Use Norton equivalent circuits: Foundations — Hardware deep dive

This page assumes you know nothing. We build every symbol the parent note (the Norton topic) leans on, one brick at a time, each brick resting on the one before it.

1. A "terminal" — the two wires we peek through

The picture: imagine a sealed metal box with only two screws poking out of it. You are forbidden to open the box. Everything you will ever learn about that box, you learn through those two screws.

Why the topic needs it: the Norton equivalent describes a network as seen from its terminals. If we could open the box and stare at every resistor, we would not need an "equivalent" at all. The whole game is: summarise the box using only what the two screws reveal.

2. Current $I$ — the flow of charge

The picture: think of water in a pipe. Current is "how many cups of water rush past this cross-section each second." A fat, fast flow is a big $I$ ; a trickle is a small $I$ .

Why the topic needs it: the Norton current source (the current that flows when we join the terminals with a wire) is measured in these same amperes. We cannot speak of it until "current" means something concrete.

3. Voltage $V$ — the push

The picture: back to water. Voltage is the height difference between two tanks. High tank above low tank ⇒ big push ⇒ water wants to flow. No height difference ⇒ no push ⇒ still water. In a circuit, $V$ across the two terminals is the pressure difference between them.

Why the topic needs it: the Norton model's whole behaviour is a relationship between terminal voltage $V$ and terminal current $I$ . Both must be defined before that relationship can exist.

4. Sign conventions — which way is "positive"?

Before any formula that mixes $V$ and $I$ can be trusted, we must agree on which direction counts as positive. A number like " $3 A$ " is meaningless until you say which way those amps flow.

The picture (see figure): an arrow drawn on the wire and a $+$ / $-$ pair on the resistor. Reverse either and the reported number changes sign — the physics is the same, only our bookkeeping flips.

Why the topic needs it: every later equation — Ohm's law, KCL, the Norton terminal law — silently assumes these choices. "Current in minus current out" only makes sense once "in" and "out" are pinned to arrows.

5. Resistance $R$ and Ohm's Law — push, flow, and thickness

The picture: a thin, clogged pipe is high resistance; a wide open pipe is low resistance.

These three ideas lock together into one rule you will use on every line of the parent note (using the passive sign convention from §4):

Why the topic needs it: the parent's "leak" term (the current that leaks back through the equivalent resistor when a voltage sits across it) is literally Ohm's law applied to that resistor. Without Ohm's law that term is meaningless. We will name that resistor $R_{N}$ in §9.

6. Kirchhoff's Current Law — nothing vanishes at a junction

The picture: a road fork. However many cars enter the junction per minute, exactly that many must leave — no cars magically appear or disappear at the intersection.

Why the topic needs it: we need KCL immediately — the next section's parallel rule splits current at a node, and later the Norton terminal law is pure KCL at one node. With the current-arrow convention from §4 fixing "in" and "out", the source pushes its full current into the node, some leaks back through the equivalent resistor, and the rest must exit to whatever we attach. We introduce it here, before the derivations that call on it.

7. Series and parallel — two ways to wire resistors

The picture (see figure): series is a single road with two toll gates in a row; parallel is a road that splits into two lanes and rejoins.

Why the topic needs it: in Worked Example 1, killing the source puts $R_{1}$ and $R_{2}$ in parallel, giving the Norton resistance $\frac{4 \cdot 6}{4 + 6} = 2.4 Ω$ . You cannot follow that line without knowing what "parallel" means and why we use product-over-sum.

8. The short circuit and the open circuit — the two extreme moves

The picture (see figure): the short is a thick bridge — everything crosses freely, no height difference builds up. The open is a broken bridge — nothing crosses, so pressure piles up but no flow moves.

Why the topic needs it: these are the two probes of the whole method.

Short the terminals → $V = 0$ → the current you read is the short-circuit current (this will be named $I_{N}$ in §10).
Open the terminals → $I = 0$ → the voltage you read is the open-circuit voltage (this will be named $V_{T h}$ in §10).

The short and open are how we "interrogate the sealed box" from §1.

9. Deactivating a source — turning off without removing

Why the topic needs it: the Norton resistance is the resistance of the box with all pushes silenced, so only the resistors' geometry remains. If we left sources on, they would shove extra current through our measurement and spoil a pure resistance reading (that is the parent's first Common Mistake).

10. Naming the Norton symbols — $I_{N}$ , $R_{N}$ , $V_{T h}$ , $R_{T h}$ , $I_{L}$

Now every underlying idea is grounded, we can finally attach names to the four Norton/Thévenin quantities and the load current — each is just one of the concepts above with a label.

The terminal law from KCL. With the current arrow of §4 pointing out to the load, KCL from §6 at the parallel node reads: $pushed in I_{N} = leaks back through R_{N} \frac{V}{R _{N}} + out to load I_{L} ⟹ I_{L} = I_{N} - \frac{V}{R _{N}} .$ Every symbol here was defined earlier: $I_{N}$ (§8/§10), $V$ (§3), $R_{N}$ (this section), $I_{L}$ (this section), and the $\div$ is Ohm's law (§5).

