EC-1.2 Network Theorems

What this is and why it exists

These theorems all say one useful thing in different words. A linear network is far simpler seen from outside than it is inside.

Reduce everything except the part you care about to one source and one impedance, and a page of algebra becomes a line of it. That is the move a working engineer makes without thinking about it.

The vocabulary

• Linear network — one whose response scales with its sources and adds when sources are combined.
• Superposition — finding the response to several sources by adding the responses to each alone.
• Suppressing a source — replacing a voltage source with a wire and a current source with a gap.
• Thevenin equivalent — one voltage source in series with one impedance.
• Norton equivalent — one current source with one impedance beside it.
• Dependent source — a source whose value is controlled by a voltage or current elsewhere in the circuit.
• Maximum power transfer — the load condition that draws the most power from a source.
• Reciprocity — swapping the excitation and the measurement gives the same reading.

The mental model

Superposition is the foundation the rest stands on. In a linear network, the response to several sources is the sum of the responses to each one acting alone. While one source acts, the others are suppressed. Suppressing a voltage source means making it zero volts, which is a wire. Suppressing a current source means making it zero amps, which is a gap.

That word linear is not decoration. Superposition is the first theorem to fail the moment any element is nonlinear, and every theorem below inherits that condition. Remembering the condition matters more than remembering the statement.

Thevenin's theorem is the workhorse. Look into any two terminals of a linear network and it behaves exactly like one voltage source in series with one impedance. The source value is the voltage across those terminals with nothing connected. The impedance is what you see looking back into the terminals with the independent sources suppressed. Those are two different calculations, and treating them as one is a common error.

Norton's theorem is the same fact stated the other way: one current source with one impedance beside it. The two forms carry the same information and convert by a single division. Choosing the form that matches what the rest of the circuit needs saves most of the work.

Dependent sources change the procedure. A dependent source cannot be suppressed, because its value is controlled by something still happening in the circuit. So the impedance cannot be found by inspection. Instead, apply a test source at the terminals and take the ratio of voltage to current. That is the version examinations reach for, and it always works.

Maximum power transfer says the load draws the most power when it matches the source impedance. It is true, and it is a poor design rule for a power supply. At the matched point exactly half the power is burnt inside the source itself, so the efficiency is fifty per cent. That is the right answer in a signal chain, where the signal is scarce and the power is tiny. It is the wrong answer wherever efficiency is what matters.

Reciprocity is a checking tool. In a linear network with no dependent sources, excite one pair of terminals and measure at another. Swap the two and you read the same number. When an answer feels wrong, this is a strong and quick test of it.

What you should now be able to explain or do

State the condition every theorem here depends on, and say which element breaks it. Find a Thevenin equivalent, keeping the voltage and the impedance as two separate calculations. Convert between Thevenin and Norton forms and choose the more convenient one. Find the equivalent impedance of a network containing a dependent source, using a test source. Say why maximum power transfer is the wrong rule for a power supply.

Check yourself