AUCKLAND · FACULTY OF ELECTRICAL ENGINEERING

ELECTENG291 · Fundamentals of Electrical Engineering

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Chapter 5 of 10 · ELECTENG 291

Thévenin & Norton Equivalents

This Module 1 chapter of University of Auckland ELECTENG 291 shows how any linear two-terminal network collapses to a simple equivalent: the Thévenin form (an open-circuit voltage V_TH in series with R_TH) or the Norton form (a short-circuit current I_N in parallel with R_N), linked by R_TH = R_N = V_TH / I_N. It covers source transformation and the maximum-power-transfer idea. Thévenin/Norton reductions are a recurring, high-value test and exam technique, and the course deliberately includes cases with dependent sources where R_TH can even be negative.

In this chapter

What this chapter covers

01Thévenin equivalent: open-circuit voltage V_TH in series with R_TH
02Norton equivalent: short-circuit current I_N in parallel with R_N
03The link R_TH = R_N = V_TH / I_N, and source transformation (V-source + series R ⇄ I-source + parallel R)
04Finding R_TH with only independent sources: zero them and find the equivalent resistance
05Finding R_TH with a dependent source present: apply a notional (test) source and take R_TH = V_test / I_test
06R_TH can be zero or negative with dependent sources — mathematically correct even when physically odd
07Maximum power transfer: a load draws maximum power when R_L = R_TH (confirm emphasis against current teaching)

Worked example · free

Reduce a network to its Thévenin equivalent, then load it

Q [5 marks]. For a linear two-terminal network, measurements give an open-circuit terminal voltage V_TH = 45 V and a short-circuit terminal current I_N = 22.5 mA. A load made of a 4 kΩ resistor is then connected across the terminals. Find R_TH and the load voltage V_o.

+1Thévenin resistance from the open-circuit voltage and short-circuit current: R_TH = V_TH / I_N = 45 V / 22.5 mA = 2 kΩ.
+1Replace the network by its Thévenin equivalent: a 45 V source in series with 2 kΩ, driving the 4 kΩ load. This is now a simple series loop.
+1The load and R_TH are in series across V_TH, so a voltage divider applies: V_o = [R_L/(R_TH + R_L)]·V_TH.
+1Substitute: V_o = [4 kΩ/(2 kΩ + 4 kΩ)]·45 = (4/6)·45 = (2/3)·45 = 30 V.
+1Sanity check via current: loop current i = V_TH/(R_TH + R_L) = 45/6 kΩ = 7.5 mA, so V_o = i·R_L = 7.5 mA × 4 kΩ = 30 V ✓.

R_TH = 2 kΩ and V_o = 30 V. Once the network is a 45 V source in series with 2 kΩ, the 4 kΩ load simply divides the Thévenin voltage: (2/3)·45 = 30 V.

Sia tip — V_TH is the open-circuit voltage, I_N the short-circuit current, and their ratio is always R_TH — one clean way to get the resistance without zeroing sources. When a dependent source is present you can't just zero sources; apply a notional test source and take R_TH = V_test/I_test, and don't be alarmed if R_TH comes out zero or negative — that is a real, if unphysical, result. Ask Sia to reduce any two-terminal network to its equivalent.

Glossary

Key terms

Thévenin equivalent: The reduction of any linear two-terminal network to a single voltage source V_TH (the open-circuit terminal voltage) in series with a resistance R_TH. It reproduces the network's behaviour for any external load.
Norton equivalent: The dual reduction: a single current source I_N (the short-circuit terminal current) in parallel with R_N. It is related to the Thévenin form by I_N = V_TH/R_TH and R_N = R_TH.
Thévenin resistance (R_TH): The equivalent resistance seen from the terminals, equal to V_TH/I_N. With only independent sources you find it by zeroing them; with a dependent source you apply a notional test source and take R_TH = V_test/I_test.
Source transformation: The interchange of a voltage source with a series resistance and a current source with a parallel resistance (same R). It lets you simplify a network by converting sources into whichever form combines more easily.
Negative Thévenin resistance: A legitimate outcome when dependent sources are present: R_TH can be zero or negative, meaning increasing terminal current increases terminal voltage. It is mathematically correct even though a passive resistor could never behave that way — a deliberate teaching point of the course.
Maximum power transfer: The result that a resistive load draws the most power from a linear source network when the load resistance matches the Thévenin resistance, R_L = R_TH. Confirm how far this is emphasised in the current teaching on Canvas.

FAQ

Thévenin & Norton Equivalents FAQ

How do I find the Thévenin voltage and resistance?

V_TH is the voltage across the open-circuited terminals. R_TH is the equivalent resistance seen from those terminals: with only independent sources, zero them (voltage sources → short, current sources → open) and reduce the resistor network; if there is a dependent source, instead apply a notional test source at the terminals and compute R_TH = V_test/I_test. A quick alternative is R_TH = V_TH/I_N using the short-circuit current.

What is the difference between Thévenin and Norton equivalents?

They are two views of the same two-terminal network. Thévenin uses a voltage source V_TH in series with R_TH; Norton uses a current source I_N in parallel with R_N. They convert into each other by V_TH = I_N·R_TH and R_N = R_TH, and a source transformation moves you between them whenever it makes the algebra easier.

Can the Thévenin resistance really be negative?

Yes — when the network contains dependent sources. A negative R_TH means increasing the terminal current increases the terminal voltage, which no passive resistor could do, but it is a mathematically valid equivalent. The course includes this deliberately to show that an equivalent model can be physically odd yet correct; find R_TH with a test source in these cases.

Can Sia help me with Thévenin and Norton problems in ELECTENG 291?

Yes, as a study aid. Sia can compute V_TH, I_N and R_TH (test-source method included), convert between Thévenin and Norton, and check your load calculation. It explains and drills you on fresh numbers; it does not do graded assessment for you, and University of Auckland academic-integrity rules apply — confirm what is permitted on Canvas.

Study strategy

Exam move

Make Thévenin reduction a reflex for any 'find the load voltage/current' question: get V_TH (open-circuit voltage), get R_TH (zero independent sources, or use a test source when a dependent source is present, or take V_TH/I_N), then attach the load and finish with a simple divider or loop. Practise the dependent-source case hard — the notional-test-source method and the possibility of a zero or negative R_TH are exactly where the course sets traps. Keep source transformation in your kit for collapsing mixed networks quickly. Always cross-check the final load result two ways (divider vs loop current), and remember that Thévenin, Norton, nodal, mesh and superposition must all agree. Show the method — the marks reward a clear V_TH/R_TH derivation. Confirm assessment details on Canvas.

Working through Thévenin & Norton Equivalents in ELECTENG 291? Sia is AskSia’s AI Electrical Engineering tutor — ask any ELECTENG 291 Thévenin & Norton Equivalents question and get a clear, step-by-step explanation grounded in how ELECTENG 291 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.

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