## Operational Amplifiers - Active Filter Design

dB/decade. This means that a first order low pass filter has an attenuation of -20. dB/decade, while a second order filter should have -40 dB/decade rolloff and on.

EE 3305 Operational Amplifiers Continuing the discussion of Op Amps, the next step is filters. There are many different types of filters, including low pass, high pass and band pass. We will discuss each of the following filters in turn and how they are used and constructed using Op Amps. When a filter contains a device like an Op Amp they are called active filters. These active filters differ from passive filters (simple RC circuits) by the fact that there is the ability for gain depending on the configuration of the elements in the circuit. There are some problems encountered in active filters that need to be overcome. The first is that there is still a gain bandwidth limitation that arises. The second is the bandwidth in general. In a high pass filter there is going to be high frequency roll off due to the limitations of the Op Amp used. This is very hard to overcome with conventional op amps. The mathematical operations discussed in the previous lab (the integrator and differentiator) are both types of active filters. As for now, the discussion will focus mainly on the low pass (LP), high pass (HP) and band pass (BP) filters. There is also a band stop filter that can be created from the band pass filter with a simple change of components. Low Pass Filters The low pass filter is one that allows low frequencies and stops (attenuates) higher frequencies, hence the name. The design of a low pass filter needs to take into consideration the maximum frequency that would need to be allowed through. This is called the cut off frequency (or the 3 dB down frequency). Based on the type of filter that is used (e.g. Butterworth, Bessel, Tschebyscheff) the attenuation of the higher frequencies can be greater. This attenuation is also based on the order (e.g. 1st, 2nd, 3rd…) of the filter that is used. Based on the order of the filter the rolloff of the filter can be calculated using the formula –n*20 dB/decade. This means that a first order low pass filter has an attenuation of -20 dB/decade, while a second order filter should have -40 dB/decade rolloff and on down the list for higher orders. Shown in Figure 1 is the basic active 1st order low pass filter (in the non-inverting configuration) with unity gain.

Figure 1: Basic 1st Order Bessel LP Filter in the Non-Inverting Configuration with Unity Gain

EE 3305 The equation in (1) is used to calculate the value of the capacitor needed based on a chosen value for cutoff frequency and R1 (or vice versa if a value for C1 and a cutoff frequency are chosen then the value of R1 can be found). There is unity gain in this configuration because of the non-inverting properties of the Op Amp. To change the gain, the feedback network must be changed to include two other resistors (R2 and R3). The gain is then found to be 1 + R3/R2 because of the noninverting configuration. The circuit with a non-unity gain is shown in Figure 2. 1 fc = (1) 2πR1C1

Figure 2: Basic 1st Order Bessel LP Filter in the Non-Inverting Configuration with Non-unity Gain

There are times when a higher order filter might be better. If this is the case the 2nd order Bessel filter can be used. This changes the circuit of Figure 1 by adding another resistor-capacitor pair to create a fast rolloff at frequencies above fc. The circuit of Figure 3 shows the 2nd order Bessel LP filter with unity gain.

Figure 3: 2nd Order Bessel LP Filter with unity gain

To find the value of the capacitors needed the equations listed in (2) are helpful. Notice that the values of the resistors in the circuit of Figure 3 and in (2) are equal. The odd coefficients of the equations in (2) come from finding the transfer function and then solving for the desired cutoff frequency. These are sometimes referred to as the frequency normalization coefficients. As the need to go to higher orders arises, the need to cascade filters comes out. To get a 4th order

EE 3305 Bessel filter one would cascade two 2nd order Bessel filters. Based on the cutoff frequency chosen and the values of resistors available, the values of the capacitors can be calculated. .9076 C1 = 2πf c R (2) .6809 C2 = 2πf c R High Pass Filters If creating a low pass filter was easy, then creating a high pass filter is even easier. In the case of the 1st order Bessel LP filter the capacitor and resistor only need to be interchanged with each other and the result is a high pass filter. The same equation holds for finding the cutoff frequency and is shown in (1). The circuit shown in Figure 4 is that of a 1st order Bessel HP filter with unity gain. The gain can be adjusted to the non-unity case by adding the feedback network resistors in the same location as the LP circuit of Figure 2.

