Foundations — Use Thevenin equivalent circuits

This is the ground floor of Use Thevenin equivalent circuits. If the parent note used a symbol without explaining it, we explain it here, from zero. Read top to bottom: each idea is a brick the next one stands on.

1. Two terminals — the "two wires you care about"

Imagine a sealed box with only two wires poking out. You are not allowed to open it. Everything you will ever learn about that box, you learn from those two wires. Thevenin's whole promise is: those two wires are enough to fully describe the box.

Read the figure: the lavender box is the "inside you never open," and the two coral dots are the only contact points — terminal $A$ (top) and terminal $B$ (bottom). Notice the coral arrow: only these two wires reach the outside. Fix this picture in mind; every symbol below lives on these two dots.

Why the topic needs this: the theorem only works for a two-terminal network. The picture of "a box with exactly two contact points" is the stage on which every other symbol will perform.

2. Voltage $V$ — the "push"

Picture a water tank up on a tower. The height of the water is like voltage: higher tank → stronger push on the water below. Voltage is always a difference between two points — just like height only makes sense relative to the ground.

Why the topic needs it: the whole theorem is about a voltage measured across the two terminals of Section 1. Everything downstream is built on this word. (The special value that voltage takes when nothing is plugged in gets its own name, $V_{th}$, once we have earned it in Section 6.)

3. Current $I$ — the "flow"

Back to the water tower: the height is voltage (the push), but the litres per second flowing out of the pipe is current. A tall tank with a pinched pipe can push hard yet deliver little flow.

Why the topic needs it: $I$ is the current the outside world forces through the terminals.

3b. Sign convention — which way is "positive"?

Before we ever write $V$ and $I$ in the same equation, we must fix arrows. Otherwise a minus sign is meaningless.

So $I$ is the current drawn by whatever we plug in, entering the box at its $+$ side. With this one choice locked, a drop inside the box appears with a minus sign — which is exactly why the terminal law will read $V=V_{th}-I{R}_{th}$ and not $+$. Every figure on this page uses these same arrow directions.

Why the topic needs it: the terminal law has a minus sign; that sign only means something once we have declared which direction of $I$ and which polarity of $V$ we call positive.

4. Resistance $R$ and Ohm's law — the "speed-bump"

Water picture: resistance is a narrow, gravelly stretch of pipe. For a given tank height (voltage), a narrower pipe (bigger $R$) lets less water through (smaller $I$).

Read the figure: the lavender tank on the left is the push $V$ (the coral double-arrow measures its height); the gravelly butter-yellow patch in the pipe is the resistance $R$; the mint arrow leaving the pipe is the flow $I$. The caption spells out $V=IR$: raise the tank (more $V$) or widen the pipe (less $R$) and more flows.

Why the topic needs it: the "$-IR$" style term in the terminal law is literally an Ohm's-law voltage drop across the box's internal resistance.

5. Nodes, then series and parallel — how resistors combine

Before "same two end-points" can mean anything, we need the word node.

Why "parallel is smaller": two roads carry more total flow than one, so the combined stiffness is less than either alone. This is exactly why, in the parent's Example 1, shorting the source ties the tops of $R_1$ and $R_2$ to the same node, making them parallel and giving $1.33\Omega$ — smaller than either.

Read the figure: the top row shows series — one road, current passes through $R_1$ then $R_2$, and the box on the right reminds you they add to $R_1+R_2$. The bottom row shows parallel — the wire splits into two roads sharing the same left and right nodes, and the result $R_1{R}_2/(R_1+R_2)$ is flagged in coral as smaller than either.

Why the topic needs it: finding the box's internal resistance almost always ends in a series/parallel reduction. Miss the series-vs-parallel call and it is wrong — this is the parent's mistake #3.

6. Open circuit, short circuit, and the names $V_{th}$, $I_{sc}$

Now — and only now — we can name the two star quantities of the whole topic, each tied to one extreme test:

These two tests probe the box's straight-line behaviour at its two ends:

• Open → gives $V_{th}$ (the open-circuit voltage $V_{oc}$).
• Short → gives $I_{sc}$ (the short-circuit current), which will yield the internal resistance $R_{th}=V_{th}/I_{sc}$ once $R_{th}$ is defined in Section 8.

Why the topic needs it: the parent's whole recipe is "OPEN for voltage, DEAD for resistance," plus the short-circuit route. All three lean on these two idealized tests and the two names just defined.

7. Independent vs dependent sources

Why this distinction is life-or-death for the topic: to find the internal resistance you deactivate sources (voltage source → short, current source → open). You may only deactivate independent ones. Killing a dependent source would erase the very relationship that defines the circuit — so instead you use the test-source method. This is the parent's mistake #1.

Why the topic needs it: the recipe for the internal resistance branches entirely on whether dependent sources are present.

8. Linearity — and finally, $R_{th}$ and the terminal law

Now we earn the last symbol and the terminal law, using superposition (Section 9) and the sign convention (Section 3b).

Read the figure: the lavender line is $V=V_{th}-I{R}_{th}$ plotted with $V_{th}=4\text{ V}$, $R_{th}=\frac{4}{3}\Omega$. Where the line hits the vertical axis ($I=0$) is the coral intercept $=V_{th}$, the open-circuit test. Where it hits the horizontal axis ($V=0$) is the green short-circuit point, current $=I_{sc}$. The line's tilt is the slope $-R_{th}$. Both tests of Section 6 are just the two ends of this one line.

Why the topic needs it: linearity is what makes superposition legal, and superposition is what derives the terminal law. Remove linearity and the theorem collapses.

9. Superposition — the tool that made Step 3 legal

That is exactly what Section 8 did: response to internal sources alone ($V_{th}$) plus response to the forced external current alone ($-I{R}_{th}$), added to get the full terminal law. This addition is only valid because every element is linear — closing the loop back to Section 8. See Superposition Theorem for the full machinery.

Why the topic needs it: it is the engine of the derivation $V=V_{th}-I{R}_{th}$.

How these foundations feed the topic

Equipment checklist