ENGG 1203 Tutorial. Solution. Op Amps 7 Mar Learning Objectives. Determine V o in the following circuit. Assume that the op-amp is ideal.

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1 ENGG 03 Tutorial Q Op Amps 7 Mar Learning Objectives Analyze circuits with ideal operational amplifiers News HW Mid term Revision tutorial ( Mar :30-6:0, CBA) Ack.: MIT OCW 6.0 Determine V o in the following circuit. Assume that the op-amp is ideal. Solution Q Since V - = V +, V - = V. So there must be /A flowing left through the two 6 ohm resistors. There must be a corresponding / A flowing to the left through the ohm resistor. V o is then the sum of V - = V and the V across the ohm resistor. Determine the current I x when V = V and V = V. Determine the voltage V A when V = V and V = V. Determine a general expression for V A in terms of V and V. 3

2 Solution Q3 When V = V and V = V, I x = A When V = V and V = V, V A = V A general expression for V A : 3 - Use a single op-amp and resistors to make a circuit that is equivalent to the following circuit. = V n 6 Q Q Use the ideal op-amp model (V + = V - ) to determine an expression for the output current I o in terms of the input voltage V i and resistors R and R. Determine R so that V o = (V V ). v i +v x v i +v x v x 7 8

3 Solution Q6 No current in +ve or -ve inputs: Ideal op-amp: A proportional controller that regulates the current through a motor by setting the motor voltage V C to V C = K(I d I o ) K is the gain (ohms) I d is the desired motor current I o is the actual current through the motor. 9 0 Solution Q7 Consider the circuit inside the dotted rectangle. Determine V as a function of I o. V + = / x I o = V - V - = 00/( ) x V V = / x I o x 00 The shaft angle of the output pot tracks that of the input pot If the person turn the left potentiometer (the input pot), then the motor will turn the right potentiometer (the output pot) Determine the gain K and desired motor current I d. KCL at -ve input to right op-amp:

4 Solution Solution Pot resistances depends on shaft angle Lower part of the pot is αr Upper part is ( α)r, where R = 000Ω. α is from 0 (most counterclockwise position) to (most clockwise position) If α i >α o, then the voltage to the motor (V M+ V M ) is positive, and the motor turns clockwise (so as to increase α o ) i.e., positive motor voltage clockwise rotation. Determine an expression for V M+ in terms of α i, R, and V S. The output of the voltage divider is The op-amp provides a gain of, so V M+ = V +. 3 The following circuit produces a voltage V o that depends on the position of the input pot. Determine an expression for the voltage V o in terms of α i, R, R, R, and V S. The following circuit produces a voltage V o that depends on the positions of both pots. Determine an expression for V o in terms of α i, α o, R, and V S. The positive input to the op-amp is connected to pot so that The positive input to the op-amp is connected to a voltage divider with equal resistors so The output pot is on the output of the op-amp, so The input pot is on the output of the op-amp, so In an ideal op-amp, V + = V so In an ideal op-amp, V + = V so 6

5 Assume that we are provided with a circuit whose output is α i /α o volts. We want to design a motor controller of the following form so that the motor shaft angle (which is proportional to α o ) will track the input pot angle (which is proportional to α i ). Assume that R = R 3 = R = 000Ω and V C = 0. Is it possible to choose R so that α o tracks α i? If yes, enter an acceptable value for R. Assume that R = R 3 = R = 000Ω and V C = 0 If R 3 = R then the right motor input is V. If α i = α o then the gain of the left op-amp circuit must be so that the motor voltage is 0. The gain is R + R /R, so R must be 000Ω Assume that R = R 3 = R = 000Ω and V C = V If R 3 = R then the right motor input is V. If α i = α o then V + = V = for the right op-amp. We need the left motor input to be V. But if the left motor input is V and V C = V then V must also be V, which leads to a contradiction. Q8 You have to design a hammer machine (i.e. using a hammer to hit a platform to see how strong the participants are). The design goal is to generate an output voltage (V o ) which is proportional to the force (F) applied on the hammer, i.e. V o = m x F + C, with m > 0 and C > 0. (a) You found a force-sensitive resistor (FSR) from the catalog, which can be modeled by RFSR = 0kΩ F You then design a circuit as a potential divider. Will this circuit correctly implement? No, because V o is not linearly proportional to F. 9 0

6 (b) Find the gain of the following circuit: At the two op-amp inputs, V - = V + = V i. Since V i is related to V o through the two resistors such that V o R = (R 3 + R ), R Vo = + R 3 Vi (c) Design (by using the non-inverting amplifier circuit) a circuit such that the output voltage (V o ) is directly proportional to the input force (F). Replace R by the FSR. We then have R F R V V V F V 3 3 i o = + i = + i V o is a linear function of F. R F R V V V F V 3 3 i o = + i = + i R F R V V V F V 3 3 i o = + i = + i (d) The system requires that when the force F = 0 N, V o = V; when F = 0 N, V o = V. Construct the circuit designed in (c) using only one FSR, one op-amp, one V power supply, and k ohm resis. When F = 0N, V o = V i = V. We can use R = k ohm and R = k ohm. k ohm resistors can be made by two k ohm resistors in series. When F = 0N, R3 = 0 + R3 = kω (e) Using the above circuit, what is the value of V o when someone hits the hammer too hard, generating a force of 00 N? V (f) Suggest modification(s) such that the max. allowable force to the circuit is 60 N. Change R 3 to /3 k ohm. This can be done by parallel composition of three k ohm resistors.

7 (Appendix) Q9 Solution Fill in the values of R and R required to satisfy the equations in the left column of the following table. The values must be non-negative (i.e., in the range [0, ]) R R V o = V - V V o = V - V V o = V - V 3 rd : Negative R i.e. Impossible R R V o =V -V 0kΩ 0kΩ V o =V -V 0kΩ 0kΩ V o =V -V Impossible Impossible 6 (Appendix) Q0 (Appendix) Q What is V o? V o = 0 V o = V V V 3 V 3 + V V 3 + V Students Kim, Pat, Jody, Chris, and Leon are trying to design a controller for a display of three robotic mice in the Rube Goldberg Machine, using a 0V power supply and three motors. The first is supposed to spin as fast as possible (in one direction only), the second at half of the speed of the first, and the third at half of the speed of the second. Assume the motors have a resistance of approximately Ω and that rotational speed is proportional to voltage. For each design, indicate the voltage across each of the motors. 7 8

8 Solution (Jody s Design) Solution (Chris s Design) P.D. of motor = 0V P.D. of motor = 0.0V P.D. of motor 3 = 0V Wrong design P.D. of motor = 0V P.D. of motor = 0.V P.D. of motor 3 = 0V Wrong design 0 0. Eq. R. (Red): K+~ K Eq. R. (Blue): K//K// ~ 0 Eq. R. (Red): 00K+~ 00K Eq. R. (Blue): K//00K// ~ Solution (Pat s Design) Solution (Kim s Design) P.D. of motor = 0V P.D. of motor = V P.D. of motor = 0V P.D. of motor = V P.D. of motor 3 = V Wrong design P.D. of motor 3 =.V Correct design 0 0 Eq. R. : K // K = /3K Eq. R. : 00 // 00K = ~

9 Solution (Leon s Design) P.D. of motor = 0V P.D. of motor 3 =.V P.D. of motor = V Correct design