EE-2302 Passive Filters and Frequency Response Objective The student should become acquainted with simple passive filters for performing highpass, lowpass, and bandpass operations. The experimental tasks also provide an introduction to Complex Frequency Response and Phasor Algebra. Discussion Filters are used to remove unwanted frequency components from a signal. Filters can be implemented using a combination of active elements (independently powered elements like operational amplifiers) and passive elements (non-powered circuit elements like resistors, capacitors, and inductors). Filters can also be constructed solely using passive elements. In this experiment, you will construct several simple passive filters and evaluate their performance. The frequency response is a complex gain as a function of frequency. The frequency response magnitude is the amplitude gain for a sinusoidal input at the given frequency. The frequency response's argument is the phase shift obtained for a sinusoidal input at the given frequency. The frequency response can be obtained by replacing the capacitor with the impedance 1/jωC and the inductor with the impedance jωL. The circuit is then analyzed treating these impedances as normal resistors. The output voltage divided by the input voltage Vo(jω)/Vi(jω) (typically a complex number for given circuit element values) is the frequency response. Pre-lab Determine the frequency response, Vo(jω)/Vi(jω), for each of the following filters. Let R, C, and L be variables in these frequency responses. R + + Vi C Vo - - Filter 1 R + + Vi L Vo - - Filter 2 L C + + Vi Vo R - - Filter 3 Procedure 1. Select the following components: a 470 Ω resistor, a 1 µF capacitor, and a 20 mH inductor. These values are only approximate. Measure the actual resistance, capacitance, and inductance of these components, and record your answers on the data sheet. 2. Construct each of the filters (one at a time) using the selected components. 3. Connect the NI 5411 Function Generator output to the input terminals of each filter and Channel 1 of the NI 5102 Oscilloscope. Connect the filter output to channel 2 of the oscilloscope. Select a sinusoid output for the NI 5411 Function Generator with an amplitude of 200 mV Vp. 4. Set the Vertical Volts/Div control and Horizontal seconds/division controls for ease of measurement. 5. Measure the voltage gain (magnitude) and the phase shift of the output of each filter for the frequencies indicated on the data sheet. The voltage gain is the amplitude of the output signal (Vo) divided by the amplitude of the input signal (Vi). To find the phase shift, measure the time difference between the zero crossing (going positive) of the two waves. This time difference is denoted ∆t. The sinusoid’s period, denoted T, is the time between successive zero crossings (going positive) of a single wave. This can be measured of computed from the frequency. The phase shift is to 360° in the same ratio as ∆t is to the period: phase shift = 360° ∆t/T. If the input signal’s zero crossing occurs before the output signal’s zero crossing, the output signal is said to lag the input, and the phase shift will be negative. If the output waveform's zero crossing occurs before the input waveform’s, it is said to lead and the phase shift is positive. Its important to note which signal’s zero crossing occurred first when measuring ∆T. Filter 1: R_____________ C____________ 50 Hz 100 Hz 200 Hz 300 Hz 400 Hz 500 Hz 800 Hz 1500 Hz. 50 Hz 100 Hz 200 Hz 300 Hz 400 Hz 500 Hz 800 Hz 1500 Hz. Vo Vi Gain ∆t T ∆Φ Filter 2 : R__________ L____________ 200 Hz 500 Hz Vi Vo Gain 1000 Hz 2000 Hz 5000 Hz 10 kHz 20 kHz 200 Hz 500 Hz 1000 Hz 2000 Hz 5000 Hz 10 kHz 20 kHz ∆t T ∆Φ Filter 3: R_________ 200 Hz 500 Hz L_________ C_________ 1000 Hz 2000 Hz 5000 Hz 10 kHz 20 kHz 1000 Hz 2000 Hz 5000 Hz 10 kHz 20 kHz Vi Vo Gain 200 Hz 500 Hz ∆t T ∆Φ 4. 5. 6. 7. Using your measured values, plot the amplitude gain verses the frequency and the phase shift (in degrees) verses the frequency. Semi-Log graph paper can be very effective for this graph. Use the linear axis for the gain and phase data and the log axis for frequency. Substitute the measured values for R, C, and L into the theoretical frequency responses obtained in the prelab, plot the resulting theoretical amplitude gain verses the frequency and the theoretical phase shift (in degrees) verses the frequency. Compare the results of the theoretical values with the experimental values. Note that phase shift is ambiguous with respect to phase shifts of 360°, so you can add ±360° phase shifts into your experimental results to obtain better agreement. The passband of the filter is defined as the range of frequencies where the amplitude gain exceeds 0.707. Use the equation you derived for the theoretical gain to compute the passband of each filter. Can you think of applications for these different types of filters?