1. ENGG 1203 Tutorial Op Amps 8 Mar Learning Objectives News
Analyze circuits with ideal operational amplifiers
News
HW2 (18 Mar 23:55)
Mid term (22 Mar 2:30pm-3:30pm)
Revision tutorial (14 Mar 3:30pm-5:30pm, CBA)
Ack.: MIT OCW 6.01

2. Analysis of a Circuit with Op Amp (I)
Determine Vo in the following circuit. Assume that the op-amp is ideal.

3. Solution
Since V- = V+, V- = 5V. So there must be 1/12A flowing left through the two 6 ohm resistors. There must be a corresponding 1/12 A flowing to the left through the 12 ohm resistor. Vo is then the sum of V- = 5V and the 1V across the 12 ohm resistor.

4. Analysis of a Circuit with Op Amp (II)
Determine the current Ix when V1 = 1V and V2 = 2V.
Determine the voltage VA when V1 = 1V and V2 = 2V.
Determine a general expression for VA in terms of V1 and V2.

5. Solution When V1 = 1V and V2 = 2V, Ix = 1A
When V1 = 1V and V2 = 2V, VA = 4V
A general expression for VA: 𝑉 𝐴 = 𝑉 2 + 𝐼 𝑥 ×2 = 𝑉 2 + 𝑉 2 − 𝑉 1 1 ×2 = 𝑉 𝑉 2 − 𝑉 1 =−2 𝑉 1 +3 𝑉 2 2 4 2 1 -1 1

6. Multi-stage Non-inverting AmplifierVn
Use a single op-amp and resistors to make a circuit that is equivalent to the following circuit.
𝑉 𝑛 𝑉 𝑖 = 1+ 𝑅 2 𝑅 1
𝑉 𝑜 𝑉 𝑖 = 𝑉 𝑜 𝑉 𝑛 𝑉 𝑛 𝑉 𝑖 = 1+ 𝑅 3 𝑅 𝑅 2 𝑅 1
=1+ 𝑅 1 𝑅 4 + 𝑅 2 𝑅 3 + 𝑅 2 𝑅 4 𝑅 1 𝑅 3

7. Voltage-controlled Current Source
Use the ideal op-amp model (V+ = V-) to determine an expression for the output current Io in terms of the input voltage Vi and resistors R1 and R2.
𝑣 𝑥 = 𝑣 𝑖 + 𝑣 𝑥 𝑅 2 𝑅 1 + 𝑅 2
⇒ 𝑣 𝑥 = 𝑣 𝑖 𝑅 2 𝑅 1
vi +vx vx
vi +vx
𝐼 𝑜 = 𝑣 𝑥 𝑅 2 = 1 𝑅 2 𝑣 𝑖 𝑅 2 𝑅 1 = 𝑣 𝑖 𝑅 1

8. Op Amp Configurations (I)
Determine R so that Vo = 2 (V1 − V2).

9. No current in +ve or -ve inputs:
Ideal op-amp: 

10. Op Amp Configurations (II)
Fill in the values of R1 and R2 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,∞]) R1 R2
Vo = 2V2 - 2V1
Vo = V2 - V1
Vo = 4V2 - 2V1

11. 𝑉 + = 𝑅 2 10𝑘+ 𝑅 2 𝑉 2 = 𝑉 − = 𝑅 1 10𝑘+ 𝑅 1 𝑉 1 + 10𝑘 10𝑘+ 𝑅 1 𝑉 𝑜
𝑉 + = 𝑅 2 10𝑘+ 𝑅 2 𝑉 2 = 𝑉 − = 𝑅 1 10𝑘+ 𝑅 1 𝑉 𝑘 10𝑘+ 𝑅 1 𝑉 𝑜
𝑉 𝑜 = 10𝑘+ 𝑅 1 10𝑘+ 𝑅 2 × 𝑅 2 10𝑘 × 𝑉 2 − 𝑅 1 10𝑘 × 𝑉 1
3rd: Negative R  i.e. Impossible R1 R2
Vo=2V2-2V1
20kΩ
Vo=V2-V1
Vo=4V2-2V1
Impossible

12. Unusual Op Amp Configurations
What is Vo?
V3+ V1
V3+ V2 V3
Vo = 0
Vo = V1 – V2

13. Motor Control
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 10V 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 5Ω and that rotational speed is proportional to voltage.
For each design, indicate the voltage across each of the motors.

