Impedance matching — Conversion de puissance 2 : Transformateur

A voltage generator gives a sinusoidal voltage

𝑒

to the primary of a transformer through a resistor

𝑅_{1}

1/ At which condition on the transformation ratio

𝑚

is the power dissipated by the resistor

𝑅_{2}

maximal ?

Coup de pouce 1

Quelle est la tension aux bornes de

𝑅_{2}

(en fonction de

𝑅_{1}

𝑅_{2}

𝑚

𝑒

) ?

Coup de pouce 2

Ramener le primaire au secondaire puis effectuer un pont diviseur de tension.

Corrigé

The circuit is equivalent to the following one (primary side brought to the secondary side):

𝑣_{2} = 𝑚 𝑒 \frac{𝑅_{2}}{𝑚^{2} 𝑅_{1} + 𝑅_{2}}

𝑃 = \frac{𝑉_{2, eff}^{2}}{𝑅_{2}} = \frac{{(𝑚 𝑒 \frac{𝑅_{2}}{𝑚^{2} 𝑅_{1} + 𝑅_{2}})}^{2}}{𝑅_{2}} = \frac{𝑚^{2} 𝐸_{eff}^{2} 𝑅_{2}}{{(𝑚^{2} 𝑅_{1} + 𝑅_{2})}^{2}}

\frac{𝜕 𝑃}{𝜕 𝑚} = \frac{2 𝑚 𝐸_{eff}^{2} 𝑅_{2} {(𝑚^{2} 𝑅_{1} + 𝑅_{2})}^{2} - 𝑚^{2} 𝐸_{eff}^{2} 𝑅_{2} 4 𝑚 (𝑚^{2} 𝑅_{1} + 𝑅_{2})}{{(𝑚^{2} 𝑅_{1} + 𝑅_{2})}^{4}}

\frac{𝜕 𝑃}{𝜕 𝑚} = 0 \Rightarrow 2 𝑚 𝐸_{eff}^{2} 𝑅_{2} {(𝑚^{2} 𝑅_{1} + 𝑅_{2})}^{2} = 4 𝑚^{3} 𝐸_{eff}^{2} 𝑅_{2} 𝑅_{1} (𝑚^{2} 𝑅_{1} + 𝑅_{2})

(𝑚^{2} 𝑅_{1} + 𝑅_{2}) = 2 𝑚^{2} 𝑅_{1}

𝑅_{2} = 𝑚^{2} 𝑅_{1}

𝑚 = \sqrt{\frac{𝑅_{2}}{𝑅_{1}}}

Impedance matching ★★