Thevenin's and Norton's Theorems

Learning Objectives

• State Thevenin's and Norton's theorems and what "equivalent circuit" means.
• Calculate V_th and R_th for a given two-terminal resistive network.
• Calculate I_N and R_N for the same network and verify the Thevenin-Norton relationship.
• Convert between Thevenin and Norton equivalents using source transformation.
• Apply the Maximum Power Transfer Theorem to find the load resistance that delivers maximum power.
• Explain why deactivating sources (not just removing the load) is essential for finding R_th.

Quick Answer

Thevenin's and Norton's theorems let you replace any complicated linear circuit, viewed from two output terminals, with one simple equivalent: a single voltage source in series with a resistor (Thevenin) or a single current source in parallel with a resistor (Norton). This matters because engineers rarely care about every detail of a power supply or amplifier — they care how it behaves when a specific load is attached. Instead of re-solving the whole circuit for every possible load, you find the equivalent once and then just apply Ohm's Law to any load you connect. Both theorems are core tools for load analysis, impedance matching, and maximum power transfer problems.

Thevenin's Theorem

Statement: Any linear two-terminal network can be replaced by an equivalent voltage source V_th in series with a resistance R_th.

• V_th = the open-circuit voltage measured across the terminals (with the load removed).
• R_th = the resistance seen looking back into the terminals with all independent sources deactivated (voltage sources replaced by short circuits, current sources replaced by open circuits).

Steps:

1. Identify the two terminals of interest.
2. Remove the load; calculate the open-circuit voltage V_oc across the terminals. V_th = V_oc.
3. Deactivate all independent sources; calculate the resistance looking into the terminals. That's R_th.
4. Reconnect the load to the simple V_th–R_th equivalent and use Ohm's Law.

Worked Example 1: A 30V battery is in series with a 4Ω resistor, then a 6Ω resistor, and a load R_L is connected across the 6Ω resistor. Find the Thevenin equivalent.

Step 1 — Find V_th (remove R_L, use the voltage divider across the 6Ω resistor):
V_th = 30 × 6 / (4 + 6) = 30 × 0.6 = 18 V

Step 2 — Find R_th (short the 30V source; 4Ω and 6Ω are now in parallel):
R_th = (4 × 6) / (4 + 6) = 24 / 10 = 2.4 Ω

Thevenin equivalent: V_th = 18V, R_th = 2.4Ω. For any load R_L, the current through it is simply:

I_L = V_th / (R_th + R_L) = 18 / (2.4 + R_L)

Norton's Theorem

Statement: Any linear two-terminal network can also be replaced by an equivalent current source I_N in parallel with a resistance R_N.

• I_N = the short-circuit current through the terminals (short them together and measure the current).
• R_N = same calculation as R_th — deactivate sources, find resistance from the terminals. R_N = R_th always.

Worked Example 2: Using the same circuit as Example 1, find the Norton equivalent.

Step 1 — Find I_N (short across the 6Ω resistor; all 30V drops across the 4Ω):
I_N = 30 / 4 = 7.5 A

Step 2 — R_N = R_th = 2.4 Ω (same as before)

Verification: V_th should equal I_N × R_N: 7.5 × 2.4 = 18V, which matches the Thevenin voltage found earlier — a useful cross-check that your calculations are consistent.

Thevenin ↔ Norton Conversion

Since both equivalents describe the same physical circuit, converting between them is direct:

Thevenin → Norton	Norton → Thevenin
I_N = V_th / R_th	V_th = I_N × R_N
R_N = R_th	R_th = R_N

You never need to re-analyze the original circuit — just convert.

Maximum Power Transfer Theorem

A direct application of Thevenin's theorem: a load draws the maximum possible power from a source when the load resistance equals the source's Thevenin resistance.

Condition: R_L = R_th

Maximum power: P_max = V_th² / (4 × R_th)

Worked Example 3: Using V_th = 18V and R_th = 2.4Ω from Example 1, find the load resistance for maximum power transfer and the value of that maximum power.

