The source that is not a source

An ideal voltage source is defined by what it refuses to do. Whatever current is drawn from it, its terminal voltage is unchanged: one ampere, a thousand amperes, the answer is the same. Drawn on a graph of voltage against current it is a horizontal line, and it is the flattest line anywhere in this subject.

It is also an idealisation with a boundary, and the boundary is a current.

Where the idealisation comes from #

Nothing that stores or converts energy can supply current without something inside it opposing the flow. A battery has electrolyte with a resistance; a bench supply has a regulator with a finite output impedance and wires from it to the terminals; a signal generator has a deliberate fifty ohms in series with its output. In every case there is a resistance between the mechanism that sets the voltage and the terminals where it is measured, and current through that resistance produces a potential difference that the outside world never sees separately.

The consequence is a straight line rather than a flat one: the terminal voltage falls linearly with the current drawn, at a rate equal to the internal resistance. Extrapolate that line to zero volts and the current there is the short-circuit current, which for the nine-volt, half-ohm source in the figure is eighteen amperes.

That single resistance carries all the information. Two sources of the same voltage and different internal resistance are different objects in every way that matters, exactly as two dividers of the same ratio and different magnitude are, and for the same reason: the quantity that decides the behaviour is the one the simple description discards.

The edge, as a number #

The ideal-source model is one per cent wrong when the current drawn multiplied by the internal resistance reaches one per cent of the source’s voltage. For the source in the figure, that is 180 milliamperes — a current an indicator lamp would draw without anyone thinking about it.

Three points on the same curve are worth having:

• at 180 mA the terminal voltage is 8.91 V, and the ideal answer is 1% high
• at 1.8 A it is 8.10 V, and the ideal answer is 10% high
• at 9 A — half the short-circuit current — it is 4.50 V, and the ideal answer is double the truth

The last one is the sharpest way to put it. At half the short-circuit current, an ideal source is not slightly optimistic; it is wrong by a factor of two, and any calculation built on it has stopped describing the circuit in front of it.

The same arithmetic as the divider, wearing a different hat #

A reader who has met the loaded divider will recognise all of this, and the recognition is exact rather than an analogy. A source with internal resistance driving a load is a divider, with the internal resistance as its upper arm and the load as its lower one. The terminal voltage is the supply times the load over the sum, which is the same fraction, and the one-per-cent boundary is the same computation with the arms relabelled.

This is worth making explicit because it means the two rules collapse into one. A source is accurate to one per cent when the load it drives is about a hundred times its internal resistance; a divider is accurate to one per cent when the load it drives is about fifty times its own two resistances in parallel. The factors differ by two only because the divider’s fraction is a half rather than one, and the underlying statement is identical: an interface is faithful only when the impedance on one side of it is far larger than the impedance on the other.

That single sentence covers loading, measurement error, filter section interaction and probe compensation, and it is the reason all of them behave the same way when they go wrong.

Why maximum power is at the worst place #

There is a classical result here that is usually presented as a curiosity and is better understood as a warning.

The power delivered to a load is the product of the terminal voltage and the current. At very high load resistance the current is negligible and so is the power; at very low load resistance the voltage has collapsed and so, again, is the power. Somewhere in between there is a maximum, and it sits exactly where the load resistance equals the internal resistance.

That point is worth locating precisely because of where it is: it is the point at which the terminal voltage is half its unloaded value. The condition for getting the most power out of a source is therefore the condition under which the ideal-source model is wrong by a factor of two, and the efficiency there is fifty per cent — half the energy is being dissipated inside the source.

Nothing about that is contradictory, but it does explain why the result is misapplied so often. It is the right criterion for a receiver extracting a weak signal from an antenna, where the signal is all there is and the source’s dissipation is irrelevant. It is entirely the wrong criterion for anything supplying power, where the objective is efficiency and the correct arrangement is the opposite one — a load resistance far above the source’s, in the region where the ideal model still holds.

Measuring the resistance that nobody drew #

The internal resistance is not marked on anything and cannot be measured directly, since it sits behind terminals that only ever present the two quantities together. It can, however, be inferred in one step, and the method is worth stating because it is the same method the site uses everywhere: take two measurements the model relates, and solve for the parameter that relates them.

