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I'm having some troubles understanding how to solve complex circuits with opamps like the following:

enter image description here

During my lab course (I'm a physics major) the professor showed us the solution of very simple circuits like the following:

enter image description here

I'm having some troubles even understanding where to start in the first image I uploaded, while with the second one the solution is straightforward: $$V_+ = 0, \ V_- = \frac{Z_C}{Z_C+R}V_S+\frac{R}{Z_C+R}V_\text{out}$$ and for the short circuit principle of ideal opamps, we get $$V_d = 0 \implies V_\text{out} = - \frac{Z_C}{R}V_S = -\frac{1}{j\omega RC} V_S$$ hence the transfer function of the circuit (which is the usual requirement for "solving the circuit") is given by $$A_V = - \frac{1}{j\omega RC}$$.

What I'm asking for is an example of finding \$V_{BP}(\omega)\$ in function of the input voltage \$V_S\$ in the first image uploaded, so that I can easily find the transfer function of the circuit. I'm asking this because I have no idea of how to that by myself and the professor didn't give us any example of a solution of a more complex circuit. By the way, I recognized that the block in the upper right-hand corner is an inverting summing amplifier, but I don't know how to it behaves with the rest of the circuit.

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  • \$\begingroup\$ Do you know how to apply KCL in analyzing circuits? Physics classes should be continually heavy on the mathematics side. So diff-eq, through annihilators and then Laplace methods, plus linear systems of equations and matrix theory should be deep in your soul (see anything by Dr. Strang -- recently retired.) How far along are you there? Graph theory captures all valid circuits, where KCL is found in the rowspace and right nullspace and Ohm's Law and KVL is found in the columspace and left nullspace. You should be comfortable with the four related spaces in linear algebra theory. \$\endgroup\$ Commented Jul 3 at 23:39
  • \$\begingroup\$ But the main thing I'd like to know is if you are comfortable with KCL and solving systems of linear equations. Or not. I already know you don't recognize the filter pattern of the dashed box part of the circuit. So knowing KCL and matrix solutions will be step 1 in being able to analyze these things on the fly and without any prior experience. \$\endgroup\$ Commented Jul 3 at 23:42

2 Answers 2

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I've grabbed up the part you seem to be asking about and labeled an additional node.

block1 from questioner

I'll be using SageMath/SymPy for the symbolic KCL analysis. It flows about like this:

var('vs vbp iout r1 r2 r3 c1 c2 va vb s')    # iout is the output current of the opamp
ea = Eq( va/r1 + va/r2 + va*s*c1 + va*s*c2, vs/r1 + vb*s*c1 + vbp*s*c2 )  # KCL node A
eb = Eq( vb/r3 + vb*s*c1, vbp/r3 + va*s*c1 )                              # KCL node B
ebp = Eq( vbp/r3 + vbp*s*c2, vb/r3 + va*s*c2 + iout )                     # KCL VBP
eeq = Eq( vb, 0 )                                                         # opamp ensures this
ans = solve([ea,eb,ebp,eeq],[va,vb,vbp,iout])
simplify(ans[vbp]/vs)
-c1*r2*r3*s/(c1*c2*r1*r2*r3*s**2 + c1*r1*r2*s + c2*r1*r2*s + r1 + r2)

You should immediately recognize this as a 2nd order bandpass of some kind.

Put in standard form, it is:

tf2(simplify(ans[vbp]/vs))
'2nd order bandpass'
omega0: sqrt(r1 + r2)/(sqrt(c1)*sqrt(c2)*sqrt(r1)*sqrt(r2)*sqrt(r3))
zeta: sqrt(r1)*sqrt(r2)*(c1/2 + c2/2)/(sqrt(c1)*sqrt(c2)*sqrt(r3)*sqrt(r1 + r2))
Av: -c1*r3/(r1*(c1 + c2))

2nd order transfer functions only have those three parameters.

You know that \$R_1=R_2\$ form a divider. \$R_3=2\cdot R_1\$, too. And \$C_1=C_2\$. This can greatly simplify the analysis:

tf2(simplify(ans[vbp]/vs).subs({r2:r1,c2:c1,r3:2*r1}))
'2nd order bandpass'
omega0: 1/(c1*r1)
zeta: 1/2
Av: -1

This just means that the \$Q=1\$, the gain is \$A_v=-1\$, and the frequency is entirely determined by your choice of \$C_1=C_2\$ and \$R_3=2\left(R_1=R_2\right)\$. So long as the ratios are maintained, it is very easy to adjust \$\omega_{_0}\$ without modifying its gain or shape.

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It's called an MFB band-pass filter and is well-known in the industry. Here are a few references that get you started on what it's all about.

I'm sure if you dig deep enough, you'll find a derivation (quite possibly on this site). Or maybe solve it yourself by considering this in the red box: -

enter image description here

Once you have found the MFB BP transfer function, the final output will be the sum of the input to the MFB and the output from the MFB but overall inverted.

Given that the MFB type is a band-pass filter and, it's output is also inverting, what you end up with at the final output (\$V_{OUT}\$) is a notch-filter with an inverted output.

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    \$\begingroup\$ First of all, I wanted to thank you for spending time in replying to my answer and for your suggestions, I'll definitely try to solve it by myself and see where I get, by first solving the red box circuit, which should be an ideal differentiator if I'm not wrong. Since I know pretty much nothing about electronics since it is taught so bad to us, do you have any source where these kind of circuits are fully solved analytically so that I can understand how it is done? \$\endgroup\$
    – deomanu01
    Commented Jul 3 at 20:47
  • \$\begingroup\$ @deomanu01 the best source I can recommend is this site. I say this because I learned my electronics (if I ever stopped!!) starting when I was a kid back in the late 1960s so, I have no recommendations that I can make but, I hear the art of electronics is a book that a lot of people use. If we are done here, please take note of this: What should I do when someone answers my question. If you are still confused about something then leave a comment to request further clarification. \$\endgroup\$
    – Andy aka
    Commented Jul 3 at 22:07

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