Maximum power transfer

YOU WILL BE ABLE TO

Derive the condition for maximum power delivery to a load.
Compute the maximum available power from a Thevenin source.
State the efficiency cost of matched operation.
Know when to match and when deliberately not to.

A source with internal resistance cannot deliver unlimited power. Given a Thevenin source V_Th, R_Th driving an adjustable load R_L, what load extracts the most power? Too small and the load voltage collapses; too large and the current dries up. Somewhere between is a maximum — and calculus (this is where the Calc I prerequisite pays off directly) puts it exactly at the matched condition.

The worked example: a 9 V, 3 kΩ Thevenin source drives a load R_L. Which R_L captures the most power?

p_L = \frac{V_{Th}^2 \, R_L}{(R_{Th} + R_L)^2}

load power as a function of R_L

Setting dp_L/dR_L = 0 and solving yields a single stationary point, and checking either side confirms it is the maximum: the load should equal the source's own resistance. Substituting back gives the maximum available power.

R_L = R_{Th} \quad \Rightarrow \quad p_{max} = \frac{V_{Th}^2}{4R_{Th}}

maximum power transfer

WORKED EXAMPLE

Matched load, start to finish

The figure's source: V_Th = 9 V, R_Th = 3 kΩ (Lesson 2's network, one more time).
Matched load: R_L = 3 kΩ.
p_max = 9²/(4 × 3 kΩ) = 81/12 = 6.75 mW.
Sanity checks either side: R_L = 1 kΩ gives 81 × 1/16 ≈ 5.06 mW; R_L = 9 kΩ gives 81 × 9/144 ≈ 5.06 mW. Both less, symmetrically. ✓

R_L = 3 kΩ, p_max = 6.75 mW

WORKED EXAMPLE

The theorem's fine print

The theorem answers: "R_L varies, source fixed — choose R_L." It does NOT say a designer should ever want a large R_Th.
If instead the LOAD is fixed and you may choose the source resistance, the answer is always: make R_Th as small as possible — p_L = V_Th²R_L/(R_Th + R_L)² only grows as R_Th shrinks.
Misreading this is a classic exam trap: "matched is best" is only true from the load's side of the fence.

Match the load to the source; never the source to the load.

PRACTICE

A source has V_Th = 12 V, R_Th = 6 kΩ. What load extracts maximum power, and how much?

In practice problem 1 at the matched point, what is the efficiency, and where does the rest of the power go?

A fixed 1 kΩ load is driven by a source whose R_Th you get to choose. Should you pick R_Th = 1 kΩ "for maximum power transfer"? What's the right choice?

An antenna interface presents V_Th = 4 V, R_Th = 50 Ω. Find the power delivered to a matched 50 Ω load.