Design and characterization of a Band-pass Filter.pdf

University of Pennsylvania Department of Electrical and Systems Engineering ESE206 Design and characterization of a Band-pass Filter Goals: • •

Design and build an active band-pass filter Measure the frequency response (magnitude and gain), the poles and bandwidth of the filter.

Background Filters are electric circuits that selectively pass signals of certain frequencies. There are several type of filters such as “Low Pass”, “High Pass” and “Band Pass” filters. As the name implies a “Low Pass” filter is a circuit that passes low-frequency signals and blocks high-frequency ones. A “High Pass” filter on the other hand passes high-frequency signals and blocks low-frequency ones. A “Band Pass” filter passes signals whose frequency lies in a certain frequency band. Filters are very important components and are used extensively in electronic and communication systems. For audio applications for instance, a filter can be used to emphasize certain frequencies and de-emphasize others. Or, you may use a filter to block out noise, e.g. a 60 Hz signal. The principle behind filters is quite simple, although the actual implementation can become complicated, depending on the specifications of the filter. Let us discuss a very simple filter, i.e. a first-order low pass filter. We will make use of the fact that the impedance value of capacitors (and inductors) is a function of the frequency: ZC= 1/jωC

(1)

Consider now the following circuit.

Figure 1: First-order low-pass filter

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The output voltage Vo is measured over the capacitor. Since the impedance of the capacitor decreases for higher frequencies, according to the expression (1), the output voltage will decrease as well. This implies that high frequency signals will be blockedout and that only low-frequency signals pass. The actual expression of the output voltage as a function of frequency can be found by using the voltage divider expression: 1 1 1 jωC Vin = Vin = Vin Vo = 1 jωCR + 1 1 + j 2πfRC R+ jω C

(2)

Or the transfer function can be written as, G ( jω ) =

Vo 1 1 1 1 = = = = Vin 1 + jωCR 1 + jωτ 1 + jω / ω b 1 + jf / f b

(3)

in which τ=RC, called the time constant and ωb=1/RC the bandwidth or –3dB point. As one can see, from the above expression (3), the transfer function is equal to 1 for very low frequencies and goes to 0 for very high frequencies (ω >>ωb). Thus, the circuit of Fig. 1 is a low-pass filter. Figure 2 shows the frequency response of the magnitude and phase (Bode plot) of the transfer function. The bandwidth is defined as the frequency at which the gain has decreased by a factor √2 or (or 3dB).

Figure 2: Bode plot of a first order low-pass filter A high-pass filter can be constructed from a resistor and capacitor as shown in Fig. 3. In this case, the signals at very low frequencies will be blocked because the capacitor acts as 2

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an open circuit. For high frequencies on the other hand, the capacitor can be considered a short circuit, passing high frequency signals.

Figure 3: First order high-pass filter A band pass filter can in principle be constructed by combining a low and high pass filter in cascade. Figure 4 shows such a filter. The first part (C1R1) will pass high frequency signals while the second part (C2R2) will pass the low frequencies (or rejects high frequency signals). However, the filter cannot be considered as a simple cascading of a high and low pass filter since the second part loads the first part. As a result, the overall transfer function is not simply the product of the individual transfer functions of the high and low pass sections (see pre-lab).

Figure 4: A passive two-pole band pass filter. The above examples of filters are called passive filters since they do not make use of amplifiers. Among the disadvantages of such a filter is that there is no gain and that the load resistor RL influences the transfer characteristic. A better way to build filters for low to mid-frequency applications is to use operational amplifiers. Such filters are called active filters. The advantage of active filters is that one can provide amplification, and have a filter whose characteristic is independent of the load. The goal of this lab is to build a passive and an active band-pass filter and to measure the transfer characteristic. A simple active band pass filter (two pole system) is shown in Fig. 5 below. The frequency response (Bode graph) of the gain is given in Fig. 6.

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Figure 5: A two-pole active band pass filter.

Figure 6: Frequency response of a bandpass filter.

Pre-lab assignment 1. Passive band pass filter of Fig. 4: a. Derive the general expression of the transfer function (Vo/V1) of the passive band pass filter of Fig. 4. Assume that RL is not present. b. Use the following values for the resistors and capacitors: R1=2kOhm, R2=20kOhm, C1=1E-6F or 1µF (1 microFarad) and C2=0.22nF. Give the expression of the transfer function and plot the Bode diagram of the gain and phase. You can sketch it by hand or use Matlab. Indicate the values of the – 3dB points and the mid frequency gain (i.e. the gain in the passband, where you can assume that the frequency f1