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RUTGERS Cardiovascular Dynamics Lab • Department of Biomedical Engineering 617 Bowser Road • Piscataway • New Jersey 08854-8014 • 732/445-3727 • FAX: 732/445-3753 • e-mail: [email protected] Thevenin Equivalent Circuit Aim: In this experiment we study the use of the Thevenin equivalent circuit and how it can be applied to gain a better understanding of our lab function generator Theoretical Introduction The Thevenin equivalent circuit The general circuit for the Thevenin equivalent circuit model is provided in figure 1. The idea behind any equivalent circuit is that a simpler well-understood circuit can be used to replace a more complex circuit between two nodes of the circuit. In applying this circuit technique, we are only requiring that the voltage and current external to the equivalent circuit are identical at the two nodes. Given this definition, it should be clear that the equivalent circuit may be a completely different circuit between the nodes that are being replaced by the thevenin circuit. In this lab, we will be using the Thevenin circuit to model the waveform generator output circuit. As you might expect, the function generator internal circuitry is much more complex than that of Figure 1. QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Figure 1. General Thevenin circuit model. There are only two parameters that must be known in order to describe the Thevenin equivalent. They are the Thevenin voltage and the Thevenin resistance or impedance. A simple two step procedure can be used to discover these two parameter values from external circuit variables. First, we measure the no load or open circuit voltage. As you see from figure 1, if there is no external current, then the internal Thevenin current must also be zero. In this case, the voltage at the external nodes must be equal to the Thevenin voltage Vth, since there is no voltage drop in the Thevenin impedance. Now that Vth is known, the Thevenin impedance Zth can be found by applying an external load. In the simplest case we can apply a short circuit to the Thevenin circuit. This yields the short circuit current Iss. Then by Ohm's Law, we can solve for Zth as follows: Zth Vth Is s Now, with Vth and ZTH known, the function generator can be replaced with the its Thevenin equivalent circuit in all future experiments. 2. Lab Experiments Equipment 1. PC with Biopac system and software 2. Simulink software 3. RC substitution box 4. Tektronics function generator 5. 1 Biopac BNC input cable, T-connector, and clip connector A Biopac Experiment (Measure Function generator Thevenin circuit) Procedure: Begin by calibrating the Biopac to read voltage on channel 1. Then, connect one channel of the Biopac to the circuit in figure 2. QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Figure2. Lab Circuit Set the Biopac software to the following settings: -Channel gain X100, DC input mode -Acquisition rate of 100Hz Save to hard drive once. Record for 15 seconds duration Set channel one to acquire data and Plot. Set RBOX=10 M Adjust the function generator to 1.0 Hz and adjust the output so that you can record a 5 volt peak to peak Sinusoid The data that you record in this experiment will help you to calculate Z th and Vth Create a table of voltage measurements by varying the RBOX down from a 100 starting value in 10 decrements down to 10. For each new value of R, use the Biopac software to measure the peak to peak voltage and add this value to your table of R and V Lastly, remove the resistance from the circuit entirely. Note that now the MP30 input is connected directly to the waveform generator Measure the peak to peak voltage at 10 and 100 Hz. For this condition, notice that the only load on the Waveform generator is the MP30. During most of our experiments, we assume that the MP30 input current is minimal. In fact, it is near only 1-10 Micro Amps. For one volt this is the same as a 1M Load. Hence, let's assume that this is equivalent to open - circuit conditions, providing us with Vthevenin. With Rbox removed, measure the peak to peak voltage at 10 and 100 Hz. and record the values. Notice that we have not measured the short circuit condition yet. But, you will recall that we measured the output of the generator with a 10 Load. In this case we will assume that the 10 load is much smaller than the ZTH Therefore, we assume that it approximates a short circuit condition. Solve for R Thevenin, using Ohm’s law. B SIMULINK EXPERIMENT For the Simulink Model it is required to create a model the predicts the MP30 input voltage given the v thevenin , R thevenin and R box , referring to figure 3. You will find that the MP30 voltage is really the output of a voltage divider circuit. So, using the voltage divider rule you can find that Vthevenin VMP100 (Rthevenin Rbox ) (refer to your classroom text) In this experiment, you need to input a sinusoidal Thevenin voltage source of the same magnitude and frequency as in the Biopac experiment. Then use the Simulink blocks to calculate the MP30 input voltage using the above voltage divider formula. A working model should give you a sinusoid of magnitude that follows your table of measured R and V according to the values in the divider equation. Use the V thevenin and R thevenin that you measured in the Biopac experiment. Then vary the value of Rbox to predict the same measurements that you obtained for R box. using your Simulink model. Record the values of the MP30 voltage for 5 values of R box that you did in the Biopac experiment. Just measure the peak to peak values of V MP30 REPORT 1. Print a graph for a single value of Rbox for the Biopac and Simullink. Create a graph of voltage (Peak -peak) versus the box resistance for both frequencies measured. 2. Plot the MP30 Voltage that was obtained from the Simulink experiment on the graph from question #1. Do an error analysis and discuss sources of discrepancy. Does frequency affect the result? Is Z Thevenin an impedance or just a resistance? Explain. 3. What was Vthevenin and indicate it on the resistance graph. Assuming that the 10 resistance is a short circuit condition, use this condition to find Rthevenin from the theory and using Ohm's Law. 4. Given the values of V an R thevenin and the circuit model of Figure 3, perform a loop analysis to solve for the Output voltage input to the MP30 for any value of load resistance Rbox. Plot this equation and compare your result to the data in answer #1. 5. Provide an error analysis for question 4. 6. What Rbox value caused the output voltage to be 0.5 VThevenin? Show this from a loop analysis of the Thevenin circuit. 7. Describe another method that uses question 7 to find the Thevenin resistance. 8. Derive a formula that predicts the MP30 voltage as a function of R thevenin and R box. 9. Provide a written summary that discusses what you learned in this experiment relative to your lecture course material. Did the experiment validate the theory? 10. If the input resistance of the MP30 is 1 M show how this will effect your results. Provide a new circuit model using this information.