- #1

ChasW.

Gold Member

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## Homework Statement

I am calculating matching impedances for a source and its load using 2 different L-network topologies.

One is:

source-parallel-to-C, load-series-to-L

and the other is:

source-parallel-to-L, load-series-to-C.

I go through all the calculations and notice that the 2 sets of answers from the quadratic part each yield a set of values that if I intuitively adjust signs for, I get a correct impedance match. Of course only 1 set of answers is appropriate for each of the above 2 L-network cases.

With signs adjusted by me, I verify the match by comparing the resulting impedance of the source paralleled with its element with the resulting impedance of the load series combined with its element. I've done this quite a few times and the results are always the same. Right values, wrong signs.

It would seem to me that if I am applying the formula correctly that at least one set of results from the quadratic equation should yield a correct set of values that are signed properly. Again, I am not sure and hope somebody can enlighten me on this point so I can know whether to continue pursuing a sign error on my part or correct my assumptions with the application of the formula.

## Homework Equations

The representations of the above two L-network cases I am using are:

parallel L to series C

[tex]{R_T+jX_T \over 1+jB_L(R_T+jX_T)}-jX_C=R_A-jX_A[/tex]

where,

R

_{T}is the source resistance

X

_{T}is the source reactance

R

_{A}is the load resistance

X

_{A}is the load reactance

B

_{L}is the inductive susceptance of the parallel element

X

_{C}is the capacitive reactance of the series element

parallel C to series L

[tex]{R_T+jX_T \over 1+jB_C(R_T+jX_T)}+jX_L=R_A-jX_A[/tex]

where,

R

_{T}is the source resistance

X

_{T}is the source reactance

R

_{A}is the load resistance

X

_{A}is the load reactance

B

_{C}is the capacitive susceptance of the parallel element

X

_{L}is the inductive reactance of the series element

Z

_{T}=150+j75Ω and Z

_{A}=75+j15Ω and the frequency of interest is 2GHz.

Note that for both cases R

_{A}-jX

_{A}is the conjugate of the actual load impedance.

## The Attempt at a Solution

The two sets of answers I get are:

B

_{L}

^{-1}= 108.712, -258.712

X

_{C}= -76.8559, 106.856

B

_{C}

^{-1}= 108.712, -258.712

X

_{L}= 76.8559, -106.856

For both sets above, the comma delimited values are each results from the positive and negative version of the quadratic.

but for the parallel L to series C, the correct values are:

B

_{L}

^{-1}= 258.712

X

_{C}= -106.856

and for the parallel C to series L, the correct values are:

B

_{C}

^{-1}= -108.712

X

_{L}= 76.8559

My questions are somewhat general in nature:

1) Am I wrong for expecting resulting values for the inductive reactance to be positive and the capacitive to be negative? Just to be clear here, I realize that the actual reactances, inductive or capacitive, are positive, but what I am referring to are the complex values as they may apply or not apply to what the result of the quadratic equation yields.

2) The book I am using says to use the positive case for the quadratic part when R

_{T}>R

_{A}, but that does not as of yet seem to be consistently be the right thing to do. Is there a rule of thumb for this?

3) For the first network I am subtracting the element in series because it is capacitive and the second one I am adding the element in series because it is inductive. Was this correct to do?

I did not post the long list of calculations because of the general nature of my questions, but will happily do so upon request.

The general overview of how I am solving can be stated briefly, however:

Separate the real and imaginary parts of the Z

_{SOURCE}. i.e. Solve for R

_{T}and X

_{T}. From the R

_{T}part, solve for the series element and substitute that into the X

_{T}part. This results in a quadratic equation for the parallel element. Plugging values into that equation yields two answers, and plugging the parallel element value into the series element equation yields the answer for the series element.

Thank you much in advance for any assistance and patience with the lengthy post.