Cauan (2)

NATIONAL COLLEGE OF SCIENCE AND TECHNOLOGY Amafel Building, Aguinaldo Highway Dasmariñas City, Cavite

Experiment No. 3

ACTIVE LOW-PASS and HIGH-PASS FILTERS

Cauan, Sarah Krystelle P. July 14, 2011

Signal Spectra and Signal Processing/BSECE 41A1 Score:

Engr. Grace Ramones

Instructor

OBJECTIVES:

Plot the gain-frequency response and determine the cutoff frequency of a second-

order (two-pole) low-pass active filter.

Plot the gain-frequency response and determine the cutoff frequency of a second-

order (two-pole) high-pass active filter.

Determine the roll-off in dB per decade for a second-order (two-pole) filter.

Plot the phase-frequency response of a second-order (two-pole) filter.

SAMPLE COMPUTATIONS:

Step 3 Computation of voltage gain based on measured value:

AdB = 20 log A

4.006 = 20 log A

Step 4 Computation of voltage gain based on circuit:

Q in Step 4 Percentage Difference

Step 6 Computation of cutoff frequency:

Q in Step 6 Percentage Difference

Q in Step 7 Roll –Off

-36.146 dB – 0.968 dB = -37.106 dB

Step 14 Calculated the actual voltage gain (A) from the dB gain

A = 1.54

Step 15 Computation of expected voltage gain based on circuit:

Q in Step 15 Computation of percentage difference:

Step 17 Computation of expected cutoff frequency:

Q in Step 17 Computation of percentage difference:

Q in Step 18 Roll –Off

-36.489 dB – 0.741 dB = -37.23 dB

DATA SHEET:

MATERIALS

One function generator

One dual-trace oscilloscope

One LM741 op-amp

Capacitors: two 0.001 µF, one 1 pF

Resistors: one 1kΩ, one 5.86 kΩ, two 10kΩ, two 30 kΩ

THEORY

In electronic communications systems, it is often necessary to separate a specific

range of frequencies from the total frequency spectrum. This is normally accomplished

with filters. A filter is a circuit that passes a specific range of frequencies while rejecting

other frequencies. Active filters use active devices such as op-amps combined with passive

elements. Active filters have several advantages over passive filters. The passive elements

provide frequency selectivity and the active devices provide voltage gain, high input

impedance, and low output impedance. The voltage gain reduces attenuation of the signal

by the filter, the high input prevents excessive loading of the source, and the low output

impedance prevents the filter from being affected by the load. Active filters are also easy to

adjust over a wide frequency range without altering the desired response. The weakness of

active filters is the upper-frequency limit due to the limited open-loop bandwidth (funity) of

op-amps. The filter cutoff frequency cannot exceed the unity-gain frequency (funity) of the

op-amp. Ideally, a high-pass filter should pass all frequencies above the cutoff frequency

(fc). Because op-amps have a limited open-loop bandwidth (unity-gain frequency, funity),

high-pass active filters have an upper-frequency limit on the high-pass response, making it

appear as a band-pass filter with a very wide bandwidth. Therefore, active filters must be

used in applications where the unity-gain frequency (funity) of the op-amp is high enough so

that it does not fall within the frequency range of the application. For this reason, active

filters are mostly used in low-frequency applications.

The most common way to describe the frequency response characteristics of a filter

is to plot the filter voltage gain (Vo/Vin) in dB as a function of frequency (f). The frequency

at which the output power gain drops to 50% of the maximum value is called the cutoff

frequency (fc). When the output power gain drops to 50%, the voltage gain drops 3 dB

(0.707 of the maximum value). When the filter dB voltage gain is plotted as a function of

frequency using straight lines to approximate the actual frequency response, it is called a

Bode plot. A Bode plot is an ideal plot of filter frequency response because it assumes that

the voltage gain remains constant in the passband until the cutoff frequency is reached, and

then drops in a straight line. The filter network voltage gain in dB is calculated from the

actual voltage gain (A) using the equation

AdB = 20 log A

where A = Vo/Vin.

An ideal filter has an instantaneous roll-off at the cutoff frequency (fc), with full

signal level on one side of the cutoff frequency. Although the ideal is not achievable, actual

filters roll-off at -20 dB/decade or higher depending on the type of filter. The -20

dB/decade roll-off is obtained with a one-pole filter (one R-C circuit). A two-pole filter has

two R-C circuits tuned to the same cutoff frequency and rolls off at -40 dB/decade. Each

additional pole (R-C circuit) will cause the filter to roll off an additional -20 dB/decade. In a

one-pole filter, the phase between the input and the output will change by 90 degrees over

the frequency range and be 45 degrees at the cutoff frequency. In a two-pole filter, the

phase will change by 180 degrees over the frequency range and be 90 degrees at the cutoff

frequency.

