EE 322

Lab 8: Voltage-Controlled Voltage Source Filters

In this lab we will experiment with a simple 2-pole low pass filter, as implemented by the `voltage-controlled voltage source' (VCVS) configuration. The VCVS filter is discussed in Sections 5.06 and 5.07 of Horowitz and Hill; you have also seen it in the circuit theory lab. It has the nice feature that the pole locations of the transfer function in the s-domain can be varied by changing the amplifier gain (see below), without changing the radius of the poles from the origin. We will use this feature to determine the effect of varying the pole locations upon the frequency and transient response of the filter.

Pre-Lab

• Table 5.2 of Horowitz and Hill gives values of the amplifier gain K and frequency multiplier f_n (f₀=f_nf_c where f₀ is the pole radius, and f_c 3dB cutoff frequency). These values can be found from transfer functions for each filter. The gain K can be found from our in class analysis. The cutoff frequency is found by solving Abs(H(jw_c))=0.707 for w_c. Use the normalized polynomial (normalized to w₀=1) for a two-pole Butterworth filter, H(s) = 1/(s²+1.414s+1), show that K = 1.586 and f_n = 1 for the filter. Repeat for for a 2-pole Bessel, H(s) = 1/(s²+1.732s+1), and 2.0 dB Chebyshev filter, H(s) = 0.794/(s²+0.886s+1), to confirm the values of K and f_n given in Table 5.2 of Horowitz and Hill.
• Use Matlab to plot the frequency response (the plot should be linear for easy comparison to the lab measurements) and step response for each of these 2-pole filters

1. The basic configuration of the low-pass VCVS filter is as shown below.
   • First build and test the amplifier. Use a 10k pot for the feedback resistor R₂ and choose R₁ so that the gain K of the amplifier can be varied from 1 to 3. Sketch the amplifier schematic. This amplifier enable the pole locations to be varied at constant radius f₀ from the negative real axis onto the j omega axis.
   • Design and construct a single-stage filter having its poles at a radius f₀ = 10 kHz. Set the gain for a Butterworth frequency response. This circuit will be tested with your function generator, which has a 50 ohms source impedance. The source impedance will be in series with the input resistor R of the filter, so you will need either to make R large in comparison, and/or to take the source impedance into account when constructing the circuit. NOTE: the resistors for the amplifier are not equal to each other or to the filter resistors.

   
   

   • First test the amplifier by first measuring its low-frequency gain and comparing with what the gain should be.
   • Confirm that the filter has a low-pass frequency response and measure its cutoff or 3-dB frequency f_c. For a Butterworth filter, the 3-dB cutoff frequency should be the same as the pole radius, i.e., w₀ = w_c. Is it?
   • Measure and plot the frequency response (measure and plot the gain at each kHz from about 1 to 15 kHz). Compare this with theory and Matlab plot form the prelab.
   • Measure and document (carefully draw) the step response of the filter for comparison with later results. Does the step response overshoot? What is the gain after the transient response dies out, and what should the gain be?
   • Sketch the pole locations of this filter, approximately to scale. Use the transfer function for this.
2. Chebyshev and Bessel Filters

   Don't change the R's and C's that determine f_c.

   • What are the effects of varying the amplifier gain and thus the pole locations on the step response of the filter?
   • Document for the step response of a Bessel filter (K = 1.268). Compare this with the Matlab plot form the prelab.
   • Document for the step response of a 2.0 dB Chebyshev filter (K = 2.114). Compare this with the Matlab plot form the prelab.
   • Measure and plot the frequency response of the 2.0 dB Chebyshev filter (measure and plot the gain at each kHz from about 1 to 15 kHz). Compare this with theory and Matlab plot form the prelab.
   • Observe the plot of the frequency response of the 2.0 dB Chebyshev filter. Why is it peaked? The peaking constitutes the `ripple' of the (frequency response of the) filter. Is the ripple equal to 2.0 dB? Measure the 3 dB cutoff frequency and compare with the theoretical value obtained from f_n = 0.907. (f_n = f₀ / f_c, f₀ is determined by RC, and f_c is the 3-dB frequency) Why is the cutoff frequency larger than the pole radius? Illustrate using a sketch of the pole locations similar to that of part 1. Compare this with the Matlab plot form the prelab.
   • Measure and plot the frequency response and f_n for the Bessel filter and compare with theory (measure and plot the gain at each kHz from about 1 to 15 kHz). Does the Bessel filter have ripple? Sketch its pole locations, as before. Compare this with the Matlab plot form the prelab.
   • Compare the filters. Does the Butterworth filter have the flattest response? Which has the fastest falloff at high frequencies? Which has the best step response?
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