Op Amps II, Page 1 Op Amps II Op-amp relaxation oscillator Questions indicated by an asterisk (*) should be answered before coming to lab. Build the relaxation oscillator shown in Figure 1 above. The output should be a square wave with a frequency about 1/(2RC). Resistor R1 can be any value between 1kΩ and 1MΩ. Resistor R is one side of a potentiometer. Examine the voltages at (+) and (-) inputs and at the output and follow the action of the switching. It is useful to display v+ and v- simultaneously on the same scale to illustrate that the switching occurs at the crossover of v+ and v-. *Show that the transfer function for the low pass resonant filter, shown in Figure 2, is given by: H(ω) =11− x + x(1 + jωτ)3 (1) Figure 1: Relaxation Oscillator Figure 2: Low-pass Resonant FilterOp Amps II, Page 2 where ω refers to the angular frequency of an oscillator connected to the non-inverting input of the first (leftmost) opamp, τ = RC and x is the ratio of R1 to the total pot resistance R1 + R2. Here R1 is the part of the pot resistance between the output and the inverting input of the first opamp and R2 is the part of the pot resistance between the inverting input and output of the first opamp. [Hint: Begin by naming the output voltages of each op amp, from left to right, as v1 through v4. Then use the infinite gain assumption to show that: 2114)()(RvvRvvinin−=− (2) Next, use what you know about RC filters to find v4 in terms of v1.] The resonance depends on both 211RRRx+= and RCωωτ=. Figure 3 shows the gain versus ωτfor four different values of x. It can be shown (you do not have to do this) that the real part of the denominator of equation 1 vanishes when()132=ωτx . Furthermore, the gain is sharply peaked when 3=ωτand91=x . Figure 3Op Amps II, Page 3 When you understand the equation for the transfer function, build the circuit. It is convenient to use a TL084 with four op amps in a package. Choose RC so that the resonant frequency is 2 to 5 kHz. Tune the pot until the circuit nearly oscillates. See how close you can get. Notice how oscillations grow and die exponentially. Find the resonant frequency by feeding in a sine signal from a function generator. (You may need to decrease the input voltage considerably to avoid saturating the filter near resonance.) Check the high frequency roll off. It should be proportional to 1/ω3. Estimate the gain at resonance. Make a Bode plot of the transfer function. (Spend your time wisely here by starting with a survey to find the frequencies where important features occur. Important features include resonance, high-frequency roll off and low-frequency constant
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