The source that must not be zeroed

Two solves that add is careful about one thing and states it twice. “Set the other source to zero” does not mean delete it. A voltage source with no voltage in it is a short circuit and it stays in the netlist, contributing its own equations, because the topology must not change between the three solves. That care is the reason the essay’s superposition came out exact to one unit in the last place of a double rather than approximately.

There is a second element in the analogue vocabulary that the same instruction is applied to, and the careful treatment of it is the opposite one. A dependent source must not be zeroed at all — neither deleted nor set to zero — and the reason is that it is not a source in the sense the theorem is about.

What superposition is actually a theorem about #

The theorem is usually stated as an instruction and it is worth restating as a claim, because the instruction is where the confusion lives.

A network of elements whose laws are linear relations between their own voltages and their own currents, driven by some set of independent sources, has a solution that is a linear function of those sources’ values. That is the whole of it. It follows from the matrix being linear in the right-hand side and nothing else, and it is why the essay before it could say that superposition needs no assumption about frequency, topology, element values or the number of sources.

Read that way, the instruction to “zero the other sources” is not a step in a proof; it is a way of extracting the coefficients of a linear function by evaluating it at basis vectors. The sources are the argument of the function and the elements are the function.

A dependent source is an element. Its law is $v = A\,v_x$ — a linear relation, which is why the network is linear and the theorem applies — and it belongs to the function rather than to the argument. It is no more a thing to be zeroed than a resistor is, and for the same reason: zeroing a resistor would be a statement about a different circuit.

That distinction is not merely a matter of bookkeeping, because the elements a dependent source stands in for are the ones every active circuit is made of. A transistor’s transconductance, an amplifier’s gain, a transformer’s ratio, a gyrator: all of them are controlled sources in a netlist, so the question of what superposition does with them is the question of whether superposition is usable on any circuit with a device in it. How small is small signal is where the licence to treat a device as a controlled source at all is priced, at 7.3 millivolts for one per cent of error.

The same care, applied to the other three elements #

Superposition’s instruction mentions sources, and a netlist has four kinds of element that the instruction could plausibly be read as covering. It is worth going through them, because three of the four are unambiguous and the fourth is the one this essay is about.

An independent voltage source is zeroed by setting its value to nought, which leaves a short circuit in the netlist. It must not be deleted, because deleting it would open the branch and change the topology. This is that essay’s own care and it is the case everybody gets right after being told once.

An independent current source is zeroed the same way, and a current source of nought amps is an open circuit. So the two kinds of independent source are zeroed by the same instruction and become opposite things, which is worth noticing because it is the reason the instruction is stated as “set to zero” rather than as “short out” — the latter is right for one and wrong for the other.

A passive element is not mentioned by the instruction at all, and nobody proposes zeroing a resistor.

A controlled source looks like the first two and behaves like the third, and that is the whole difficulty. It has the word “source” in its name, it is drawn with a source’s symbol, and it is an element whose law involves a voltage or current elsewhere in the network. The instruction’s word picks it out and the theorem’s meaning does not.

The test that settles it is short, and it is the one that would have prevented the mistake without knowing anything about this network. Ask whether the element’s value is an input to the problem or a property of the circuit. An independent source’s value is an input: it is what the linear function is a function of. A controlled source’s gain is a property of the circuit, and its value is an output — computed from the solution rather than given. Nothing whose value is computed from the solution can be zeroed to extract a coefficient, because zeroing it is a statement about a different function.

Which is also why the mistake is hard to make in a matrix and easy to make in a diagram. Assembling the network’s equations, a controlled source contributes a term to the matrix’s left-hand side and an independent one contributes to the right — so the two are already in different places, and the instruction “zero the sources” means “zero the right-hand side except one entry”, which cannot reach a controlled source at all. It is only when the circuit is manipulated as a picture that the two look alike.

What zeroing it does, and it is not a small thing #

Zeroing a controlled voltage source means setting $A$ to nought, which makes it a voltage source of nought volts — a short circuit from its output to ground. That essay’s own care applies: the element stays in the netlist and contributes its equation, and its equation now says that its output node is at zero.

So the wrong route solves a circuit in which the dependent source’s output is grounded. Measured on this network:

The first two agree to one part in $10^{16}$, which is the theorem. The third is 62.5 per cent wrong at the node and a hundred per cent wrong at the output, and the hundred per cent is not interesting — nought volts is what a nought-volt source produces. The node is where the wrong answer is finite, and its character is the finding.