Figure 4: 1st Order Bessel HP Filter with unity gain

If the circuit is modified to allow another resistor-capacitor pair, the filter type can be changed from 1st order to 2nd order. The modified circuit will then look like that of Figure 5. Take note that in this configuration the values that need to be calculated are the values of the resistors R1 and R2, and the value of the capacitor can be chosen between 4.7 – 10 nF.

Figure 5: 2nd Order Bessel HP Filter with Unity Gain

EE 3305 The values of the components used are calculated from (3). These values again arise from the transfer function and then solving for each of the coefficients. To obtain a higher order filter the cascade technique will have to be used. Therefore to make a 4th order HP filter two 2nd order HP filters need to be cascaded.

R1 =

1.1017 2πf c C

1.4688 R2 = 2πf c C

(3)

Band Pass Filters The final type of filter to be discussed here is that of a band pass filter. The band pass filter takes advantage of the low pass configuration as well as the high pass configuration. The two of these combine to for a range of frequencies that is called the pass band. Below the lower cutoff frequency the signals are stopped as well as above the higher cutoff frequency. The difference between these two frequencies is called the bandwidth of the filter. The logic behind the cutoff frequencies is a little misleading. The lower cutoff frequency is controlled by the high pass filter part of the band pass filter. On the same type of idea, the upper cutoff frequency is controlled by the low pass filter part of the band pass filter. The circuit shown in Figure 6 is that of a basic pass band filter. Notice the combination of the low pass and high pass connections. The combination of a 1st order HP and a 1st order LP creates a 2nd order band pass. If the trend were to continue a 2nd order HP and a 2nd order LP create a 4th order band pass.

Figure 6: Band Pass filter with Low Pass and High Pass Connections

The determination of the center frequency of the band pass filter can be found from either the low pass or the high pass filter. The transition from the low pass to the band pass is easier, and only requires that the transfer function be 1 1 modified from s to ( s + ) . Once this change is made, the new transfer ∆Ω s function will give the transfer function for the BP filter. The value of ∆Ω is the bandwidth of the filter (the distance between the two 3 dB down points).

EE 3305

Hand In Requirements 1. 2. 3. 4.

Pre lab exercises sheet(at the beginning of class) Simulation output waveforms from pre lab (at beginning of class). Data sheet with TA’s signature Lab report with detailed answers to post lab exercises.

EE 3305 Pre Lab Exercises 1. Using any technique available find the transfer function of the first order Low Pass Bessel filter shown in Figure 1. Show your work for credit.

2. Now, find the transfer function (and simplify) for a band pass filter by 1 1 changing the s term in Part 1 to ( s + ) , where ∆Ω = 25 kHz. ∆Ω s

3. Using PSPICE Simulate the 1st order Low Pass Bessel Filter in Figure 1 using (1) to find the value needed for the capacitor. Plot the magnitude (dB) using a VAC input source (set to 1 V) with AC SWEEP from 1 Hz to 100 kHz. The cutoff frequency should be 15 kHz and the value of R = 4.7 kΩ. 4. Using PSPICE simulate the 2nd order Low Pass Bessel Filter in Figure 3 using (2) to find the values of the components. The value of R should be 4.7 kΩ and the cutoff frequency should be 15 kHz. Comment on how the cutoff frequency and the slope of the roll off compare to that of Part 3 above.

5. Using PSPICE simulate the 2nd order High Pass Bessel Filter in Figure 5 using (3) to find the values of the components. Use a value of 15 kHz for fc and C = 10 nF. Plot the magnitude (dB) of the response and comment on the slope outside of the cutoff frequency. How does this compare to the plot in Part 4.

6. Simulate a band pass filter from either Figure 6 or make one from a cascade of a low pass and a high pass filter. Use a center frequency of 15 kHz and a bandwidth of 25 kHz. Plot the magnitude (dB) and comment on the shape of the plot.

