PHY 405: Electronics Lab Lab # 5 Lab # 5: Calculus with circuits Operational amplifiers were originally developed as the building blocks of analog computers. In this lab, you will construct circuits that simulate the action of linear differential operators. Note: remember that the solderless breadboards can have ∼ nF capacitances between neighbouring sockets. You are welcome to implement your circuits on a solderable breadboard if you prefer. Background preparation • Operational amplifier basics: AOE, Chapter 4, 225-227. • Op Amps for Everyone, by Ron Mancini. Available from the course webpage. Differentiator a) In addition to amplifying signals (as you saw last week), an opamp can be configured to work as a differentiator. In the configuration shown in Figure 1, the output dVin(t) voltage in the time domain is Vout(t) = −RC dt . Construct this differentiator circuit using an LF356, choosing component values of R ≈ 1 kΩ and C ≈ 1 uF. Set the HP power supply to tracking mode to create +15 V and −15 V power supplies (V+ and V−) for the opamp (and don’t forget the decoupling capacitors). Connect the function generator to Vin, and apply a variety of input waveforms (sine wave, square wave, triangle wave) with f ≈ 100 Hz. For each waveform verify using the oscilloscope that the differentiator’s behaves as expected. In your lab report, include a plot of the observed input and output waveforms. Note: in order to get the differentiator to operate stably, you may need to place a small capacitor (∼ 1 nF) in parallel with the feedback resistor, and a resistor (∼ 1k) in series before the input capacitor. Figure 1: The capacitor on the input and the resistor in the feedback cause this opamp to output a signal dVin(t) proportional to the derivative of the input signal: Vout(t) = −RC dt . 1 PHY 405: Electronics Lab Lab # 5 b) As in previous labs, we can define the gain (or transfer function) of the differentiator in the frequency domain: Vout(f) Hd(f) = (1) Vin(f) Use the function generator to apply sine wave inputs, and measure the transfer function of the differentiator. In your report, explain how this transfer function in the frequency-domain is consistent with the circuit’s action as a derivative operator in the time-domain. (45 mins) Integrator In this task, you will construct an integrator: the inverse of a differentiator. a) Devise an opamp circuit that acts as an integrator in the time-domain Z Vout(t) / dt Vin(t) (2) b) Construct this circuit using a second LF356 chip, and verify that it acts as an inte- grator using a variety of waveforms from the function generator. c) Measure the transfer function of the integrator, Hi(f). Explain how this is consis- tent with the integrating action of the circuit. Note: you may find that the output of your integrator slowly and steadily rises (or falls) towards one of the power rails and gets stuck there. You can fix this by placing a large resistor (& 1 MΩ) between the output and the inverting input of the opamp. You may also need to trim the input offset voltage of your opamp with a 10k trimming potentiometer (“trimpot”), as shown in Figure 2, to center the output of the integrator around zero volts. Figure 2: Offset adjustment with a 25k trimpot (Figure 55 in the LF356 opamp datasheet). (45 mins) 2 PHY 405: Electronics Lab Lab # 5 Cascaded integrators and differentiators a) Construct a cascaded circuit, with the integrator’s input connected to the differ- entiator’s output. Feed the differentiator with a triangle wave, and describe the action of the cascaded circuit. Be as quantitative as possible. Explain any non-ideal behaviour that you observe, based on the transfer functions that you measured above. b) Rewire the integrator as a differentiator, so that you now have two consecutive differentiators. Feed the cascaded differentiators with a triangle wave. Observe the output of the first, and second differentiator circuits. Using an accurate sketch or a plot, describe if this behaviour is consistent with what you expect. Be as quantitative as possible. (30 mins) Design problem (Extra credit) a) Design a circuit that simulates the differential equation of a simple harmonic oscillator, with a resonance frequency of 100 Hz. Verify, using Qucs, if your design works. Implement it on a breadboard if you can. 3.