Exercise 11.1 Transmit power

The figure below shows a (quite generic) setup of an RF transmitter, consisting of a power amplifier (PA) that is driven by a modulated oscillator and that drives an antenna. Between the PA and the antenna there may be an extra matching network. The complex antenna impedance is $Z_{\mathit{antenna}}=R_a+j{X}_a$. The maximum output current of the PA is $I_{\mathit{max}}$ while the maximum voltage swing is $V_{\mathit{max}}$.

a)

Explain (briefly & clear) why an antenna that actually transmits has a finite non-zero real part $R_a$ in the antenna impedance $Z_{\mathit{antenna}}=R_a+j{X}_a$.

b)

The impedance of a(n about) quarter-wave monopole antenna is purely real. Connecting this antenna directly to the output node of a PA – having both the intended RF signal and a non-zero DC-(bias) voltage – without coupling capacitor does not result in any DC power into the antenna. Explain this.

c)

Derive an expression for the maximum transmit power of the system described above the figure.

d)

Derive 2 expressions for the ratio between the transmit power (at the same PA output voltage swing) when on one hand a short antenna with $Z_{\mathit{antenna}}=10\mathrm{\Omega }-j1000\mathrm{\Omega }$ and compared to using:
1) a short antenna with “match” for which $Z_{\mathit{antenna},\mathit{matched}}=10\mathrm{\Omega }$ (1st expression)
2) a $\lambda ∕4$ antenna with $Z_{\mathit{antenna},\lambda ∕4}\cong 50\mathrm{\Omega }$ (2nd expression)