this particular low-pass filter
circuit is the addition of the capacitor, set in parallel with R2.
Voltage is supplied with a power supply set to +-15V. A
function generator is use to supply the input signal through the
circuit and to a NI USB-6008 data acquisition device to
convert the along signals to digital signals for MatLab and
Simulink. The Simulink model records both the input and
output voltages so that the magnitude ratio and phase lag may
be calculated from the recorded data.
Figure 2. Inverting Low Pass Filter Circuit
Starting with a very low frequency Sine wave of 0.1Hz, the
model was ran long enough to produce the full amplitude and
period of the input and output voltages on the scope display.
Two important differences between the input and output
voltages had to be recorded. The peak voltage differences of
the input and output, along with the time delay between the
peaks of the output and input were logged. This process was
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repeated in given increments with faster frequencies each time,
maxing out at 500Hz.
With raw data obtained, the next step of the experiment was
to generate plots of the theoretical and experimental magnitude
ratios vs. the frequency, and the phase lag vs. the input
frequency.
III. RESULTS
The plots generated using MATLAB show the experimental
results obtained and the theoretical values.
Figure 4. Phase shift, or the phase lag
Plotting the theoretical and experimental phase lags for the
data reveals that they are practically identical. The
experimental data curve falls perfectly in line with the
experimental curve.
IV. DISCUSSION AND CONCLUSIONS
For the frequency vs. magnitude ratio, if the plots were
properly graphed, they would almost line up identically.
Again, the reason was not found. However, for the phase lag,
the experimental and theoretical data seemed to match almost
perfectly, with no sign of any difference.
This filter is called a low pass because it only allows the
passage of certain frequencies or voltages to flow to the output
signal. This is governed by the cutoff frequency, which can be
manipulated to any value, thus allowing any range of low
frequencies to be passed to a certain cutoff point.
Possible sources of random errors in the experimental data
would mostly be random noise associated with the input
voltage. A second source would be the accuracy of recording
the input and output values from the Simulink software, only
approximating the values by zooming and judging with the
naked eye.
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REFERENCES
[1] LM358 Op-Amp Data Sheet
http://cache.national.com/ds/LM/LM158.pdf
Figure 3. Frequency vs. Magnitude ratio plot.
This plot shows the theoretical magnitude ratio in blue, and
the experimental magnitude ratio in red crosses. With this plot,
there was an error somewhere that couldn’t be corrected. The
problem is that the experimental plot is off quite a bit on the y
axis, only showing the bottom portion of the actual plot. The
reason for this was not found. Although the experimental curve
should have been closer to the theoretical plot curve.