Networks, and how a solve is checked

The reduction that stays finite

The bootstrap's Thévenin pair diverges at A = 2: its resistance and its open-circuit voltage both pass through infinity and the solve that finds them refuses. Its Norton pair does not. The short-circuit current is Vs/R₁, 0.5 mA, at every gain to a part in a billion, and the conductance is a straight line through zero. A bench that drives a current and reads a volt loses the network for every gain from 1.992 to 2.008; a bench that drives a volt and reads a current sees it to a nanosiemens throughout. A 0.1% error in the gain moves the conductance by a constant 0.1 µS and the resistance by an amount that diverges at the pole — and four hundred bootstraps built from 0.1% parts to sit on it have conductances within ±0.3 µS of zero and resistances that are a median 10.6 MΩ, 196 of them negative.

16 min read 5 figures Two routes to a number Refused, not extrapolated The schematic is a label

Assumes: Exact outside and wrong within

Every linear network seen from two terminals is a source and an impedance, and there are two ways to write that down. Thévenin’s puts a voltage source in series with the impedance; Norton’s puts a current source in parallel with it. Exact outside and wrong within reduced a network of six elements to the first form and found it indistinguishable from the original by any load, to the last bit, and the textbook treatment of the second form is a single line: divide the open-circuit voltage by the resistance and the two are interchangeable.

They are interchangeable wherever both exist, and the resistor that is not made of the resistors built a network where one of them does not. Its bootstrap has a Thévenin resistance of $R_1 \parallel R_2/(1 - A)$ , which passes through infinity at $A = 1 + R_2/R_1$ and comes back negative; its open-circuit voltage passes through infinity at the same gain. That essay noticed that the short-circuit current, alone among the reduction’s quantities, did not move, and asked whether the Norton form — which is built from that current — is the better form near the pole or merely moves the difficulty somewhere else. The pole that crosses with the resistance then found that the gain where the resistance changes sign is a gain where the network’s own pole crosses into the right half-plane, which makes the neighbourhood of the pole the most interesting place the network has.

This essay compares the two forms there: as numbers, as bench measurements, as targets for a tolerance, and as tools for computing what a load sees.

Two coordinates for one network

Reduce the bootstrap both ways at every gain from nothing to three, with two ten-kilohm resistors and a five-volt source, and draw everything in units of $R_1$ so that one axis carries it.

The bootstrap reduced both ways: Thévenin's resistance diverges at A = 2, Norton's conductance crosses zero there. computed by solving, not by drawing at 121 gains, R₁ = R₂ = 10 kΩ. The Norton current is Vs/R₁, 0.500 mA, at every gain to a part in a billion (dashed, at 1), because shorting the node also shorts the bootstrap's input. The Norton conductance times R₁ is a straight line, 2 at A = 0 through zero at A = 2 to −1 at A = 3 (solid). The Thévenin resistance over R₁ is its reciprocal (faint, clipped at ±3): 0.5, 1 at A = 1, infinite at A = 2, −1 at A = 3. One network, two coordinate systems, and only one of them is singular exactly where the network is interesting. — Fig. 1 The bootstrap reduced both ways at 121 gains. The Norton current (dashed) is $V_s/R_1$ at every gain to a part in a billion. The Norton conductance times $R_1$ (solid) is a straight line, 2 at A = 0, zero at A = 2, −1 at A = 3. The Thévenin resistance over $R_1$ (faint, clipped at ±3) is its reciprocal: 0.5, 1 at A = 1, infinite at A = 2.

The Norton current is 0.5 milliamperes at every gain, to a part in a billion. Shorting the output node also shorts the bootstrap’s input — the controlled source drives its far end with $A$ times zero — so the current into the short is the source voltage over $R_1$ whatever the gain. The Norton conductance is a straight line: $1/R_1 + (1 - A)/R_2$ , twice $1/R_1$ at no gain, zero at $A = 2$ , minus $1/R_1$ at $A = 3$ . Neither has anything special happening at the pole; the conductance merely passes through zero there, as a straight line will.

The Thévenin pair is the same information in other coordinates: the resistance is the conductance’s reciprocal, and the open-circuit voltage is the current over the conductance. Both pass through infinity at $A = 2$ because both divide by a quantity that is zero there, and both are finite and exact everywhere else. The reduction that finds the Thévenin resistance — look into the node with the sources dead — refuses at exactly $A = 2$ , with the conductance matrix singular, while the two Norton solves, the current into a short and the current a test voltage draws, solve cleanly at every gain including that one. The network is not singular. One of the two coordinate systems is, and it is singular at the one gain where the network is most worth describing.

