Tuesday, April 7, 2015

Revisiting L-Network Equations and Constraints

[ I first mentioned this topic within my much longer post on L-Networks.  Because it addresses a commonly-held misconception regarding the application of L-Networks, I thought it deserved a post of its own.  

- Jeff, K6JCA  7 April 2015 ]


The selection of one L-network configuration over another (series-parallel versus parallel-series) is based upon a relationship between the load impedance and the "target" resistance that the load is to be transformed to.

This relationship typically assumes a resistive load, and thus the selection constraints are also based upon a resistive load, rather than being based upon a more universal "complex" load.

These constraints are commonly defined to be:
  • