How can R1 and R3 be expressed in terms of R2 and R4 in an op-amp circuit?

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Discussion Overview

The discussion revolves around determining the values of resistors R1 and R3 in terms of R2 and R4 within an operational amplifier (op-amp) circuit, specifically to achieve a desired output voltage relationship of vo = v1 - 5v2. The scope includes theoretical analysis and mathematical reasoning related to circuit design.

Discussion Character

  • Technical explanation
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents initial equations based on node analysis but expresses difficulty in isolating R1 and R3 without additional unknowns.
  • Another participant suggests using gain formulas for non-inverting and inverting op-amp circuits, indicating that the superposition theorem could simplify the problem.
  • Some participants argue that the task may be poorly formulated, emphasizing that only the ratios of resistors matter, leading to an infinite number of possible resistor values that satisfy the conditions.
  • A participant proposes a specific relationship between the resistors, suggesting R1 = R3 and R2 = C * R4, where C is a constant to be determined.
  • One participant mentions successfully using coefficient comparison to derive relationships between the resistors, indicating a potential solution path.
  • Another participant provides specific values for R1 and R3 in terms of R2 and R4 but acknowledges possible calculation mistakes.
  • There is a reiteration that the results can only yield a ratio of resistors, with no unique solution provided.

Areas of Agreement / Disagreement

Participants express differing views on the formulation of the problem and the approach to finding solutions. While some agree on the importance of resistor ratios, others challenge the clarity of the task and the methods proposed. No consensus is reached regarding a definitive solution.

Contextual Notes

Limitations include the dependence on assumptions regarding the op-amp behavior and the potential for multiple valid resistor configurations that satisfy the output condition. The discussion reflects uncertainty in deriving specific resistor values without additional constraints.

Who May Find This Useful

This discussion may be useful for individuals interested in circuit design, particularly those exploring operational amplifier configurations and the mathematical relationships between circuit components.

ichabodgrant
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In the following circuit, compute the values of R1 and R3 in terms of R2 and R4, such that vo is always equal to v1 - 5v2.I have marked 3 nodes, A, B and C. v+ = v- as assumption for the op-amp.
Consider node A.
v- = v+ = v1 × (R2 / (R1+R2))

Consider node B and C.

(v- - v2) / R3 = (vo - v-) / R4

I now stick at here...
I can write R1 in terms of R2, but there exist v- and v1.
And for R3, the same problem exists.

First, is there anything wrong in my above computations?
Second, can I express R1 and R3 in terms of R2 and R4 without using other unknowns?
 

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The two given expressions (formulas) are correct. However, I would use another approach.
Do you know the gain formulas for (a) non-inverting and (b) inverting opamp circuits?
Your circuit is a combination of both and the most simple solution uses the above mentioned formulas. This is allowed because of the superposition theorem.
That means: The output voltage is Vo = V1- 5V2 = V1*k*gain1 - V2*gain2.
It shouldn`t be a problem to solve the system (starting with gain2).
 
LvW said:
The two given expressions (formulas) are correct. However, I would use another approach.
Do you know the gain formulas for (a) non-inverting and (b) inverting opamp circuits?
Your circuit is a combination of both and the most simple solution uses the above mentioned formulas. This is allowed because of the superposition theorem.
That means: The output voltage is Vo = V1- 5V2 = V1*k*gain1 - V2*gain2.
It shouldn`t be a problem to solve the system (starting with gain2).

That's good stuff and all, but how do you get R1 and R3 in terms of R2 and R4?

This is a tough one!
 
"...compute the values of R1 and R3 in terms of R2 and R4,"

I think the task is not well formulated. For all opamp amplifiers it is only the ratio of resistors that matters.
Therefore, we have - in theory - an infinite number of resistor values which all fulfill the required ratios.
The answer can only be R2/R1=C*R4/R3 and we can select - as one possible solution - R1=R3 and R2=C*R4.
 
ichabodgrant said:
In the following circuit, compute the values of R1 and R3 in terms of R2 and R4, such that vo is always equal to v1 - 5v2.I have marked 3 nodes, A, B and C. v+ = v- as assumption for the op-amp.
Consider node A.
v- = v+ = v1 × (R2 / (R1+R2))

Consider node B and C.

(v- - v2) / R3 = (vo - v-) / R4

I now stick at here...
I can write R1 in terms of R2, but there exist v- and v1.
And for R3, the same problem exists.

First, is there anything wrong in my above computations?
Second, can I express R1 and R3 in terms of R2 and R4 without using other unknowns?

You have 2 equations. both equations have extra variables (V+ and V-). so plug one equation into the other getting rid of V+ and V-. Then solve for Vo.

The equation will then look like this

Vo=XV1-YV2

you know X=1 and Y=5

so solve for the two equations and you will get the relationship between all the resistors in the form of two equations. take it one step further and solve in the form of

R1=Z and R3=T
 
LvW said:
"...compute the values of R1 and R3 in terms of R2 and R4,"

I think the task is not well formulated. For all opamp amplifiers it is only the ratio of resistors that matters.
Therefore, we have - in theory - an infinite number of resistor values which all fulfill the required ratios.
The answer can only be R2/R1=C*R4/R3 and we can select - as one possible solution - R1=R3 and R2=C*R4.
1. In general the goal of the exercise is to find that C value
2. I don't want to just give the answer to the OP, but I do want to point for his/her sake that your solution is incorrect.
 
I solve it...
I had never thought of using comparing coefficients... turns out to be not too difficult...Thanks everyone
 
Can you tell us your solution?
 
R1 = 5R2
R3 = (1/5)R4

there might be some calculation mistakes...
 
  • #10
Yes - is correct. And - as I have expected - no other results than a RATIO only (with infinite alternatives for resistor values9.
 

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