Two bench routes, and the one that sees the pole

The same asymmetry shows up on a bench, where it is no longer a matter of notation. There are two ways to measure the resistance looking into a node: drive a known current and read the voltage, or drive a known voltage and read the current.

The bootstrap's resistance measured near its pole by two bench routes, both converted to a conductance. computed by solving, not by drawing for the true value, then read through two instruments. Current in (10 µA) and voltage read on a ±10 V meter to a millivolt (dots): the reading leaves the meter's range for every gain from 1.992 to 2.008, a resistance above a megohm in either sign, and returns nothing there. Voltage in (1 V) and current read to a nanoamp (solid): the conductance comes back to a nanosiemens at every gain, through zero at A = 2 and negative beyond it. The pole is where one route has nothing to say and the other has the easiest reading it ever makes. — Fig. 2 The bootstrap’s conductance near its pole, read by two bench routes. Current in (10 µA) and voltage read on a ±10 V meter to a millivolt (dots): no reading for gains from 1.992 to 2.008, where the resistance exceeds a megohm in either sign. Voltage in (1 V) and current read to a nanoamp (solid): the conductance to a nanosiemens at every gain, through zero and negative beyond.

Drive ten microamps into the node and read the voltage on a ten-volt meter that resolves a millivolt. Away from the pole that is a fine measurement of the resistance, to a hundred ohms. Near the pole the voltage climbs past the meter’s range, and for every gain from 1.992 to 2.008 the meter returns nothing — the resistance is above a megohm in one sign or the other, and at $A = 2$ exactly the solve of the bench itself refuses, because no finite voltage carries a current into a node with no conductance to ground.

Drive one volt into the node and read the current on an ammeter that resolves a nanoamp. The reading is the conductance directly, to a nanosiemens at every gain, and the pole is the easiest reading the instrument ever makes: at $A = 2$ the node draws no current at all, and past it the current reverses. Plotted as conductance, the voltage-driven route draws a straight line through zero where the current-driven route has a gap. This is the measurement the earlier essay asked about — whether driving a voltage and reading a current is better near the pole — and the answer is that it is the only one of the two that works there.

The rule generalises beyond this network. A quantity that passes through zero is measured well by an instrument whose reading is proportional to it; a quantity that passes through infinity is measured by nothing. Near a pole of the resistance, measure conductance; near a zero of it — a node that is nearly a short — measure resistance, and drive a current.

A tolerance is a constant in one coordinate

A real bootstrap’s gain is set by components with tolerances, and the useful question is what an error in the gain does to each form.

What a 0.1% change in the gain does to the Thévenin resistance, against how far the gain is from the pole. computed by solving, not by drawing at each gain and at the gain plus a thousandth. The Norton conductance moves by ΔA/R₂ = 0.1 µS at every gain, to a part in a million — a straight line shifted. The Thévenin resistance moves by that times itself squared, so its relative change is Rₜₕ·ΔA/R₂, inversely proportional to the distance from the pole: 0.050% at A = 0 and 5% at A = 1.98. A tolerance that is a constant in one coordinate is a divergence in the other. — Fig. 3 The relative change in the Thévenin resistance when the gain moves by 0.1%, against the gain’s distance from the pole. The Norton conductance moves by $\Delta A/R_2$ = 0.1 µS at every gain; the Thévenin resistance moves by that times its own square, 0.050% of itself at A = 0 and 5% at A = 1.98.

Move the gain by a thousandth. The conductance moves by $\Delta A/R_2$ , 0.1 microsiemens, at every gain, to a part in a million: the straight line shifts sideways and keeps its slope. The resistance moves by that shift times its own square, so its relative change is $R_\text{th}\,\Delta A/R_2$ — 0.050 per cent of itself at $A = 0$ and 5 per cent at $A = 1.98$ , and without bound as the gain approaches the pole. A tolerance that is a constant in one coordinate is a divergence in the other.

That is why a specification near the pole should be written in conductance. “Output resistance greater than ten megohms” is a statement about how close to zero a conductance is held, in the coordinate where closeness to zero is measured in microsiemens and moves linearly with every error source. Written as a resistance it sounds like a number to be met; written as a conductance it is a tolerance band around zero, of a width each component’s error contributes to additively.

A current source built to sit on the pole

The bootstrap tuned to its pole is a current source: an infinite output resistance means the current delivered is the same into any load. That is the Howland current source, and it is built by setting the gain with a resistor ratio.

