Question about matching network

I am now working on circuit (a)





It seems that I have done some mistakes that I could not reach equation (2.117) with w=zero imaginary component condition

Any idea ?

When you defined Zin you wrote

R*(1-L*C*w)^2

instead of

R*(1-L*C*w^2)

Thanks albbg

For circuit (a), how can we infer that Re(Zin) = L/(C*R) is less than R ?

From [2.178] Re(Zin) = R/(1+Q^2)

1+Q^2 is alwas > 1 then

Re(Zin)= R/(1+Q^2) < R

The last equation [2.181], from which Re(Zin) = L/(C*R) is derived as an approximation if Q^2>>1 so in this case Re(Zin)= R/(1+Q^2) << R

I am trying to apply circuit (c) to **broken link removed** using the formula Re(Zin) = L1/(C1*RL) computed below with RL = 75 ohm and Re(Zin) = 150 ohm

However when I did the calculation, it does not fit with the values of L1=6.12 nH and C1=0.73 pF given, meaning the antenna impedance is not matched with transmitter side. Which part of calculation did I mess up ?

Your calculation doesn't seem to consider the complex source and load impedances.

@FvM

Please refer to the wxmaxima computation for the matching network circuitry

Where is zin in the network? Where is the complex load in your formula?

Thanks FvM for pointing out that the formula needs to be modified for the case of complex load.



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Do I just put R=a+%i*b which implies complex load into the Zin equation ?

OK, let's call the transmitter impedance ZT=a+jb where a=150 ohm and b=75 ohm
then let's call the antenna impedance ZA=m+jn where m=75 ohm and n=15 ohm

Furthermore we know that Xc=1/(wC) and XL=wL.

Let's calculate the impedance seen by the antenna. The transmitter has a Thevenin generator in series with the ZT. Short this generator.
We have ZT in parallel with the matching capacitor, all this in series with the inductor, then the impedance seen by the antenna will be:

Zx=(a+jb)*jXc/[a+j(b+Xc)] + jXL

It's quite easy to find the real and inmaginary part of Zx that is:

real(Zx)=a*Xc^2/[a^2+j(b+Xc)^2]
imag(Zx)=[a^2*Xc+b*Xc*(b+Xc)]/[a^2+j(b+Xc)^2]

we can notice that the real part doesn't depend from XL, but only from Xc. So we can solve with respect to Xc:

real(Zx)=m

I obtained Xc=-108.8 ohm, that is C=0.73 pF @ 2GHz

now we have to solve

imag(Zx)=n

after substituting in Zx the value of Xc previously calculated I've found XL=106.9 ohm, that is L=8.5 nH @ 2GHz

This value is different from that calculated in the solution you gave. But if n=-15 ohm, instead of +15 ohm I've found XL=76.9 ohm, that is L=6.12 nH @ 2GHz