Solving Diff. Eqn & Designing Circuits: An Understanding

[HansBu]

Homework Statement

Design a circuit that satisfies this differential equation using op-amps, resistors, and capacitors.

Relevant Equations

v_{out} = -RC \dfrac{dv_{in}}{dt}

where: 𝑣𝑖𝑛(𝑡)=0.3𝑐𝑜𝑠100𝑡

I have read all over the internet that this differential equation can be solved by isolating the term with the highest degree on one side of the equation. After doing so, I integrated it. However after integrating, I don't know that the next step is. Can anyone help me understand the concept on designing circuits with this differential equation? Specifically the concepts of integrator, summer, and differentiator operational amplifiers. Have tried to understand example on linear circuit analysis but their steps were way of a shortcut, making it hard for me to grasp the concept. A help from you guys will be really appreciated.

 

[Delta2]

I am not so familiar with the theory of op-amps but

Is the equation you write at relevant equations for the output voltage of an op-amp?

if so something tells me that in the circuit you ll have to use two chained op-amps(the output of one op-amp into the input of the other ) in order to make the term $\frac{d^2v_{in}}{dt^2}$ appear in the differential equation that describes the circuit.

 

[Joshy]

I know we can't post the answer, but maybe a little nudge :)?

Do you recall the transfer function for this op-amp? Your professor might tell you to use KCL to solve for it, but if you're having a hard time then it's a classic (inverter) you'll typically see with resistors; however: What does its transfer function look like if you replace $Z_1$ and $Z_2$ with one resistor and one capacitor (in the frequency domain)? Try it like this:

One like this:
$$\begin{align} Z_1 = R \nonumber \\ Z_2 = C \nonumber\end{align}$$

The other like this (swap $C$ and $R$):
$$\begin{align} Z_1 = C \nonumber \\ Z_2 = R \nonumber \end{align}$$

I would guess you've learned about Laplace transforms. I would look at the Laplace transform for differential equation and for an integrator. If you solve the two above op-amps and you see something on the Laplace transform table that kind of looks like it, then you might be able to massage each term in your equation to an op-amp and combine it for the overall system.

 

[Delta2]

I am not sure @Joshy but I think the circuit you give will have infinite series $$\sum_n\frac{d^nV_{in}}{dt^n}$$ in the differential equation that describes it.

EDIT: Sorry guys this is wrong by me i didnt know basic stuff about op-amps when i wrote this.

 

[HansBu]

I know we can't post the answer, but maybe a little nudge :)?

View attachment 262754

Do you recall the transfer function for this op-amp? Your professor might tell you to use KCL to solve for it, but if you're having a hard time then it's a classic (inverter) you'll typically see with resistors; however: What does its transfer function look like if you replace $Z_1$ and $Z_2$ with one resistor and one capacitor (in the frequency domain)? Try it like this:

One like this:
$$\begin{align} Z_1 = R \nonumber \\ Z_2 = C \nonumber\end{align}$$

The other like this (swap $C$ and $R$):
$$\begin{align} Z_1 = C \nonumber \\ Z_2 = R \nonumber \end{align}$$

I would guess you've learned about Laplace transforms. I would look at the Laplace transform for differential equation and for an integrator. If you solve the two above op-amps and you see something on the Laplace transform table that kind of looks like it, then you might be able to massage each term in your equation to an op-amp and combine it for the overall system.

Hello there, Joshy! I appreciate you reply. We were not introduced to Laplace Transforms yet. This makes the problem even harder to work out. If I only know when and how integrators, summers, and differentiation op-amps are used to solve the differential equation, then I can possibly work out the problem.

 

[HansBu]

I am not so familiar with the theory of op-amps but

Is the equation you write at relevant equations for the output voltage of an op-amp?

if so something tells me that in the circuit you ll have to use two chained op-amps(the output of one op-amp into the input of the other ) in order to make the term $\frac{d^2v_{in}}{dt^2}$ appear in the differential equation that describes the circuit.

Hi! Yes, that is what I am thinking of right now. Two chained op-amps because of the order of the differential equation. If only I could find resources to work out the problem.

 

[Joshy]

I can see why this would be really hard. Did your professor have a different way of teaching it?

I'd hate to point to Wikipedia, but they've got a definition on there and the table that I'm okay with using: https://en.wikipedia.org/wiki/Laplace_transform

The formal definition looks okay. I'm an electrical engineer in my class I was only required to solve a few before using the table. The strategy is to massage your equation (its format) into something that looks like something on the table. You'll see the table has a time domain and s domain. In any coverage of your circuit did they cover the impedance of capacitors and inductors? I'll give the example for inductors you might see something like $Z_L = j \omega L$ and sometimes they replace $j\omega$ suspiciously with $s$, and so you'll see instead sometimes $Z_L = sL$. Laplace transforms are used often in circuit analysis.