EE 3305 Lab Exercises 1. Construct the low pass filter with unity gain of Figure 1 for a cutoff frequency of 15 kHz with a capacitor value of .01 µF. Have your TA initial the Data Sheet. 2. Take measurements of the input voltage, output voltage and calculate the gain for various frequencies. Record this information on the Data Sheet. 3. Construct the high pass filter with unity gain of Figure 4 using the same component values from Part 1. Have your TA initial the Data Sheet. 4. Take measurements of the input voltage, output voltage and calculate the gain for various frequencies. Record this information on the Data Sheet. 5. Construct the second order low pass filter with unity gain of Figure 3 using the equations to determine what values of capacitors to use. Use a value of R = 4.7 kΩ and a cutoff frequency of 15 kHz. Have your TA initial the Data Sheet 6. Take measurements of the input voltage, output voltage and calculate the gain for various frequencies. Record this information on the Data Sheet. 7. Construct the band pass filter of Figure 6 (or of a cascade of a low pass and a high pass) with a center frequency of 15 kHz and a bandwidth of 25 kHz. Use values from the Pre Lab for components or items that are comparably close. Have your TA initial the Data Sheet. 8. Take measurements of the input voltage, output voltage and calculate the gain for various frequencies. Record this information on the Data Sheet.

EE 3305 Data Sheet 1. Circuit Construction Figure 1. TA INITIAL _______ 2. Measure Input, Output, Gain of the 1st order Bessel LP Frequency

Input Voltage

Output Voltage

dB = 20 log(Gain)

10 Hz 100 Hz 1 kHz 5 kHz 10 kHz 15 kHz 20 kHz 50 kHz 100 kHz 3. Circuit Construction Figure 4. TA INITIAL _______ 4. Measure Input, Output, Gain of the 1st order Bessel HP Frequency

10 Hz 100 Hz 1 kHz 5 kHz 10 kHz 15 kHz 20 kHz 50 kHz 100 kHz

Input Voltage

Output Voltage

dB = 20 log(Gain)

EE 3305 5. Circuit Construction Figure 3. TA INITIAL _______ 6. Measure Input, Output, Gain of the 2nd order Bessel LP Frequency

Input Voltage

Output Voltage

dB = 20 log(Gain)

10 Hz 100 Hz 1 kHz 5 kHz 10 kHz 15 kHz 20 kHz 50 kHz 100 kHz 7. Circuit Construction Figure 6. TA INITIAL _______ 8. Measure Input, Output, Gain Band Pass Frequency

10 Hz 100 Hz 1 kHz 5 kHz 10 kHz 15 kHz 20 kHz 50 kHz 100 kHz

Input Voltage

Output Voltage

dB = 20 log(Gain)

EE 3305 Post Lab Questions 1. Determine the cutoff frequency of the First Order LP Filter and compare to the theoretical value. Draw a graph and note the cutoff frequency of the measured data. 2. What is the gain of the First Order LP Filter in the pass band? How does this compare to the theoretical value? What is the roll off rate of the filter in the stop band? 3. Determine the cutoff frequency of the First Order HP Filter and compare to the theoretical value. Draw a graph and note the cut off frequency of the measured data. 4. What is the gain of the First Order HP Filter in the pass band? How does this compare to the theoretical value? What is the roll off rate of the filter in the stop band? 5. How does the roll off of the Second Order LP Filter compare to that of the First Order LP Filter. Draw a graph of each on the same axis noting the cutoff frequencies and the slope in the stop band. 6. Draw a graph of the Band Pass Filter and show the two cutoff frequencies. Comment on how they compare to the theoretical frequencies and the bandwidth of the filter. 7. Look up the definition of a Ground Loop and explain on how it can effect measurements made on the filters input signal and output signal. Explain how these might be able to be avoided, or minimized. Do you feel that there could be any Ground Loop problems in the circuits that were constructed in the lab?