The answer that does not depend on the gain #

Sweep the gain and the true answer moves. The wrong one does not.

4.0625 volts at every gain, to a part in $10^{12}$, because the route has deleted the only element in the network that knows what the gain is. That is the clearest possible sign that a different circuit was solved, and it is a better diagnostic than the size of the error — which is six per cent at a gain of one and nineteen hundred at three hundred, and would be mistaken for a rounding difference at the low end.

The lesson generalises past this particular mistake, and it is worth stating because it is a habit rather than a rule. An answer that does not depend on a parameter the circuit depends on is evidence that the parameter was removed. That test is used elsewhere too: that essay’s own check normalises a difference against the larger contribution rather than against the result, because at 180 degrees the result is zero and a relative error against it reports $10^{-14}$ of nothing as a catastrophe. Both are the same discipline — look at what the answer is a function of, not only at how large it is.

Why the feedback matters to the size of it #

The network has a resistor from the dependent source’s output back to the node that controls it, and that choice is deliberate rather than incidental.

Without the feedback resistor the dependent source is a bystander: it reads the node and drives a load, and zeroing it changes what the load sees and leaves the node alone. The node’s voltage would then be the same by both routes and the error would appear only at the output — which is a real error and an uninteresting one, because a reader would locate it immediately.

With the feedback resistor the dependent source is in the network that sets the node. Its output pulls the node down through ten kilohms in proportion to the node’s own voltage, which is why the true node voltage falls from 3.82 volts to 0.206 as the gain rises: the element is doing the work of an inverting amplifier’s feedback path. Zeroing it grounds the far end of that resistor, so the resistor becomes a load to ground rather than a feedback path, and the node rises to the value a plain three-resistor divider gives. The quantity that has been deleted is the loop gain itself, which what is left at crossover measures by cutting the loop deliberately — and a cut loop is a different circuit on purpose, with the difference recorded rather than accidental.

Which is the case that matters, because it is what every real circuit is. An amplifier’s gain block is inside its own feedback loop by construction, a transistor’s transconductance is inside whatever biasing network sets its operating point, and a transformer’s ratio couples both ways. What a network answers is where the solver and its two standing checks are set out, and the reason both checks are rebuilt from the element laws — the current law at every node, and the energy counted two ways — is exactly that a controlled source’s contribution to them is not separable from the passive network’s.

Two routes, and what the correct one costs #

The figure computes the answer three ways and two of them are meant to agree, which makes the third a control rather than a comparison.

The first route is the whole netlist solved at once: seven elements, one matrix, one solution. The second is the superposition done properly — two solves, each with one independent source at its value and the other at zero, and the dependent source present and live in both. They agree to one part in $10^{16}$, which is the arithmetic’s floor and is the same agreement the earlier measurement reports for its own two-source network.

The third is the mistake, and putting it in the figure rather than describing it is the point. A check that demonstrates the right answer says nothing about a wrong one being available.

It is worth noting what the correct route costs, because it is not free and the cost is the reason the mistake is tempting. The number of solves is the number of independent sources, and the dependent source appears in every one of them — so superposition on a network with two independent sources and five controlled ones is two solves of a seven-element network rather than two solves of a two-element one. There is no decomposition of the controlled elements: they are the network, and the network is solved in full each time.

Which means superposition on an active circuit buys understanding rather than arithmetic. One solve of the whole thing is cheaper than two, and what the two give is the separation of contributions — this much of the output is the signal and this much is the disturbance — which is a statement a designer wants and a single solve does not provide. It is the same reason exact outside and wrong within keeps a reduction and the network it reduces side by side: the reduction is exact about what leaves the port and silent about everything inside it. That essay’s closing section names three places in this collection where exactly that separation is being relied on without being called superposition, and every one of them has controlled sources in it.

Where the separation is actually wanted #

The essay before it closes by naming three results in this collection that are superpositions being relied on without being called one, and every one of them has controlled sources in it — which is the reason this essay’s distinction is load-bearing rather than pedantic.

What gets through from the rail sets the reference to zero and drives the rail, which is a superposition exactly: the essay’s whole result is a statement about one term of a sum, and the circuit whose term it is contains a pass device modelled as a controlled source inside its own feedback loop. Zeroing that device along with the reference would have deleted the loop and returned a disturbance rejection that is a property of a resistive divider.