Is this helping? Please share if you can a little bit of what you believe is relevant your professor may have shown you maybe there's a completely different way I'm less familiar with? Also maybe if you know the impedance of capacitors I think that'll be helpful too.

 

[berkeman]

Hi! Yes, that is what I am thinking of right now. Two chained op-amps because of the order of the differential equation.

It's easier to just do it with 4 opamp circuits, at least as a first solution. You could then look at simplifications if you like. However, the problem statement that you wrote does not ask for a minimized circuit or limit how opamps and other components you use.

The output opamp will be a summing opamp circuit. What other 3 opamp circuits can you say will feed that output summing amplifier?

 

[Delta2]

@berkeman I haven't study opamps as summing circuits but up to how many inputs the simplest summing op-amp circuit can add? I suppose only 2?

 

[berkeman]

up to how many inputs the simplest summing op-amp circuit can add? I suppose only 2?

Lots. Using the "virtual ground" property of opamps, the multiple inputs to a summing stage stay separated, but contribute to the overall summed output.

https://www.electronics-tutorials.ws/opamp/opamp_4.html

 

[berkeman]

And come to think of it, I think I have it down to 3 opamps now...

 

[HansBu]

Hello guys, I appreciated so much of your efforts. Can you at least come up with an initial starting point for the solution or design of the circuit so I can work my way out of it? It would be a great help tho. Thank you.

 

[HansBu]

And come to think of it, I think I have it down to 3 opamps now...

How can we reduce the number of op amps used in this circuit? I suppose reducing the number of op amps will complicate the solution of the problem? as well as the design?

 

[HansBu]

I can see why this would be really hard. Did your professor have a different way of teaching it?

I'd hate to point to Wikipedia, but they've got a definition on there and the table that I'm okay with using: https://en.wikipedia.org/wiki/Laplace_transform

The formal definition looks okay. I'm an electrical engineer in my class I was only required to solve a few before using the table. The strategy is to massage your equation (its format) into something that looks like something on the table. You'll see the table has a time domain and s domain. In any coverage of your circuit did they cover the impedance of capacitors and inductors? I'll give the example for inductors you might see something like $Z_L = j \omega L$ and sometimes they replace $j\omega$ suspiciously with $s$, and so you'll see instead sometimes $Z_L = sL$. Laplace transforms are used often in circuit analysis.

Is this helping? Please share if you can a little bit of what you believe is relevant your professor may have shown you maybe there's a completely different way I'm less familiar with? Also maybe if you know the impedance of capacitors I think that'll be helpful too.

Here is a sample explanation on the differential equation. However, I can't even guess what is happening right here. Why is there a need to feed the input voltage to an integrator?

 

[Joshy]

I can completely understand the confusion. I feel confused reading it too. I think that text is trying to say more specifically how you can achieve the second derivative in two different ways (forward with differentiators or backwards with integrators).

I was imagining 3 op-amps too. I'm not sure if what I'm about to share is close to what berkeman had in mind, but here is what I had in mind. After a couple more thoughts you might need an inverting op-amp because the signs change so not all of them are addition neither are all of them subtraction. I think you'll have to explore that on your own so that we don't deprive you of the learning experience, which is also why I've made it very simplified not including the op-amp configuration/details and most passives.

Do you think you could work with this? This is probably really pushing it.

 

[HansBu]

I can completely understand the confusion. I feel confused reading it too. I think that text is trying to say more specifically how you can achieve the second derivative in two different ways (forward with differentiators or backwards with integrators).

I was imagining 3 op-amps too. I'm not sure if what I'm about to share is close to what berkeman had in mind, but here is what I had in mind. After a couple more thoughts you might need an inverting op-amp because the signs change so not all of them are addition neither are all of them subtraction. I think you'll have to explore that on your own so that we don't deprive you of the learning experience, which is also why I've made it very simplified not including the op-amp configuration/details and most passives.

View attachment 262803

Do you think you could work with this? This is probably really pushing it.

This is also what I think about. Yes, I think I can get more of it now by learning more. Thank you so much, Joshy! Learning is indeed fun!

 

[Delta2]

@Joshy with your post #3 and post #15 you basically reveal 80-90% of the solution! You bad boy !

 

[HansBu]

@Joshy with your post #3 and post #15 you basically reveal 80-90% of the solution! You bad boy !

Nope, he didn't tho. He is just trying to explain the concept behind. Anyway, thank you for your help too Delta 2!

 

[berkeman]

Here is a sample explanation on the differential equation. However, I can't even guess what is happening right here. Why is there a need to feed the input voltage to an integrator?

Those circuits are using integrators. To implement your equation that you posted, you need differentiators. Try swapping the positions of the R and C components in the first 2-opamp circuits that you posted and calculate $\frac{V_{out}}{V_{in}}$ for the two opamp outputs with respect to the input to the first opamp. Then look at the summing opamp circuit that I posted above, and well, you know...