Measure the terminal voltage at two different known currents. The difference in voltage divided by the difference in current is the internal resistance — the slope of the straight line, obtained without ever knowing where the line starts. For the source in the figure, readings of 8.91 V at 180 mA and 8.10 V at 1.8 A give a slope of 0.81 V per 1.62 A, or half an ohm, which is the value that went in.

Two details make the difference between that working and not working. The two currents must be far enough apart that the difference in voltage is well above the resolution of the meter, which is why a single reading near no load tells nobody anything. And the source must be genuinely linear over the range used — a battery is not, once its chemistry starts to matter, and a regulated supply is emphatically not once its current limit engages, at which point the terminal characteristic stops being a straight line and turns down almost vertically.

That last case is worth naming because it is where the linear model ends. A bench supply in current limit is a current source: it holds the current constant and lets the voltage be whatever the load requires. It is the same instrument, described by a different model, on the other side of a boundary that is drawn on its own front panel.

The measurement problem, which is the same problem #

Any instrument connected to a circuit is a load on it. A voltmeter of ten megohms across the output of a divider made from two 1 MΩ resistors changes that output by a fifth, and the reading is a correct measurement of a circuit that no longer exists.

The rule follows directly from the boundary above and requires no separate theory: an instrument disturbs a measurement by less than one per cent when its input resistance is about a hundred times the output resistance of what it is measuring. A ten-megohm meter is therefore trustworthy on anything below about a hundred kilohms, and progressively less so above.

There is a version of this that catches people at high frequency rather than high impedance, and it is worth flagging here because the mechanism is identical: an oscilloscope probe has capacitance as well as resistance, and the capacitance loads the circuit more and more as frequency rises. The resistance sets a boundary in ohms; the capacitance sets one in hertz. The same sentence, in different units.

What a solver has to refuse #

The ideal source is also the element that produces the most instructive failures, and a solver that does not refuse them is not to be trusted with the cases that do have answers.

Two ideal sources in parallel, of different voltages. Each insists on the potential difference between the same pair of nodes and they disagree. There is no set of currents that satisfies both, and there is nothing to solve for: the current circulating between them is not determined by anything in the description. A real pair of batteries connected that way does have an answer, and the answer comes entirely from their internal resistances — which is to say, from the very quantity the ideal model removed.

A loop of ideal sources. The same failure with more steps in it. Going round the loop, the voltage law either restates something already known or contradicts it, and either way one equation has been lost.

An ideal source short-circuited. The current is infinite, which is a way of saying the model has no answer. Add the internal resistance and the answer is eighteen amperes.

Every one of those failures is fixed by the same thing: the resistance the ideal model left out. That is a general pattern in this subject and it is worth stating as such. The idealisation that makes a model tractable is usually the same one that makes it fail, and the failure is usually cured by putting back exactly what was removed.

The dual, and why it is not a curiosity #

Everything above has an exact counterpart with voltage and current exchanged. An ideal current source holds its current whatever voltage is required, its imperfection is a resistance in parallel rather than in series, and it fails when the load resistance becomes comparable with that parallel resistance rather than with a series one.

The two descriptions are not competing accounts. A source of nine volts behind half an ohm and a source of eighteen amperes across half an ohm are indistinguishable from outside: they produce the same voltage at every current, the same current at every voltage, and no measurement made at the terminals can tell them apart. That is Norton’s observation sitting beside Thévenin’s, and the equivalence is exact for any linear network.

Which description to use is therefore a question about convenience rather than about truth, and the useful guidance is that each is clumsy at the other’s easy end. A source with a very small internal resistance is naturally described as a voltage source, since the resistance is a small correction; as a current source it would be an enormous current across an almost-short-circuit, with the answer emerging as a small difference between two large numbers. The reverse holds for a transistor’s collector, which is naturally a current source with a large parallel resistance and awkward as a voltage source of hundreds of volts.