The rail the load moves measures one amplifier’s load step reaching a second amplifier that shares nothing with it but a wire — 3.84 microvolts per ampere or 17.3 millivolts depending on where one capacitor returns. The two amplifiers are controlled sources and the separation being relied on is between the disturbing channel’s contribution and the disturbed one’s own signal.

And the millivolts in the wire is the same construction with the disturbance being somebody else’s load current, where the arrangement’s own verification is a superposition — the interfering current computed on its own, times the shared impedance — agreeing with the full solve to four parts in ten million. The residue there is physical rather than arithmetic: the sensor’s own return current, which the second route does not contain.

The pattern across the three is worth naming. What superposition is used for in practice is not solving a network but attributing an answer, and attribution is exactly the operation that needs the controlled elements left alone — because the question being asked is “how much of this output came from that input”, and a circuit with its gain blocks deleted has a different answer to it for every input.

Where this leaves the diode #

The essay before it ends its linear section with a nonlinear one: the same two-source arrangement with a diode in it, solved by Newton’s method and then solved twice more with one source zeroed each time, giving an added node voltage 43.2 per cent high and an added diode drop of 1.35 volts, which is a voltage no silicon diode has ever had.

It is worth being explicit that this essay’s mistake and that one are different failures, because both produce a wrong answer from three correct solves.

Here the network is linear and the method was misapplied. Superposition holds; the practitioner zeroed an element instead of a source. Applied correctly it is exact to the last bit, and the repair is free.

There the network is nonlinear and the method does not hold. No care in applying it helps, because the theorem’s premise is false: the diode’s law is not a linear relation between its own voltage and its own current. A bias point is a solution, not a choice is where that is established, and the only thing superposition can be used for in such a network is the small-signal problem about a solved operating point — where the diode has been replaced by a resistance and is therefore a linear element again.

The two failures are easy to conflate and the diagnostic separates them cleanly. A misapplied superposition on a linear network gives an answer that does not depend on something it should. A correctly applied superposition on a nonlinear one gives an answer that depends on everything correctly and is still wrong, and nothing in the arithmetic complains — each of the three solves converges, each satisfies the current law to a part in $10^{15}$, and adding three correct numbers gives a wrong one.

Still open: the transistor stage, and the transformer that couples both ways #

The same measurement on a device rather than on a gain block. The controlled source here is an ideal voltage-controlled voltage source, which is the simplest case and the least like anything on a board. A transistor’s small-signal model is a controlled current source with a resistance across it, and zeroing a controlled current source makes it an open circuit rather than a short — the opposite topological change, so the error should go the other way. Measuring both on one figure would say whether the sign of the mistake depends on the kind of source, which is a thing a reader would want to know before trusting an intuition about it.

The current-source version, which should fail the other way. Zeroing a controlled current source leaves an open circuit rather than a short, so the wrong route’s answer should move in the opposite direction — and whether it is still independent of the gain is the question, since an open circuit removes the feedback path entirely rather than grounding it.

A network with several controlled sources in a loop. With one dependent source the wrong route deletes one path. With three in a cascade, zeroing them deletes a chain, and whether the errors compound or partly cancel is not obvious from anything above — and a three-stage amplifier is the ordinary case.

And the coupling that goes both ways. A transformer in a netlist is two controlled sources referring to each other, so neither can be zeroed without the other becoming meaningless. That is a sharper version of this essay’s point and the magnetics field has the elements for it: the question is whether the mutual term can be treated as an excitation at all, or whether it is unambiguously part of the network.

What is checked #

Superposition with the dependent source left in is required exact, to a part in $10^{14}$ of the answer, at the node and at the output. That is the claim that nothing is wrong with the theorem and that the essay is about a misapplication of it.

The wrong route’s answer is required to be the same at every gain, to a part in $10^{12}$ across seven settings from nought to three hundred. That is the diagnostic the essay is built on, and it is a stronger statement than the size of the error.

And the true answer is required to move by more than a factor across the same gains, because the constancy of the wrong answer means nothing unless the right one is a function of the parameter. The pair is the check; either alone is a number.

The size of the error is required to exist rather than to be large. Six per cent at a gain of one and nineteen hundred at three hundred — so “wrong by tens of per cent” holds over most of the slider and failed at its first setting, which is this ledger’s most frequently recorded defect: a requirement with a number in it is usually a fact about the first placement rather than about the family.