One source and one resistor, hiding in plain sight
In guide 2 you learned that any cluster of resistors looks, from the outside, like one resistor — its equivalent resistance. Now comes the grown-up version of that idea, and it is one of the most useful tricks in all of electronics. Thevenin's theorem (Thevenin's theorem) says that any two-terminal network of voltage sources, current sources, and resistors — however knotted inside — behaves, as seen from those two terminals, exactly like a single voltage source V_th in series with a single resistor R_th. Nothing the outside world does can tell the difference.
Picture a sealed metal box with two terminals sticking out. Inside might be one battery, or a dozen sources and fifty resistors. You are not allowed to open it; you may only poke the terminals with a meter and test loads. Thevenin's promise is that everything you could ever measure is captured by just two numbers: the open-circuit voltage V_th (what you read with nothing connected) and the Thevenin resistance R_th (how stiff or droopy the box is when you draw current). That is why we say a real battery, a bench supply, or the output of an amplifier each has an internal source resistance — every real source is secretly a Thevenin box.
Finding the Thevenin equivalent
The recipe has just two unknowns, so you only need two measurements (or two calculations). V_th is the voltage across the open terminals with the load removed — nothing is drawing current, so it is the circuit at rest. R_th is the resistance you would measure looking back into those same terminals if you reached inside and turned every independent source off: replace each voltage source with a wire (a short, since an ideal voltage source holds 0 V change) and each current source with a gap (an open, since an ideal current source allows 0 A). What is left is pure resistors, which you already know how to collapse.
- Disconnect the load and mark the two terminals A and B — these define what 'the outside' sees.
- Find V_th, the open-circuit voltage across A-B, using whatever tool fits: a voltage divider, KVL, or nodal analysis.
- Find R_th by turning off every independent source (voltage source to short, current source to open) and reducing the remaining resistor network seen from A-B.
- As a cross-check, instead short A-B, find the short-circuit current I_sc, and use R_th = V_th / I_sc — the two methods must agree.
- Redraw the whole box as V_th in series with R_th, reconnect the real load, and solve the now-trivial series circuit.
Walk one through with numbers. A 12 V source drives a 4 k resistor (call it R1), and from that node a 2 k resistor (R2) returns to ground; the load hangs off that middle node. With the load removed, no current flows out, so R1 and R2 form a plain voltage divider: V_th = 12 times 2 / (4 + 2) = 4 V. Now zero the source — replace the 12 V with a wire — and R1 and R2 are simply in parallel from the terminals: R_th = (4 times 2) / (4 + 2) = 8/6 ≈ 1.33 k. The whole left-hand circuit is now just '4 V behind 1.33 k', and any load you bolt on is a one-line divider calculation.
Norton: the same box, current-side out
There is a second, equally valid way to draw the very same box. Norton's theorem (Norton's theorem) says it also behaves exactly like a single current source I_n in parallel with one resistor R_th. Where Thevenin pushes with a voltage behind a series resistance, Norton injects a current that splits between its own parallel resistance and your load. The Norton current is simply the short-circuit current of the box: dead-short the terminals and I_n is whatever flows. The resistance R_th is identical to the Thevenin one — same box, same internal stiffness.
Converting between the two views is a one-liner, because both must produce the same open-circuit voltage and the same short-circuit current: V_th = I_n times R_th. In our example I_n = V_th / R_th = 4 V / 1.33 k = 3 mA, so the box is equally well described as '4 V behind 1.33 k' or '3 mA across 1.33 k'. Use whichever simplifies the work: voltage sources in series love the Thevenin form, while current sources and parallel branches collapse more neatly in the Norton form. This back-and-forth is called a source transformation, and it is a quiet workhorse for taming messy schematics.
THEVENIN form NORTON form
Rth A o---+-------+---o A
A o--/\/\--+ | |
| (In) Rth
(Vth) | |
| B o---+-------+---o B
B o--------+
Vth = In x Rth In = Vth / Rth Rth is the SAME
(open-circuit V) (short-circuit I) (sources turned off)Superposition: one source at a time
Finding V_th when a box has several sources is much easier with a divide-and-conquer idea called superposition. In any linear circuit, the response (a voltage or current somewhere) caused by many sources acting together equals the sum of the responses caused by each source acting alone. To use it, turn all sources off but one — voltage sources to shorts, current sources to opens, exactly as for R_th — solve that simple circuit, then repeat for each source and add the partial answers.
A quick taste: suppose at some output node a 6 V battery acting alone would set up 2 V, and a separate 3 V battery acting alone would add 1 V there. With both connected, superposition says the output is simply 2 + 1 = 3 V — no simultaneous equations needed. The honest fine print: superposition handles only the sources, never nonlinear quantities. You cannot superpose power, because power depends on voltage squared (or current squared), and squares do not add that way. Find the total voltage or current first, then compute power once at the end.
Loading, and the honest truth about maximum power
Now the payoff. Once a stage is reduced to V_th behind R_th, connecting a load R_L is just a voltage divider, and you can see the loading effect (loading effect) instantly. The load receives V_th times R_L / (R_th + R_L), so a load far larger than R_th barely sags the voltage, while a load comparable to R_th steals a big share. That R_th is exactly the output resistance of the stage, and R_L is the input resistance of the next stage. The whole craft of connecting blocks together is this conversation between a source resistance pushing and an input resistance pulling back.
This leads to a famous and often-misunderstood result, the maximum power transfer theorem (maximum power transfer). A load draws the most power from a Thevenin source when R_L exactly equals R_th — too small and the voltage collapses, too large and almost no current flows; the sweet spot is the match. In our box (V_th = 4 V, R_th = 1.33 k), a matched 1.33 k load sees 2 V across it and absorbs V^2/R_L = 2^2 / 1330 ≈ 3 mW, which equals the peak formula P_max = V_th^2 / (4 R_th) = 16 / (4 times 1330) ≈ 3 mW.
Step back and notice what you have gained across this whole rung. Ohm's law gave you one resistor; series and parallel let you fold many into one; dividers and Kirchhoff's laws let you solve any resistive web; and Thevenin and Norton let you hide an entire sub-circuit behind a single source and a single resistance. That last move is how engineers actually think when joining stages — each block is just a source resistance feeding the next block's input resistance, and the equivalent is the clean handle that makes the connection predictable.