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I would like to know if there is a way you can find the impedance that has to be set between the load \$Z_L\$ when it is complex and the characteristic impedance \$Z_0\$ to have impedance matching without using the Smith Chart, since we are not allow to use it and all the methods I have search use it. It has to do something with the formula:$$Z_{in}=Z_1\frac{Z_L+jZ_1\tan\beta l}{Z_1+jZ_L\tan\beta l}=Z_0$$ and \$\beta l=(2\pi/\lambda)(\lambda/4)=\pi/2\$. The goal is to compute \$Z_1\$ and it seems always to be real. The formula can be found in Microwave Engineering, Pozar, page 79, and with a scheme like this one:

enter image description here

also found in page 92, but the book specifies that \$R_L\$ is real. The closest thing that I have found to my problem is this one: enter image description here but it uses the Smith Charts as it says so. In my problem \$Z_L=25-j175\$ and \$Z_0=36.11\$, giving a result of \$Z_1=\sqrt{75\cdot36.11}\$ but I don't know how that result has been achieved. Could it be related to a multiple media problem?

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  • \$\begingroup\$ If you express the load in polar coordinates vs cartesian then find the conjugate length to match Zo then it is like using a Smith Chart \$\endgroup\$ – Tony Stewart Sunnyskyguy EE75 Apr 13 '18 at 16:25
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When I remember right, Pozars formula should also be valid when ZL is complex. Just replace ZL real with ZL complex.

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Your question is a little unclear but I'll try to help

Find impedance matching in a quarter wave transformer

For a quarter wave transformer \$Z_{IN} = \dfrac{Z_0^2}{Z_L}\$

I would like to know if there is a way you can find the impedance that has to be set between the load ZL when it is complex and the characteristic impedance Z0 to have impedance matching

I interpret this as asking what impedance is placed between the end of the t-line and ZL to correctly terminate it? If so then the impedance has to be such that it cancels reactances i.e. 100 ohm inductive is cancelled by 100 ohm capacitive.

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