Three basic types of response characteristics that can be realized with most active

filters are Butterworth, Chebyshev, and Bessel, depending on the selection of certain filter

component values. The Butterworth filter provides a flat amplitude response in the

passband and a roll-off of -20 dB/decade/pole with a nonlinear phase response. Because of

the nonlinear phase response, a pulse wave shape applied to the input of a Butterworth

filter will have an overshoot on the output. Filters with a Butterworth response are

normally used in applications where all frequencies in the passband must have the same

gain. The Chebyshev filter provides a ripple amplitude response in the passband and a roll-

off better than -20 dB/decade/pole with a less linear phase response than the Butterworth

filter. Filters with a Chebyshev response are most useful when a rapid roll-off is required.

The Bessel filter provides a flat amplitude response in the passband and a roll-off of less

than -20 dB/decade/pole with a linear phase response. Because of its linear phase

response, the Bessel filter produces almost no overshoot on the output with a pulse input.

For this reason, filters with a Bessel response are the most effective for filtering pulse

waveforms without distorting the wave shape. Because of its maximally flat response in the

passband, the Butterworth filter is the most widely used active filter.

A second-order (two-pole) active low-pass Butterworth filter is shown in Figure 3-

1. Because it is a two-pole (two R-C circuits) low-pass filter, the output will roll-off -40

dB/decade at frequencies above the cutoff frequency. A second-order (two-pole) active

high-pass Butterworth filter is shown in Figure 3-2. Because it is a two-pole (two R-C

circuits) high-pass filter, the output will roll-off -40 dB/decade at frequencies below the

cutoff frequency. These two-pole Sallen-Key Butterworth filters require a passband voltage

gain of 1.586 to produce the Butterworth response. Therefore,

and

At the cutoff frequency of both filters, the capacitive reactance of each capacitor (C) is equal

to the resistance of each resistor (R), causing the output voltage to be 0.707 times the input

voltage (-3 dB). The expected cutoff frequency (fc), based on the circuit component values,

can be calculated from

wherein,

FIGURE 3 – 1 Second-order (2-pole) Sallen-Key Low-Pass Butterworth Filter

FIGURE 3 – 2 Second-order (2-pole) Sallen-Key High-Pass Butterworth Filter

PROCEDURE

Low-Pass Active Filter

Step 1 Open circuit file FIG 3-1. Make sure that the following Bode plotter settings are

selected: Magnitude, Vertical (Log, F = 10dB, I = -40dB), Horizontal (Log, F = 100

kHz, I = 100 Hz).

Step 2 Run the simulation. Notice that the voltage gain has been plotted between the

frequencies of 100 Hz and 100 kHz by the Bode plotter. Draw the curve plot in the

space provided. Next, move the cursor to the flat part of the curve at a frequency

of approximately 100 Hz and measure the voltage gain in dB. Record the dB gain

on the curve plot.

dB gain = 4.006 dB

Question: Is the frequency response curve that of a low-pass filter? Explain why.

The response curve shown above is a low-pass filter response. I said so

because low-pass filter only allows the frequencies below the cutoff

frequency and block the frequencies above the cutoff frequency.

Step 3 Calculate the actual voltage gain (A) from the measured dB gain.

A = 1.586

f

AdB

Step 4 Based on the circuit component values in Figure 3-1, calculate the expected

voltage gain (A) on the flat part of the curve for the low-pass Butterworth filter.

A = 1.586

Question: How did the measured voltage gain in Step 3 compared with the calculated

voltage gain in Step 4?

There is no difference between the measured voltage gain and the

calculated voltage gain.

Step 5 Move the cursor as close as possible to a point on the curve that is 3dB down

from the dB gain at the low frequencies. Record the dB gain and the frequency

(cutoff frequency, fc) on the curve plot.

dB gain= 0.968 dB

fc = 5.321 kHz

Step 6 Calculate the expected cutoff frequency (fc) based on the circuit component

values.

fc = 5.305 kHz

Question: How did the calculated value for the cutoff frequency compare with the measured

value recorded on the curve plot for the two-pole low-pass active filter

The difference between the calculated cutoff frequency and the measured

value has 0.30%. They are almost equal.

Step 7 Move the cursor to a point on the curve where the frequency is as close as

possible to ten times fc. Record the dB gain and frequency (fc) on the curve plot.

dB gain = -36.146 dB

fc = 53.214 kHz

Questions: Approximately how much did the dB gain decrease for a one-decade increase in

frequency? Was this what you expected for a two-pole filter?

The dB gain decrease approximately 37.106 dB for a one-decade increase

in frequency I am expecting 40 dB decrease per decade increase in

frequency.

Step 8 Click Phase on the Bode plotter to plot the phase curve. Change the vertical axis

initial value (I) to 180 degrees and the final value (F) to 0 degree. Run the

simulation again. You are looking at the phase difference (θ) between the filter

input and output wave shapes as a function of frequency (f). Draw the curve plot

in the space provided.

Step 9 Move the cursor as close as possible on the curve to the cutoff frequency (fc).

Record the frequency (fc) and phase (θ) on the curve.

fc = 5.321 kHz

θ = -90.941

Question: Was the phase shift between input and output at the cutoff frequency what you

expected for a two-pole low-pass filter?

