Compound feedback topologies

If the equations are exact then you should be able to apply them in either order and get the same answer.

I dimly remember being presented with the whole feedback equation thing in 3rd-year circuits and wondering if they were exact or if there were approximations being made that were only usually true. I never went back and verified, because if you're just analyzing one circuit it's easier to grind the whole thing out using mesh or nodal analysis.

Easier yet, use LTSpice and believe it 'cause the Computer Says So.

Where it gets tricky is with multiple loops within a circuit... when one loop is not totally contained within the other.

Like some DC common mode loops, with differential _signal_ feedback.

It gets quite nasty. I end up having to totally trust transient analysis and over-shoot estimations :-( ...Jim Thompson

Theorems of "block diagram algebra" say you can take a diagram with loops (which may be intersecting) and move branch points around (obviously, this is done by introducing a factor of H or 1/H in the branch, in order to move it in front of or after the block H), thus simplifying it to a diagram with no intersecting loops, which can be reduced, ultimately, to the single loop T = G/(1+GH) form (a formula which can be proven by the same rule, if you don't mind chugging one step further).

In electronic terms, you might as well write the system of equations and play with that. You can reduce it to block form if you like, it's an equivalent approach.

Tim

-- Deep Friar: a very philosophical monk. Website:

I'll often do this sort of thing with block diagrams, if I can figure out which dynamic elements to ignore. Grinding through the system of equations won't take you any more time than the block diagram way, but the block diagrams _may_ give you more insight into why the system operates the way it does.

The type of circuit I'm trying to analyze via this method is here:

(note there should be a DC blocking capacitor after R5).

Basically I'm trying to calculate the voltage gain and voltage feedback ratio, and current gain and current feedback ratio independently using superposition and a cascaded hybrid-pi model of the two transistor circuit. I'm starting to conclude that it's pretty much an equivalent amount of math to using nodal or mesh analysis, with the downside that it can't be automated via computer! The only advantage I can see is that one could analyze the effects of the current feedback and voltage feedback paths independently, while an expression for the gain and input impedance using nodal analysis will be a huge expression with everything jumbled together, making it hard to know directly what changes in either feedback network will have on the parameters.

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The gain and impedances of that circuit are discussed in "A Wide-Band Ultralinear Amplifier from 3 to 300 MHz" by Meyer, Eschenbach, and Chin in IEEE Journal of Solid Stated Circuits, vol. SC-9, no. 4 (Aug. 1974), pp. 167-175. There are some brief derivations and the resultant equations are shown.

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I think "signal flow graph" would get you a relevant google search. Next up is Mason's rules. My understanding is they haven't taught that stuff in EE courses since the late 80s, at least in any detail. I doubt anyone could make a leapfrog filter design these days since signal flow graph theory is just glossed over in colleges.

Is the original poster over complicating the question? I mean, app a voltage source to the input and compute the current, then get the impedance. Who cares if the box has feedback. Think of it as a black box.

Thanks for your replies, everyone. Miso - I think your approach would be fine if I were given the task of just analyzing a circuit that existed out in the world someplace to discover its particular behavior. However, if I actually want to design a circuit using such a topology, that method doesn't give me much intuition as to what changing various component values independently will do to the behavior of the circuit.

I've started reading up on Mason's rules and I think they might work along with superposition in analyzing the behavior of this circuit. If it has, for example, current sampling current mixing feedback, I could analyze the amplifier as a current amplifier with current feedback with Mason's rule to find the output current

Thanks for your replies, everyone. Miso - I think your approach would be fine if I were given the task of just analyzing a circuit that existed out in the world someplace to discover its particular behavior. However, if I actually want to design a circuit using such a topology, that method doesn't give me much intuition as to what changing various component values independently will do to the behavior of the circuit.

I've started reading up on Mason's rules and I think they might work along with superposition in analyzing the behavior of this circuit. If it has, for example, both current sampling current mixing and voltage sampling voltage mixing feedback I could do the following: analyze the amplifier as a current amplifier with current feedback with Mason's rule to find the current gain, use that to find the voltage gain of the amplifier with only the current feedback network, then use Mason's rule again with that value of voltage gain and the voltage feedback network.

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I would approach it with a more primitive T model and a spreadsheet. =20 =46irst calculate the operating point, it will give you nice clues to=20 the rest of the response. And finally hit it with LTspice, just to=20 compare.

Huh? You seem detached from the discussion at hand. I'm talking _feedback_stability_ analysis... and I use PSpice probably more hours per day than you are awake :-) ...Jim Thompson

They shouldn't, but they still do. I have little appreciation for algorism. Alas, such is school.

At least the E&M courses I've taken include more computational modeling than "chug the equations" stuff.

Tim

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I still think you are overcomplicating this, or I don't understand the problem. Once you solve for the impedance, the feedback network parameters should be part of that equation. Seems to me you could then solve for whatever combination give you the optimal answer. If it was just one variable, you could just sweep it and look for where it peaks (or do calculus to find the peak). For two variables, I'd have to dust off a multivariable calculus book.

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I came across this paper on signal flow graph tricks. Of course, you can derive all these tricks by math or merely inspection.

Note that you can simplify your signal flow graph by removing nodes using these rules as long as you don't need the nodes in your analysis. The classic case is massaging the flow graph to make zeros "build-able" in leapfrog designs.