Formula Why

$R_{T h} = R_{N}$ , and why $V_{T h} = I_{N} R_{N}$ (symbolic proof) By the linearity assumption (top of page), the box's terminal characteristic is one straight line relating $V$ and $I_{L}$ . Both models must reproduce that same line, so we can compare them line-to-line.

The Norton line. From the terminal law above, solve for $V$ : $I_L = I_N - \frac{V}{R_N} \;\Longrightarrow\; \frac{V}{R_N} = I_N - I_L \;\Longrightarrow\; V = I_N R_N - I_L R_N. \tag{N}$

The Thévenin line. A Thévenin model is a voltage source $V_{T h}$ in series with $R_{T h}$ ; walking from the $+$ terminal, the source lifts by $V_{T h}$ and the resistor drops by $I_{L} R_{T h}$ (Ohm's law, §5), so $V = V_{Th} - I_L R_{Th}. \tag{T}$

Match the two straight lines. Lines (N) and (T) describe the identical box, so their slopes (the coefficient of $I_{L}$ ) and their intercepts (the constant term) must be equal, term by term: $equal slopes - R_{N} = - R_{T h} ⟹ R_{N} = R_{T h}, equal intercepts I_{N} R_{N} = V_{T h} ⟹ V_{T h} = I_{N} R_{N} .$ So $R_{N} = R_{T h}$ is not a coincidence: the slope of the $I$ – $V$ line is set purely by the resistors, and deactivating the sources measures exactly that slope in both models — identical resistor geometry ⇒ identical slope. And $V_{T h} = I_{N} R_{N}$ (equivalently $I_{N} = V_{T h} / R_{T h}$ ) is forced by the intercepts matching. This is Source transformation.

11. Putting it together — the full glossary

Recall The full symbol glossary (test yourself, then open)

$I$ ::: current, charge per second, in amperes (A). $V$ ::: voltage, the electrical push, in volts (V). $R$ ::: resistance, opposition to flow, in ohms ( $Ω$ ). $I_{N}$ ::: Norton current — the current through a short across the terminals. $R_{N}$ ::: Norton resistance — resistance seen at terminals with sources deactivated. $I_{L}$ ::: load current — current delivered to whatever you attach at the terminals. $V_{T h}$ ::: Thévenin voltage — the open-circuit voltage at the terminals. $R_{T h}$ ::: Thévenin resistance — equals $R_{N}$ ; same slope, proven in §10. $∥$ ::: "in parallel with" — same two nodes, current splits. $Ω$ ::: the ohm, unit of resistance (Greek capital omega).

Prerequisite map

Equipment checklist

Read each line, answer in your head, then reveal:

I can state the one assumption Norton/Thévenin needs

The network is linear and time-invariant — only resistors and independent sources — so the terminal

I

–

V

curve is a straight line.

I can point to the two terminals of a sealed circuit and say what they are for

Yes — they are the only window into the box; the equivalent describes behaviour seen from them.

I can state current

I

in one sentence

Charge flowing past a point each second, measured in amperes.

I can state voltage

V

in one sentence

The push (pressure difference) that drives current, measured in volts.

I understand why a chosen current direction can come out negative

The arrow is picked in advance; a negative answer just means the real flow is the other way.

I can write Ohm's Law three ways

V = I R

I = V / R

R = V / I

I can state KCL

Current in equals current out at every junction.

I can explain why series resistances add

Same current through both; their voltage drops

I R_{1}

and

I R_{2}

stack, so

R = R_{1} + R_{2}

I can explain why parallel uses product-over-sum

Same voltage across both; currents

V / R_{1}

and

V / R_{2}

add, so conductances add and

R = R_{1} R_{2} / (R_{1} + R_{2})

I know what a short circuit forces

V = 0

across the terminals; it carries all the current.

I know what an open circuit forces

I = 0

; current cannot cross the gap.

I can deactivate each source type

Voltage source → wire (short); current source → gap (open).

I can say what

I_{N}

and

R_{N}

mean

I_{N}

= short-circuit terminal current;

R_{N}

= terminal resistance with sources deactivated.

I can prove

V_{T h} = I_{N} R_{N}

Both models are the same straight line

V = I_{N} R_{N} - I_{L} R_{N}

and

V = V_{T h} - I_{L} R_{T h}

; matching intercepts gives

V_{T h} = I_{N} R_{N}

and matching slopes gives

R_{T h} = R_{N}

Connections

Use Norton equivalent circuits — the parent this page prepares you for.
Ohm's Law — supplies $V = I R$ and the leak term $V / R_{N}$ .
Kirchhoff's Current Law — justifies $I_{L} = I_{N} - V / R_{N}$ .
Thévenin equivalent circuits — the voltage-source twin using $V_{T h}$ , $R_{T h}$ .
Source transformation — converts between the two using $V_{T h} = I_{N} R_{N}$ .
Superposition — why the terminal behaviour is linear/affine (a straight line).