Analytical Approach to Impedance Matching

[ChasW.]

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
$${R_T+jX_T \over 1+jB_L(R_T+jX_T)}-jX_C=R_A-jX_A$$
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
$${R_T+jX_T \over 1+jB_C(R_T+jX_T)}+jX_L=R_A-jX_A$$
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.

 

[marcusl]

I haven't worked through your equations, which would be needed for question 2, but the answers to questions 1 and 3 are are yes and yes. Are you aware that a given network topology can match only a certain range of impedances? If not, then this may be why you are unable to make your circuits match--you need instead to pick the appropriate network based on where your load is on the Smith chart.

I have seen no clearer or more complete treatment of matching and Smith Chart usage than Phillip Smith's own book "Electronic Applications of the Smith Chart" (1969). I highly recommend it. Here's a version that I found on the web of Smith's illustration showing the different network topologies and the impedances that they can match:
http://mwrf.com/site-files/mwrf.com/files/archive/mwrf.com/Files/30/15569/Figure_02.gif

 

[ChasW.]

.. but the answers to questions 1 and 3 are are yes and yes.

Thank you for your response.

If not, then this may be why you are unable to make your circuits match

True enough, but for the values and topologies I provided, a match for each case is possible.

Are you aware that a given network topology can match only a certain range of impedances?

In the general sense, yes. I did select two cases for a single topology, normal L-section, with values that are matchable given both cases. It is a normal L-section because R_T>R_A.

Aside from using the smith chart, I've typically worked with procedures that are much simpler that rely on Q calculation first before establishing component values.

What I am finding, and this goes to the ultimate premise of my original questions, is that for some cases, like for instance, when the source impedance is purely resistive, yet, still greater than the complex load, that it may be only possible to solve as if the topology was a reverse-L section. This is the case for L matches that rely on solving for Q first. So from a programmatic perspective, I have the choice to rely on the user to know what he/she wants topology wise, and give them (or not give them) an answer based on that initial assumption, or I can solve for all related topologies given source and load Z and frequency, or as it may appear, at least for some cases, I could apply the two versions of the quadratic and test the results.

For the purpose of this question, I specifically am interested in how the quadratic equation is being used to obtain values and what those values represent, particularly the signs yielded by the quadratic.

Regarding your response to question 1:

To be clear, you are saying that the quadratic equation, as it applies to finding the parallel element value, may not necessarily yield the sign that would be associated with the complex inductance or capacitance of the element. If this is definitely the case, then I am not reliant on the answer to question 2 anyway.

So, if I were to use the approach that relies on the quadratic equation, I could solve using both cases (positive and negative sign preceding the square root of), set the signs of the resulting values as dictated by the type of element it is to be associated with, inductor or capacitor, and plug those complex values back in as a test using the method I described prior for validating the match. Only 1 set of values, if any, will be expected to match.

While the quadratic approach may on the surface seem too tedious compared to solving for Q first, the quadratic approach does seem to work. Where the Q methods work only if you are willing to, for the purpose of calculation and keeping radicands positive, switch source and load impedances for odd cases where one of the impedances may be purely resistive. I could provide examples supporting this claim if there is interest.