Hi friends,
In the case of antenna matching, we are looking for a maximum of power transfer between the feeding port and the radiation resistance of the antenna.
At the frequency at which Zin is purely real (i.e. Im(Zin)=0) you can get exact matching at that port (eventually with an impedance transformer). If there are no losses, all the power is radiated.
Reactive components (feeders, stubs, etc) shift the resonant frequencies or can produce multiple resonances.
For d(Re(Zin))/df =0, I don’t agree that this is a necessary nor sufficient condition of resonance, and let me give an example.
One could think at an analog condition for the admittance Yin, say that it should be d(Re(Yin))/df=0 at resonance. (After all, the two dual models are equally valid). But the two conditions are very different: If one is satisfied at a given frequency, the other is not necessarily satisfied. What is guaranteed is that at a frequency at which Zin or Yin is purely real, the other is real as well.
Imagine a series RLC circuit. At any frequency Re(Zin)=R. At the resonant frequency Im(Zin)=0 and Re(Yin) has a maximum.
[Now the dual:
Imagine a parallel RLC circuit. At any frequency Re(Yin)=G (=1/R). At the resonant frequency Im(Yin)=0 and Re(Zin) has a maximum.]
Now consider the series circuit at the end of a lambda/8 line with Zo=R: there is a maximum of Re(Zin) at a frequency above resonance [and the maximum of Re(Yin) is at a frequency below resonance]. Nevertheless, at the resonance frequency the value of Rin is unchanged [the same happens for Gin], and the reactive part of Zin [and of Yin] is zero. There is mismatch at the frequency at which the maximum of Rin is located [the same applies to Gin].
The same happens with other line lengths, but the fact that at the resonance frequency Zin and Yin are purely real doesn’t change for any line length.
This example shows that d(Re(Zin))/df [and d(Re(Yin))/df] change although the resonant frequency (at which the circuit is matched) does not. The derivative can be different from zero at resonance, and it can be null far from resonance.
I hope this is clear. :?
Regards
Z