R_L (for max power) = R_th = 2.4 Ω
P_max = V_th² / (4 × R_th) = 18² / (4 × 2.4) = 324 / 9.6 = 33.75 W

This principle underlies speaker impedance matching in audio amplifiers, antenna matching in RF systems, and load matching in power-supply design — in each case, engineers choose or design the load to match the source's internal resistance for maximum power delivery (accepting that this is a 50% efficiency point, since half the power is dissipated inside the source itself).

Dependent Sources: A Special Case

When a circuit contains dependent (controlled) sources, finding R_th requires care:

• V_th is still found the same way (open-circuit voltage).
• R_th cannot be found simply by deactivating sources, because a dependent source's value depends on some other circuit variable — zeroing independent sources may leave the dependent source's controlling variable undefined or zero, distorting the result.
• Test source method: deactivate all independent sources; apply an external test voltage V_test (or current I_test) at the terminals; solve for the resulting current (or voltage); R_th = V_test / I_test.

Key Terms

Term	Definition
Thevenin voltage (V_th)	The open-circuit voltage at the terminals of interest
Thevenin resistance (R_th)	The resistance seen from the terminals with all independent sources deactivated
Norton current (I_N)	The short-circuit current through the terminals
Norton resistance (R_N)	Equal to R_th; the parallel resistance in the Norton equivalent
Source deactivation	Replacing voltage sources with short circuits and current sources with open circuits
Dependent (controlled) source	A source whose value depends on another voltage or current elsewhere in the circuit
Maximum power transfer	The condition R_L = R_th at which a load receives the greatest possible power from a source

Common Mistakes

1. Misconception: "To find R_th, just remove the load and measure resistance — sources stay as they are." Why it's wrong: Active sources still contribute current/voltage while measuring, which corrupts the resistance calculation. R_th specifically requires the circuit's own sources to be deactivated first. Correct: Deactivate independent sources (short voltage sources, open current sources) before calculating the resistance looking into the terminals.

2. Misconception: "Maximum power transfer means maximum efficiency." Why it's wrong: At R_L = R_th, exactly half the total power is dissipated inside the source's own resistance — efficiency at that point is only 50%. Correct: Maximum power transfer maximizes the power delivered to the load, not overall efficiency; for high efficiency you actually want R_L >> R_th (which delivers less absolute power but wastes less inside the source).

3. Misconception: "R_th and R_N are found differently, so you calculate them separately every time." Why it's wrong: Both are defined as the resistance seen from the terminals with sources deactivated, so they're identical by definition (R_N = R_th). Correct: Calculate the deactivated-source resistance once; it serves as both R_th and R_N.

Comparison and Connections

Property	Thevenin equivalent	Norton equivalent
Source type	Voltage source V_th	Current source I_N
Resistance placement	In series	In parallel
Found from	Open-circuit voltage	Short-circuit current
Relationship	R_th = R_N	V_th = I_N × R_N
Best suited for	Series-dominated (voltage-divider) loads	Parallel-dominated (current-divider) loads

Practice Questions

Recall 1: Define V_th and R_th in your own words. Answer guidance: V_th is the open-circuit voltage across the terminals; R_th is the resistance seen from the terminals with all independent sources deactivated.

Recall 2: What is the condition for maximum power transfer to a load? Answer guidance: R_L = R_th (load resistance equals the source's Thevenin resistance).

Understanding 1: Why must independent sources be deactivated (not just removed) when finding R_th, and why does a voltage source become a short while a current source becomes an open? Answer guidance: Deactivating means setting the source's own contribution to zero without breaking the circuit topology. An ideal voltage source held at 0V behaves as a short circuit; an ideal current source held at 0A behaves as an open circuit (no path for current).

Understanding 2: Explain why R_N always equals R_th for the same circuit. Answer guidance: Both are defined identically — the resistance seen from the two terminals when all independent sources inside the circuit are deactivated — so they must be numerically the same value by definition.