14. Motor Control (Jody’s Design)
P.D. of motor 1 = 10V P.D. of motor 2 = 0.05V P.D. of motor 3 = 0V Wrong design 10
0.05
Eq. R. (Red): 1K+~5  1K
Eq. R. (Blue): 1K//1K//5  ~5

15. Motor Control (Chris’s Design)
P.D. of motor 1 = 10V P.D. of motor 2 = 0.45V P.D. of motor 3 = 0V Wrong design 10
0.45
Eq. R. (Red): 100K+~5  100K
Eq. R. (Blue): 1K//100K//5  ~5

16. Motor Control (Pat’s Design)
P.D. of motor 1 = 10V P.D. of motor 2 = 4V
P.D. of motor 3 = 2V Wrong design 10 4
Eq. R. : 1K // 2K = 2/3K 4 2 2

17. Motor Control (Kim’s Design)
P.D. of motor 1 = 10V P.D. of motor 2 = 5V
P.D. of motor 3 = 2.5V Correct design 10 5 5
Eq. R. : 100 // 200K = ~100
2.5
2.5

18. Motor Control (Leon’s Design)
P.D. of motor 1 = 10V P.D. of motor 2 = 5V P.D. of motor 3 = 2.5V Correct design 10 5 5
2.5
2.5

19. Motor Control
The following circuit is a proportional controller that regulates the current through a motor by setting the motor voltage VC to VC = K(Id − Io) where K is the gain (notice that its dimensions are ohms), Id is the desired motor current, and Io is the actual current through the motor.

20. Solution
Consider the circuit inside the dotted rectangle. Determine V1 as a function of Io.
V+ = 1/2 x Io = V- V- = 100/( ) x V1 V1 = 1/2 x Io x 100
Determine the gain K and desired motor current Id.
KCL at -ve input to right op-amp: 𝑉 𝑐 − = 2.5−0.5× 𝐼 𝑜 × ⇒ 𝑉 𝑐 =50 0.1− 𝐼 𝑜

21. Position Controller
The following figure shows a motor controller. A human can turn the left potentiometer (the input pot). Then the motor will turn the right potentiometer (the output pot) so that the shaft angle of the output pot tracks that of the input pot.

22. The dependence of the pot resistances on shaft angle is given in terms of α, which varies from 0 (most counterclockwise position) to 1 (most clockwise position). The resistance of the lower part of the pot is αR and that of the upper part is (1 − α)R, where R = 1000Ω.
Notice that if αi >αo, then the voltage to the motor (VM+ − VM−) is positive, and the motor turns clockwise (so as to increase αo)—i.e., positive motor voltage clockwise rotation.

23. Determine an expression for VM+ in terms of αi, R, and VS.
The output of the voltage divider is
The op-amp provides a gain of 1, so VM+ = V+.

24. The following circuit produces a voltage Vo that depends on the position of the input pot. Determine an expression for the voltage Vo in terms of αi, R, R1, R2, and VS.
The positive input to the op-amp is connected to a voltage divider with equal resistors so
The input pot is on the output of the op-amp, so
In an ideal op-amp, V+ = V− so

25. The following circuit produces a voltage Vo that depends on the positions of both pots. Determine an expression for Vo in terms of αi, αo, R, and VS.
The positive input to the op-amp is connected to pot 1 so that The output pot is on the output of the op-amp, so In an ideal op-amp, V+ = V− so

26. Assume that we are provided with a circuit whose output is αi/αo volts
Assume that we are provided with a circuit whose output is αi/αo volts. We wish to determine if it is possible to design a motor controller of the following form so that the motor shaft angle (which is proportional to αo) will track the input pot angle (which is proportional to αi).
Assume that R1 = R3 = R4 = 1000Ω and VC = 0. Is it possible to choose R2 so that αo tracks αi? If yes, enter an acceptable value for R2.

27. Assume that R1 = R3 = R4 = 1000Ω and VC = 0
Assume that R1 = R3 = R4 = 1000Ω and VC = 0. Is it possible to choose R2 so that αo tracks αi? If yes, enter an acceptable value (a number) for R2.
If R3 = R4 then the right motor input is 5V. If αi = αo then the gain of the left op-amp circuit must be 5 so that the motor voltage is 0. The gain is R1 + R2/R1, so R2 must be 4000Ω. 5 5 1 5 1

28. Assume that R1 = R3 = R4 = 1000Ω and VC = 5V
Assume that R1 = R3 = R4 = 1000Ω and VC = 5V. Is it possible to choose R2 so that αo tracks αi?
If R3 = R4 then the right motor input is 5V. If αi = αo then V+ = V− = 1 for the right op-amp. We need the left motor input to be 5V. But if the left motor input is 5V and VC = 5V then V− must also be 5V, which leads to a contradiction. 5 5 1 5 1 5

29. Course Timeline (Tentative)
Lecture
Tutorial
Lab
Homework
Systems L1 T1
Digital systems
Combinational logic L2
T2, T3
#1, #2
HW1
Sequential logic L3
FSM
L3, L4
T3, T4
#3, #4
ADC/DAC L4
#4 (DAC), #7,8 (ADC)
Circuit L5
T4, T5
HW2
Project T5 #6
Op Amp L6 T6
#5, #8 ……

30. Tutorial Schedule (Tentative)
1/25 Introduction+System
4/5 Signal
2/1 Digital Logic
4/12 Signal
2/8 Digital Logic
4/19 Signal
2/15 N/A
4/26 N/A
2/22 Digital Logic+Circuit
5/3 N/A
3/1 Circuit+Project
5/X Computer+Revision
3/8 Circuit
3/15 Revision Tutorial
3/22 ** Mid Term **
3/29 N/A