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

[ichabodgrant]

In the following circuit, compute the values of R₁ and R₃ in terms of R₂ and R₄, such that v_o 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 = (v_o - 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?

 

[LvW]

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).

 

[psparky]

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!

 

[LvW]

"...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.

 

[donpacino]

In the following circuit, compute the values of R₁ and R₃ in terms of R₂ and R₄, such that v_o 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 = (v_o - 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

 

[donpacino]

"...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.

 

[ichabodgrant]

I solve it...
I had never thought of using comparing coefficients... turns out to be not too difficult...Thanks everyone

 

[LvW]

Can you tell us your solution?

 

[ichabodgrant]

R1 = 5R2
R3 = (1/5)R4

there might be some calculation mistakes...

 

[LvW]

Yes - is correct. And - as I have expected - no other results than a RATIO only (with infinite alternatives for resistor values9.