The reduce-and-expand method of the last lesson fails on circuits that have no clean series or parallel groups to collapse, the bridge network being the standard example, where every resistor is entangled with every other. Such circuits are not harder in principle; they only need a method that does not rely on spotting simple groupings. The method that always works is to make the unknowns the voltages at the circuit's nodes and to enforce Kirchhoff's current law at each one, turning the circuit into a small set of equations. Alongside it sits a second idea of great practical value: that any two-terminal box of sources and resistors, however complex inside, behaves from outside exactly like one source and one resistor.
Node analysis
A node is a point of the circuit at a single voltage, all the junctions connected by plain wire counting as one node. Pick one node as the reference, ground, and call it zero volts, and let the voltage at every other node be an unknown. At each unknown node, Kirchhoff's current law says the currents leaving through the connected resistors sum to zero, and each such current is written by Ohm's law as the voltage difference across the resistor divided by its resistance. Writing that balance at every node gives one equation per unknown node, and solving the set gives every node voltage at once. From the node voltages, every branch current and component voltage follows immediately. The virtue of the method is that it never needs to recognise a series or parallel group; it grinds any circuit, tangled or not, into linear equations that always have a solution.
Example. A node sits at unknown voltage . It connects through 1000 ohms to a 10 V supply, through 2000 ohms to ground, and through 2000 ohms to ground again. Find by balancing the currents.
The current law says the current in from the supply equals the current out to ground: . So , giving V. One equation, one unknown, and the node voltage drops out; every branch current then follows by Ohm's law.
Now you. A node at voltage connects through 1000 ohms to a 12 V supply and through 1000 ohms to ground, with nothing else attached. What is ?
Answer
, so and V. This is the voltage divider again, recovered from the node balance.
Thevenin's equivalent, a box seen from two terminals
Often only two terminals of a complicated circuit matter, the two where something will be connected, and everything behind them is detail. Thevenin's theorem makes that detail collapse: any network of sources and resistors, viewed from two terminals, behaves exactly like a single voltage source in series with a single resistance. The source, the Thevenin voltage, is the voltage that appears across the two terminals with nothing connected, the open-circuit voltage. The resistance, the Thevenin resistance, is the resistance seen looking back into the terminals with the sources turned off, voltage sources replaced by wires. Whatever is then attached to the terminals cannot tell the real network from this simple pair, because they deliver the same voltage and current for every possible load. The theorem is what lets an engineer treat a whole power supply, or a whole sensor circuit, as one source and one resistance when designing what plugs into it.
Example. A 12 V supply feeds two equal 1000 ohm resistors in series, and the two terminals of interest are across the second resistor. What is the Thevenin equivalent seen there?
With nothing connected, the divider puts half the supply across the second resistor, so the Thevenin voltage is 6 V. Turning off the source, the two 1000 ohm resistors appear in parallel across the terminals, giving a Thevenin resistance of 500 ohms. So the whole thing looks, from those terminals, like a 6 V source behind 500 ohms.
Now you. If instead both resistors were 2000 ohms, what would the Thevenin voltage and resistance across the second one be?
Answer
The divider still gives half, so the Thevenin voltage is 6 V, and the two 2000 ohm resistors in parallel give a Thevenin resistance of 1000 ohms.
Why the equivalent resistance matters
The Thevenin resistance is not a bookkeeping curiosity; it decides how a source behaves under load. A source with a low equivalent resistance holds its voltage nearly steady as more current is drawn, which is what a good power supply does. A source with a high equivalent resistance sags badly when loaded, which is the internal-resistance sag of a tired battery met in the voltage lesson, now named precisely: the battery's Thevenin resistance is its internal resistance, and the terminal voltage falls by the load current times that resistance. It also sets how much power a source can deliver to a load, which is greatest when the load resistance matches the source's equivalent resistance, a result called maximum power transfer that governs everything from loudspeakers to radio antennas. These methods, node voltages for solving and Thevenin equivalents for simplifying, complete the toolkit for steady direct-current circuits. What they cannot yet handle is a circuit whose currents and voltages change with time, and the next lessons add the two components that make that happen: the capacitor, which stores charge, and the inductor, which stores current's magnetism.