Phase shift between input and output at the cutoff frequency is what I

expected because the phase at cutoff frequency is 90o

f

θ

Step 10 Click Magnitude on the plotter. Change R to 1 kΩ in both places and C to 1 pF in

both places. Adjust the horizontal final frequency (F) on the Bode plotter to 20 MHz. Run the

simulation. Measure the cutoff frequency (fc) and record your answer.

fc = 631.367 kHz

Step 11 Based on the new values for resistor R and capacitor C, calculate the new cutoff

frequency (fc).

fc = 159.1549 MHz

Question: Explain why there was such a large difference between the calculated and the

measured values of the cutoff frequency when R = 1kΩ and C = 1pF. Hint: The value of the

unity-gain bandwidth, funity, for the 741 op-amp is approximately 1 MHz.

There is a large difference between the calculated and measured value

because the cutoff frequency exceed the unity-gain frequency of the op-

amp. And op-amp has a limited open-loop bandwidth that causes the

active filter to have an upper-frequency limit.

High-Pass Active Filter

Step 12 Open circuit file FIG 3-2. Make sure that the following Bode plotter settings are

selected: Magnitude, Vertical (Log, F = 10dB, I = -40dB), Horizontal (Log, F = 100

kHz, I = 100 Hz).

Step 13 Run the simulation. Notice that the voltage gain has been plotted between the

frequencies of 100 Hz and 100 kHz by the Bode plotter. Draw the curve plot in the

space provided. Next, move the cursor to the flat part of the curve at a frequency

of approximately 100 kHz and measure the voltage gain in dB. Record the dB gain

on the curve plot.

dB gain = 3.776 dB

Question: Is the frequency response curve that of a high-pass filter? Explain why.

The response curve shown above is a high-pass filter response. I said so

because high-pass filter only allows the frequencies above the cutoff

frequency and block the frequencies below the cutoff frequency.

Step 14 Calculate the actual voltage gain (A) from the measured dB gain.

A = 1.54

f

AdB

Step 15 Based on the circuit component values in Figure 3-2, calculate the expected

voltage gain (A) on the flat part of the curve for the high -pass Butterworth filter.

Av = 1.586

Question: How did the measured voltage gain in Step 14 compare with the calculated

voltage gain in Step 15?

The measured voltage gain and the calculated voltage gain has a

percentage difference of 2.98%. Yet, it is still approximately the equal.

Step 16 Move the cursor as close as possible to a point on the curve that is 3dB down

from the dB gain at the high frequencies. Record the dB gain and the frequency

(cutoff frequency, fc) on the curve plot.

dB gain = 0.741 dB fc = 5.156 kHz

Step 17 Calculate the expected cutoff frequency (fc) based on the circuit component

values.

fc = 5.305 kHz

Question: How did the calculated value of the cutoff frequency compare with the measured

value recorded on the curve plot for the two-pole low-pass active filter?

They are almost equal. The percentage difference between the calculated

and measured value is 2.89%

Step 18 Move the cursor to a point on the curve where the frequency is as close as

possible to one-tenth fc. Record the dB gain and frequency (fc) on the curve plot.

dB gain = -36.489 dB

fc = 515.619 Hz

Questions: Approximately how much did the dB gain decrease for a one-decade decrease in

frequency? Was this what you expected for a two-pole filter?

It decreases 37.23 dB per decade. It is approximate -40 dB per decade so it

was what I am expecting.

Step 19 Change the horizontal axis final setting (F) to 50 MHz on the Bode plotter. Run

the simulation. Draw the curve plot in the space provided.

Step 20 Measure the upper cutoff frequency (fc2) and record the value on the curve plot.

fc2 = 92.595 kHz

Question: Explain why the filter frequency response looked like a band-pass response when

frequencies above 1 MHz were plotted. Hint: The value of the unity-gain

bandwidth, funity, for the 741 op-amp is approximately 1 MHz

The filter frequency response appears like a band-pass filter because the

cutoff frequency exceeds the unity-gain frequency of the active filter. The

active filters have an upper frequency limit on the high-pass response.

f

AdB

CONCLUSION

After performing the experiment, I conclude that active filter uses op-amps and other

passive elements. This filter has several advantages over the passive filter such as providing a

frequency selectivity, voltage gain, high input impedance, and low output impedance. However,

the weakness of this kind of filter is having an upper-frequency limit because of the limited

open-loop funity of the op-amp.

I also notice that the frequency response curve of passive and active filters appear the

same except in high-pass response. In high-pass response, the frequency looked like a band-

pass filter because of the funity of the op-amp.

I also notice that two-pole filter which has two R-C circuits rolls-off at approximately -40

dB per decade. Furthermore, this filter has a phase frequency response of 90 degrees at the

cutoff frequency and 180 degrees over the frequency range.

Lastly, the circuit we have performed is a Butterworth filter because it has a passband

voltage gain of 1.586. That is why the curve has a flat amplitude response in the passband and

then rolls-off at approximately -40dB per decade.