Homework 29ECE201: Linear Circuit AnalysisDue in class: Wednesday, November 9, 2011Question 1The data listed in the following table was taken for the circuit shown in the following figure.a. Fill in the values for the third column of the following table. b. What is the value for RL such that the power delivered to RL is the maximum? Find the maximum power delivered to RL.c. Assume that R1=1kΩ and we now take it out of the circuit, i.e. we set R1=0, what is the value for RL such that the power delivered to RL is the maximum? Find the maximum power delivered to RL.RL (kΩ)vAB (V)iA(mA)481010Hints (you should be able to solve the previous problem without the hints. Please refrain from reading the hints unless you have invested significant effort). You are not required to write up answers for the questions in this hint.1. Do you really need to know the value of R1 to solve sub-questions a) and b)?2. Is it possible to enclose R1 together with the square box to form a new network?3. If so, please redraw the circuit (we strongly recommend you do so, as this shows the magic of abstraction that is universal for engineering).4. For the new circuit with the new network, is iA still the same, i.e. the current flowing into the network from node A?5. Is vAB the voltage across the resistor RL?6. If we know the voltage across a resistor, how do we calculate the current through it? Is vAB = RLiA (don’t make your conclusion too quickly)?7. Now that we filled the table, how do we model the new network, i.e. the network you drew in step 3? Can we use the Thevenin’s equivalent circuit to model the network?8. What is the relationship between vAB, iA, and the two variables of Thevenin equivalent circuit, Vs, and RTH?9. We have two unknown variables, Vs, and RTH, where do we find data to solve this equation?10. Assume that you have solved Vs, and RTH, which value should RL take so that it receives the maximum power?11. What is the formula for maximum power delivered to RL, assume you know Vs, and RTH? You should memorize this formula.12. Question c) is very similar to b), but involves R1. Can we still use the circuit model you drew in step 3?13. Let’s go back to the original circuit in the previous page, what can we say about the original square box, i.e. in terms of its equivalent circuit model, Vs’, and RTH’, when R1=2kΩ?14. When R1=0, the original square box should remain the same. Repeat steps 10 and 11, using Vs’, and RTH’.Question 2:The op amps in the following circuit are ideal. The voltage vIN(t) has been toggling between -0.2V and 0.2V for a long time (as shown in the plot). Find and plot vout(t). Label the axis. (note, you should show your calculation and reasoning why you got the vout plot, though you do not have to get accurate numerical values in your plot).!!!!!!!!!Hints (you should be able to solve the previous problem without the hints. Please refrain from reading the hints unless you have invested significant effort).You are not required to write up answers for the questions in this hint.1. Let’s mark iL and vL on the circuit. Be sure to mark the polarity of voltage and the direction of current, and make sure that they follow the passive sign convention for the inductor.2. What is the relationship between V+ and vL?3. We know that the input current to both input nodes of an ideal Op-Amp is zero. Now look at the wire connecting the top of the inductor and the noninverting input. Does it have current at any time? If not, what do we call a circuit segment that has no current at all time? Can you cut open such a circuit element? 4. Which input node, the inverting one, or the non-inverting one, should you start your analysis?5. Let’s start with the non-inverting node and at t=0. Assume you did what we suggested in step 2. Note that the relationship between V+ and vL still holds. Our goal is to find out the trace of vL when vIN is switching between -0.2 V and 0.2V.6. We do not know exactly vL at t=0-, since we don’t know how the input voltage changed during t<0 (strictly speaking, we know what happened, but that’s not what we learned how to analytically analyze an RL circuit).7. What we learned is that we must assume that the input voltage has stayed at a fixed value for a long time, and only at t=0 there is a voltage jump or other things happen (i.e. a switch opens or closes). This is what we typically call transient analysis. 8. Now redraw the RL circuit assuming the following, the vIN stayed at -0.2 V for all t<0 and at t=0 it jumps from -0.2 V to 0.2V, what would be vL?9. We are asked about a voltage across an inductor. Should we solve vL directly or should we solve iL first? Which value does not change when something jumps?10. Assume you decided that we should first solve iL. What is iL(0-), what is its direction, pointing up or down?11. What is iL(0+), why?12. We now need to find vL(0+), but how?13. We know that in equilibrium, an inductor can be replaced by a short circuit and a capacitor can be replaced with an open circuit. What can we replace an inductor in such a transient condition?14. Assume you decided that we can replace the inductor with a current source of the same direction and value of iL(0+), calculate vL(0+) You should a consider the subcircuit with a vIN(0+), resistor and the current source (in place of the inductor).15. Next you should calculate iL(∞), assuming vIN stays at 0.2V. What can we replace the inductor with if stays at 0.2V for a long time?16. What would be vL(∞).17. We next calculate the time constant for the RL circuit. What is the formula to calculate the time constant for the first order RL circuit? You should remember this formula at least during the semester you take ECE 201.18. Once we know vL(0+), vL(∞) and time constant, we should plot vL. Does it reach vL(∞) fairly close after 1 microsecond?19. Now we calculate vout. Let’s assume with know vL, what is V+ and V-?20. After we know V-, how do we find vout? Is there any relationship between the current flowing through the 1kΩ and 9kΩ resisters?21. Once we have the solution to our model system, i.e. vIN changes from -0.2 V to 0.2V only at t=0, we need to plot the response assume it flips between -0.2V and 0.2V. Do you think the plot you have for the model system is correct for our question at least between 0 to 1 microsecond?22. Unfortunately, what we plotted for our model system between 0 to 1 microsecond is not correct for our question.
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