That last observation is not aesthetic. Computing a small difference between two large quantities loses precision, and a solver handed the wrong form of an equivalent source can produce a visibly worse answer to an identical question. The equivalence is exact in mathematics and not in floating point, which is a distinction this collection returns to whenever a model and a machine disagree.

What the number on the label is #

A source’s stated voltage is the value it produces into nothing at all, and that is worth saying in those words rather than as “the open-circuit voltage”, because the two phrasings carry different amounts of warning.

Every practical source departs from it in proportion to what is drawn. A battery’s terminal voltage falls as its internal resistance drops the load current. A regulated supply’s falls too, by much less, because its feedback loop is measuring its own output and correcting — and the amount it still falls is the loop’s error, which is the loop gain divided into the open-loop drop.

Both cases are the same circuit with a different resistance in series, and the useful discipline is to ask what value that resistance has rather than whether it is small. A supply specified as “0.01 Ω output impedance” delivers 10 mV less at an ampere, which is 0.2% of a five-volt rail and is either negligible or the whole of a measurement’s error depending on what is being measured. The figure on this page is that arithmetic performed rather than described.

Sources with a direction #

One more kind of source belongs in this essay, because it is the reason the whole apparatus is worth building rather than a curiosity.

A controlled source produces a voltage or a current determined by what is happening somewhere else in the circuit. That is what a transistor does, what an operational amplifier does, and what a transformer does. In the matrix it looks almost exactly like an independent source, with one difference: the pattern it contributes is not symmetric, because its output does not affect its input.

The number in that caption is the essay’s conclusion in a single measurement. A device specified as a gain of fifty delivers thirty-one, and neither the device nor the specification is wrong: the missing factor is entirely the interface impedances at its two ends. That gap between the specification and the circuit is computable, it is the reason a nodal solve exists, and it is what the rest of this collection spends its time measuring.

Where the edge goes when the circuit is built to remove it #

The whole of the applied field’s regulator ladder is an answer to this essay, and the interesting part is that the answer moves the boundary onto a different axis rather than removing it.

A source below a frequency measures the circuit built specifically to hold a voltage against a current, and finds it doing so — 0.43 milliohms at direct current, four orders below the half-ohm this essay’s battery has. What it also finds is that the number is a direct-current number: the impedance has doubled by 4.81 hertz, is ten times worse by 27, and reaches 1.95 ohms at ten kilohertz, which is higher than the same circuit with its feedback loop cut. So the edge that was a current has become a frequency, and the model “an ideal voltage source” is now true below a frequency instead of below a current.

The direct voltage that is a sawtooth is the same observation one stage further back, where what is being called a source is a rectifier and a capacitor: 15.69 volts with 1.33 volts of sawtooth on it, and a mean that moves by fourteen per cent across a sweep of reservoir capacitance with nothing else changed. And the divider, and the thing it does not know about is the same failure with no source in it at all — two dividers of identical ratio giving six volts and one volt into the same load, because the quantity the ratio was built to discard is the one that decides the answer.

The three together say what this essay’s rule amounts to in practice. Every source in a schematic is a model with an edge, the edge is on whichever axis the source’s internal impedance depends on, and the only question a designer has to answer is which axis that is.

For a battery it is a current, because a chemical cell’s internal resistance is very nearly constant. For a regulator it is a frequency, because a loop’s gain is. And for a capacitor holding a rail between switching events it is a duration — which is the shape the total that has no resistor in it meets from the noise side, where a resistor charging a capacitor gives one total whatever its value because the density and the bandwidth move inversely. Three axes, one model, and nothing in the symbol to say which.

Where the ideal source is refused #

An ideal source is the flattest line in the subject and the one model this collection’s solver has to refuse. The answer that is perfect and absurd is the refusal, at a network of two ideal sources joined by something that is nearly a wire. What a network answers, and how the answer is checked is where the refusal is one of four the solver makes by name. Exact outside and wrong within is the reduction that produces an ideal source deliberately and is exact at the terminals for doing it. The divider, and the thing it does not know about is the smallest circuit in which the internal resistance matters, and Kirchhoff’s own frequency is the boundary above which none of these models is about the circuit at all.