Ideal Operational Amplifier Lab
Background: In the analysis of an ideal operational amplifier configuration, two powerful assumptions are made: 1. No current flows into the amplifier input nodes. 2. When feedback is employed, the differential input voltage is zero.
The ideal operational amplifier model:
Procedure: 1. Construct an operational amplifier capacitance multiplier. (Ref. Figure 1). Verify the value of Cff. (Hint: Construct a Bode Plot). Derive the value for Ceff assuming an ideal op amp. 2. Construct an operational amplifier virtual inductor (Ref. Figure 2). Verify the value of Leq. Assuming an ideal op amp, derive the value for the virtual inductor, Leq 3. Construct one of the operational amplifier power booster circuit from Figures 3. Predict the circuit performance. Experimentally verify the circuit operation. Write Up and Data: For the given experiment the characteristics of an ideal operational amplifier were examined using several configurations. It is assumed for the ideal case that the input impedance looking into the inverting and non inverting nodes has infinite impedance. Also when DC feedback is produced from the output of the op amp to the inputs of the inverting and non inverting terminals, the differential input voltage is assumed to be zero. Part 1: In the first part of the experiment a uA741 operational amplifier was used to construct a capacitance multiplier. For this configuration uses the characteristics of the ideal op amp to provide large capacitance values with out the use of large components (Figure 1). Figure 1: Capacitance Multiplier To develop an understanding of the circuit's characteristics hand calculations and predictions were made as to its performance. A test voltage (Vt) and test resistance (Rt) were connected to the circuits output node where the output capacitance is measured. This aided in deriving a transfer function for the circuit and allowed the predicted output capacitance (see Lab Notes page 1). From this a relationship of Vo / Nt a bode plot was constructed to simulate the circuits behavior over a specified frequency range (Figure 1a). Once an understanding of the circuit's behavior was predicted, the circuit was constructed in a laboratory environment; measurements were taken and compared to the previous predictions. It was found by taking measurements over a frequency range of 10Hz to 100kHz and plotting the results the measured data agreed within an acceptable tolerance range for the components being used (Table 1). Table 1: Measured Data for Capacitance Multiplier Part 2: In the second part of the experiment an ideal op amp was constructed in a virtual inductor configuration. In this configuration the op amp acts as an active inductor allowing the non inverting input to behave as an inductor. With such a configuration large inductance are made possible (Figure 2). Figure 2: Virtual Inductor
Again hand calculations and predictions were made to determine the outcome of the simulated inductor. A test voltage (Vt) was connected to the non inverting terminal of the op amp to provide an input AC source to the simulated inductor. With this five equations were derived and solved using the methods of determinants. This provided a mathematical representation of the predicted circuit's behavior (see Lab Notes, pg. 3 & 4). The final result in the derivation was determining the input impedance looking into the non inverting terminal. From the productions this terminal should model an inductor (L = R1 R2 C). However it was found through the derivation that the impedance at the non inverting node modeled an inductance with a series resistance (L = R1 R2 C + R1). Also one characteristic of the simulated inductor was having the operational amplifier in a unity gain configuration and matching the resistance of the positive feedback loop. Although for practical applications the resistance in the positive feedback loop needed to be slightly different in size to avoid instability at high frequencies.
To better understand the circuit's behavior a simple model was created as follows. An input test voltage (Vt) and test resistance was hooked in series with the calculated impedance of the simulated inductor. This configuration produced a simple high pass filter (Figure 2.1). In this configuration the simulated inductor could be tested for its response over a specified frequency range. Hand calculations were developed to obtain a relationship of the input test voltage to the output terminal we are calling VdB. This result produced a transfer function, which was bode plotted and compared to our expected response in Figure 2a. Figure 2.1: Test Model for Simulated Inductor Next the circuit was constructed in a laboratory environment and tested agents the predicted results. Several attempts were made at building the circuit and producing expected results but none were completely successful. To test the previous predictions and hand calculations a Pspice model was generated and plotted over the desired frequency range (Figure 2b). Figure 2b: Pspice Simulation for Virtual Inductor Throughout the second part of the experiment for the virtual inductor, all the simulations and hand calculations match the expected predictions. However the data gathered in the constructed/measured circuit was inconclusive. At low frequencies the data matched within acceptable tolerances, yet the data dropped off at approximately 100Hz. This was due to a lower break frequency of approximately 14.0Hz, and or the unmatched ratio of the components within the circuit. Part 3: For the third part of the experiment a power booster configuration circuit was examined to predict and measure the expected results (Figure 3). Figure 3: Power Booster As with the experiments specifications the circuit was examined and predications were made as to its performance. The purpose of the power booster circuit was to provide more current to the load. This would allow moderate size loads to be driven using this op amp configuration. However with closer inspection one can see the negative feedback loop is blocked by a capacitor. This capacitor would allow no negative DC feedback to the op amp, which would cause the op amp to run open loop. This was tested using Pspice (Figure 3a), and constructed in a laboratory environment to confirm this prediction. The results for both the simulation and experimental measured data matched as expected. The op ainp with no DC negative feedback caused the output to rail at the DC limit set at l5volts. Attempts were made to modify the circuit by moving the capacitor outside the feedback loop or by shorting a resistor to the across the capacitor. However neither attempt provided the intended results. To complete the experiment another circuit was examined. This time a half wave rectifier op amp configuration was constructed and predictions were made as how it would behave (Figure 4). Figure 4: Half Wave Rectifier In predicting the output it was determined that as the input AC sine wave oscillating about the zero volts offset would have all negative swing eliminated and produce a positively rectified wave. As predicted, when the circuit was constructed and the output measured it produced exactly what was expected (Figure 4a). Conclusion and Discoveries: 1. When examining the capacitance multiplier circuit over a frequency range the following is learned. When the circuit is at very low frequencies the capacitor in the positive feedback loop looks like an open circuit. At high frequencies the capacitor looks like a short circuit passing current back to the non inverting terminal. 2. The virtual inductor op amp configuration acted as a simulated inductor except there was an addition of an extra internal resistance (RI) that was seen looking in at the non inverting terminal. This extra resistance allowed for the following. At low frequencies the current would all pass through the internal resistance (RI) of the circuit. But as frequencies became large the non inverting feedback loop would pass more and more current causing the overall impedance to approach infinity. 3. With extensive trial and error in the construction and predictions for the power booster circuit it was learned that by eliminating all DC negative feedback from any circuit configuration would always produce an infinite open loop gain of the op amp. In closing the overall analysis of using hand calculations, MatLab and experimental measurements for the given laboratory experiment allowed one to develop methods of analysis and continually checking themselves. For these designs it was learned that by developing mathematical model and examining that model for effects such as open loop gain, helps in the design process for wanted or unwanted signal response.
Electrical Engineering lab key words: Matlab, ideal operational amplifier, positive feedback, internal resistance, negative feedback, model, rolloff, cutoff, frequency, uA741 Op-Amp, half wave rectifier, capacitance multiplier, inductor, differential input voltage, linear systems.