Application 1: A circuit has V_th = 12V and R_th = 3Ω. Find the current delivered to a 9Ω load. Answer guidance: I_L = V_th/(R_th+R_L) = 12/(3+9) = 1A.

Application 2: A Norton equivalent has I_N = 4A and R_N = 5Ω. Find the equivalent Thevenin voltage and the current through a 5Ω load connected to the Norton source. Answer guidance: V_th = I_N × R_N = 4×5 = 20V. Load current: since load equals R_N, current splits evenly between R_N and load, so I_load = I_N/2 = 2A (or via Thevenin: I=20/(5+5)=2A).

Analysis 1: For the circuit in Worked Example 1 (V_th=18V, R_th=2.4Ω), calculate the power delivered to loads of 1Ω, 2.4Ω, and 6Ω. Which load delivers the most power, and does this match the Maximum Power Transfer Theorem? Answer guidance: P(1Ω)=I²R where I=18/3.4=5.29A, P=28.0W; P(2.4Ω): I=18/4.8=3.75A, P=33.75W; P(6Ω): I=18/8.4=2.14A, P=27.5W. The 2.4Ω load (equal to R_th) delivers the most power, confirming the theorem.

Analysis 2: A student claims that since R_N = R_th, Thevenin and Norton equivalents are "the same thing" and choosing between them doesn't matter. Evaluate this claim. Answer guidance: Partially true — they're mathematically equivalent and always convertible, but practically the choice matters: Thevenin is more convenient for series/voltage-divider loads, Norton for parallel/current-divider loads or when the source circuit already contains current sources, since it avoids extra source-transformation steps.

FAQ

Q1: Do Thevenin and Norton theorems work for AC circuits? Yes — you use the same procedure but with complex impedances (Z) instead of pure resistances, and phasor voltages/currents instead of real numbers.

Q2: What happens if R_th = 0? Then the source behaves as an ideal voltage source with no internal resistance — all the source voltage appears directly across any load, and Norton's I_N becomes undefined (infinite), since R_N = R_th = 0 would need I_N × 0 = V_th, which only makes sense in the limit.

Q3: Can I use Thevenin's theorem on a circuit with only resistors and no sources? Then V_th = 0 (there's no source to create an open-circuit voltage), and the equivalent is just R_th — a pure resistance, no source needed.

Q4: Why is R_L = R_th for maximum power transfer, rather than R_L being very small or very large? The power delivered, P = I²R_L = [V_th/(R_th+R_L)]² × R_L, is a function that rises and then falls as R_L increases from 0 to infinity; calculus (or symmetry of the expression) shows this function peaks exactly at R_L = R_th.

Q5: How do dependent sources change the process? V_th is found the same way, but R_th requires the "test source" method — apply a test voltage or current at the terminals (with independent sources deactivated but dependent sources active) and compute R_th = V_test/I_test, since dependent sources can't simply be zeroed out.

Quick Revision

• Thevenin equivalent: V_th (open-circuit voltage) in series with R_th (deactivated-source resistance).
• Norton equivalent: I_N (short-circuit current) in parallel with R_N.
• R_N = R_th always, for the same circuit.
• Conversion: V_th = I_N × R_N, and I_N = V_th / R_th.
• Deactivating sources: voltage sources → short circuit; current sources → open circuit.
• Maximum power transfer: R_L = R_th gives P_max = V_th²/(4R_th).
• At maximum power transfer, efficiency is only 50%, not maximum.
• Dependent sources need the test-source method to find R_th (can't just zero them).
• Thevenin suits series/voltage-divider loads; Norton suits parallel/current-divider loads.
• Both theorems apply to AC circuits using complex impedance instead of resistance.

Related Topics

Prerequisites: Ohm's Law, Kirchhoff's Laws, Series and Parallel Circuits.

Related Topics: Network Theorems, Maximum Power Transfer, AC and DC Circuits.

Next Topics: Network Theorems, AC and DC Circuits.