Four hundred bootstraps built to sit exactly on the pole, their gain set by two 0.1% resistors. computed by solving, not by drawing for each of 400 seeded units whose gain is 1 + Ra/Rb with Ra and Rb equal to 0.1%. Their output conductances form a narrow, symmetric distribution about zero, 0.145 µS standard deviation (bars). Read as resistances the same units are a median of 10.6 MΩ in magnitude, with 196 of the 400 negative — current sources that will run away if left open-circuit. The specification "infinite output resistance" is a conductance of zero, and it is in conductance that the spread is a number. — Fig. 4 Four hundred seeded bootstraps built to sit on the pole, their gain 1 + Ra/Rb with Ra and Rb equal to 0.1% (one standard deviation). Their output conductances are a narrow distribution about zero, 0.145 µS standard deviation. As resistances they are a median of 10.6 MΩ in magnitude, and 196 of the 400 are negative.

Four hundred units, each with its gain set by two resistors of 0.1 per cent, have output conductances spread symmetrically about zero with a standard deviation of 0.145 microsiemens — $\sqrt2$ times the 0.1 per cent over $R_2$ , as the sensitivity figure predicts. Read as resistances the same four hundred units are a heavy-tailed scatter with a median magnitude of 10.6 megohms, and 196 of them are negative.

Those 196 are the units that the pole that crosses with the resistance said will run away if left unloaded: a negative resistance across an open circuit is unstable at any value. A current source is normally specified with its output open as one of its legitimate conditions — it is a source of current into whatever is there, including nothing — and half the units of an infinite-resistance design meet that condition by saturating against a rail. The specification in conductance says so in advance: a band of ±0.3 microsiemens about zero contains both signs, and the stable half is the half whose conductance is positive.

The repair is to aim off the pole on purpose. A gain set a few tenths of a per cent below $1 + R_2/R_1$ puts the whole distribution on the positive side, at the cost of a finite output resistance — a few megohms rather than “infinite” — which is what a carefully designed Howland source actually delivers, and the only thing a design with toleranced parts can promise.

Computing through the pole

The two forms differ, finally, in what they can compute. The point of a reduction is to find what a load sees without solving the whole network again.

Fig. 5 The voltage across a 20 kΩ load at 76 gains: the full network solved (solid) and the Norton pair (dots) agree to a part in a billion at every gain, through A = 2 where the voltage is 10.000 V. The Thévenin pair is infinite over infinite at A = 2. The loaded network’s own singularity is at A = 2.50.

Across a twenty-kilohm load, the Norton pair gives the voltage as $I_N/(G_N + 1/R_L)$ , and it agrees with the full network’s solve to a part in a billion at every gain, through $A = 2$ where the load sees a perfectly ordinary 10.000 volts. The Thévenin pair gives $V_\text{th}R_L/(R_\text{th} + R_L)$ , which agrees everywhere it can be evaluated and is infinity over infinity at $A = 2$ — a form that cannot be used at the point where it is most needed, though the voltage it is meant to compute is finite and smooth there.

Both forms do share one singularity, and it is the loaded network’s own: at $A = 2.5$ the network’s negative conductance cancels the load’s, the denominator of the Norton expression is zero, and the voltage grows without bound on either side. That is the loaded crossing of the previous essay, and it belongs to the circuit rather than to a choice of coordinates. A reduction can move a singularity that belongs to its own coordinates; it cannot remove one that belongs to the network, and the two forms agree on where that one is.

Which form to carry

The textbook’s claim that the forms are interchangeable is true and incomplete. Each is singular where its own source or impedance is: Thévenin’s where the resistance is infinite, which is where a network is nearly a current source; Norton’s where the resistance is zero, which is where a network is nearly a voltage source. The usual habit is to reduce everything to Thévenin’s form, because voltages are what oscilloscopes show and voltage sources are what bench supplies are, and for most networks the choice costs nothing.

It costs something for exactly the networks that this pair of essays is about: bootstrapped circuits, current sources, anything with positive feedback near the point where its resistance changes sign. For those, the Norton conductance is the quantity that is linear in the design parameters, finite through the interesting point, measurable on a bench, and specifiable as a tolerance; and the Thévenin resistance is its reciprocal, useful only to the extent that the conductance stays away from zero. Two ports from two one-ports makes the same choice for a two-port, where the admittance parameters exist for some networks whose impedance parameters do not, and the reason there is the same as here.

The same choice, made elsewhere

The rule that a quantity near zero is measured in the coordinate where it is near zero, and a quantity near infinity is measured in the other one, has been at work on this site in places where it was not named.

A virtual earth is a node whose impedance to ground is nearly zero, because a loop holds it there. The node that is at ground for a while measured it by driving a current into the junction and reading the voltage that appeared — the resistance route, the right one for a node that is nearly a short, and the route that would have run out of current range trying to hold a test voltage on a node the loop insists is at zero. That essay found a tenth of an ohm at direct current, a number that a voltage-driven bench would have had to infer from a current of amperes.

