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Apply conjugate matching to derive the expressions for power supplied and captured considering antenna impedance.

Question

Apply conjugate matching to derive the expressions for power supplied and captured considering antenna impedance.

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Solution

Conjugate matching is a technique used in antenna theory to maximize power transfer between the source and the load. The idea is to make the load impedance equal to the complex conjugate of the source impedance.

Here's a step-by-step guide on how to derive the expressions for power supplied and captured considering antenna impedance:

  1. Start with the basic formula for power, P = IV, where I is the current and V is the voltage.

  2. Express the current and voltage in terms of the source and load impedances (Zs and Zl respectively) and the source voltage Vs. The voltage across the load is V = Vs * Zl / (Zs + Zl) and the current through the load is I = Vs / (Zs + Zl).

  3. Substitute these expressions into the power formula to get P = Vs^2 * Zl / (Zs + Zl)^2.

  4. To maximize power, take the derivative of this expression with respect to Zl and set it equal to zero. This gives the condition for maximum power transfer: Zl = Zs*.

  5. Substitute this condition back into the power formula to get the expressions for maximum power supplied and captured: Psupplied = Vs^2 / (4Re{Zs}) and Pcaptured = Vs^2 / (4Re{Zl}), where Re{Z} denotes the real part of the impedance Z.

Remember, these formulas assume that the source and load impedances are complex numbers, and that the source voltage is a constant. If these conditions are not met, the formulas may not give accurate results.

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