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ID: past-exam-of-the-mphil-in-micro-and-nanotechnology-enterprise-course-of-the-university-of-cambridge/2022/paper-1/section-b/12/ii/solution

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Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2022 / Paper 1 / Section B / 12 / ii / Solution by

Codex  0 2026-10-20

A generator couple has a p-type and an n-type thermoelement joined electrically at the hot face, with their other ends connected through a load at the cold face. The legs are electrically in series and thermally in parallel. Electrically insulating, thermally conducting plates couple them to the hot source and cold sink; metal interconnects carry current. The Seebeck effect provides an open-circuit electromotive force

E = α_{n p} Δ T

, where the magnitude of

α_{n p}

is the difference between the two leg Seebeck coefficients.

For fixed junction temperatures, constant properties and internal resistance

R

P_{L} = I^{2} R_{L} = \frac{( α _{n p} Δ T ) ^{2} R _{L}}{( R + R _{L} ) ^{2}} .

(1)

Differentiate with respect to load resistance:

\frac{d P _{L}}{d R _{L}} = (α_{n p} Δ T)^{2} \frac{R - R _{L}}{( R + R _{L} ) ^{3}} .

(2)

The derivative changes from positive to negative at

R_{L} = R

, so

R_{L} = R, I_{P_{m a x}} = \frac{α _{n p} Δ T}{2 R}, P_{m a x} = \frac{( α _{n p} Δ T ) ^{2}}{4 R} .

(3)

The load voltage is then half the open-circuit voltage. The heat source and sink must sustain the temperature difference; an isolated hot object cools as energy is extracted. Maximum power and maximum conversion efficiency are different operating points: the efficiency-optimal load is generally larger than

R

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