# How to relate resistance/reactance to parallel/series equivalent circuits?

Problem 7A of The Electronics of Radio asks the reader to find the relationships between resistance and reactance values in any arbitrary series/parallel circuit and their counterparts in a parallel/series equivalent circuit with the same impedance; i.e., given a parallel circuit with resistance Rp and reactance Xp, what is the relationship between those values and the the resistance Rs and reactance Xs of a series circuit with the same impedance, and vice-versa.

As a starting point for solving the problem, the author reminds the reader that Qs= Xs/Rs and Qp= Rp /Xp, and that if the impedances of these circuits are to be the same, then these Q values must be same. Therefore, I thought it was logical and correct to conclude that, for example, Xs=(RpRs)/Xp.

This felt easier than it should be, so I looked it up on the internet and found relationships like Xs=(Xp Rp2)/(Xp2+Rp2) (Source) . I can’t figure out how to get from the aforementioned relationships to that equation, and the websites I can find don’t go through all the intermediate steps either. Could anybody please help me understand this problem?

(Fig 8.5c is only relevant for the later parts of the problem)

• I am aware of and am fairly certain that I understand all of the concepts that you have mentioned; I read through the entire chapter on phasors and copied out all the important equations to make sure I got the concepts in my head. However, what I seem to be having trouble with is connecting it all together, and would like to be able to understand the flaw in the reasoning for my initial solution. Commented Dec 23, 2021 at 5:45
• Do it via complex notation: $R_s+jX_s=\frac{jX_pR_p}{R_p+jX_p}$, then rationalise and equate real parts, and imaginary parts.
– Chu
Commented Dec 23, 2021 at 9:11

You know that in the parallel case it must be that $$\Z_p=R_p \mid\mid jX_p = \frac{R_pjX_p}{R_p+jX_p}\$$ and that in the series case it must be that $$\Z_s=R_s+jX_s\$$. If these are supposed to be equal, then:
\begin{align*} R_s+jX_s&=\frac{R_pjX_p}{R_p+jX_p} \\\\ &=\frac{R_pjX_p}{R_p+jX_p}\cdot \frac{R_p-jX_p}{R_p-jX_p} \\\\\ &=\frac{R_p X_p^{\,2}+j R_p^{\,2}X_p}{R_p^{\,2}+X_p^{\,2}} \\\\ &=\frac{R_p X_p^{\,2}}{R_p^{\,2}+X_p^{\,2}}+j\frac{R_p^{\,2}X_p}{R_p^{\,2}+X_p^{\,2}} \end{align*}