An ammeter is a resistance chosen small, and the ammeter that is a resistor measured its two opposing errors. A current source, its dual, is a conductance chosen small, and this essay’s Howland source is that dual met in the same terms: what an ammeter’s specification says in ohms, a current source’s says in siemens, and each is a statement about how close to zero the relevant quantity is held.

A photovoltaic cell is the clearest case of a network that is naturally Norton. The source that is not a source found that an ideal voltage source has an edge set by a current; a cell has no such edge, because over most of its range it is a current source — the photocurrent — with a large conductance in parallel only near its open-circuit voltage. The load a curve recommends reduced a cell’s curve to a Thévenin pair at its operating point, which works, and which is the less natural coordinate: the short-circuit current is the cell’s most stable parameter, proportional to the light, while its open-circuit voltage is the one that moves with temperature by the junction coefficient. A cell modelled as its Norton pair has its sources and its sensitivities in separate places.

Each of those is the same decision this essay made about the bootstrap, and each was made for the same reason: the form whose parameters are finite, smooth in the design variables and measurable at the operating point is the one to carry. The Thévenin form is not wrong anywhere it exists. It is merely the form whose parameters happen to explode in some of the most useful circuits there are.

What the pole looks like from each side

There is a last way to see the difference, which is to ask what each form says happens as the gain is swept through the pole.

The Thévenin form tells a story with a discontinuity in it. The resistance rises without bound, vanishes into infinity, and returns from minus infinity; the open-circuit voltage does the same. A designer reading those curves sees a catastrophe at $A = 2$ , and is right to be alarmed by the resistance’s sign beyond it, but is misled about the nature of the event: nothing in the network jumps.

The Norton form tells a story with no discontinuity in it. The conductance passes smoothly through zero and the current does not move at all. The event at $A = 2$ is that the network stops drawing current from a voltage applied to it — it becomes, at that one gain, a perfect current source — and beyond it the network begins to deliver current into a voltage applied to it, which is what a negative conductance is. That is an accurate description of a continuous change, and it is the description the previous essay’s poles gave too: a pole sliding smoothly along the real axis and through the origin, with nothing singular about its motion. The Norton form and the pole agree, and the Thévenin form is the one telling a different story about the same network.

How the figures were made, and what they assume

Every quantity is a solve of the five-element netlist: the open-circuit voltage by solving it as it stands; the short-circuit current by putting a zero-volt source across the node; the Thévenin resistance by looking into the node with the independent source set to zero and the controlled one live; the Norton conductance by putting a one-volt test source at the node with the independent source dead and reading its current. The pole is approached on grids that land on it exactly, so that the refusal is a measured event rather than an extrapolation. The bench routes round each solved reading to the stated resolution and drop it when it exceeds the stated range. The four hundred units draw their two resistors from a seeded generator, and the spread of their conductance is held against $\sqrt2 \times 0.1\% / R_2$ to fifteen per cent.

The controlled source is ideal and instantaneous here, as it was in the earlier reduction; the previous essay is where it acquired a lag. The resistors are exact apart from the stated tolerance and have no temperature coefficient, which for a current source built on a resistor ratio is the next error after the ratio itself.

Still open: conductance across frequency, two-port forms, and the drift of a tuned source

The Norton conductance as a function of frequency. At direct current the conductance is a straight line in the gain; with the source’s lag it is the complex admittance the previous essay drew, and a current source’s output admittance has a capacitive part that rises with frequency however well its direct-current part is tuned. The frequency at which the Howland source’s output admittance has risen to the size of its tolerance band is the bandwidth of its “infinite” resistance, and it is a solve of this netlist with a lag in it.

Two-port reductions near their own poles. A two-port has impedance, admittance, hybrid and transmission parameters, and each set is singular for a different class of network. Which set stays finite for a bootstrapped two-port — a current-feedback amplifier, a gyrator — is the same question asked with four numbers instead of one, and the answer should again be the set whose entries are linear in the design parameters.

A tuned source that drifts. A Howland source trimmed to sit a few tenths of a per cent below its pole keeps its conductance positive only while the resistor ratio holds. Resistors drift with temperature at different rates, and the drift moves the conductance linearly; how many parts per million per kelvin of ratio mismatch carry a source trimmed at 25 °C across zero at 85 °C is the tolerance figure above with temperature on its axis.

Part 4 on equivalent circuit

One argument about Equivalent circuit, and one of 4 essays on it so far, each part numbered by how much of the idea it assumes. What sits either side of it: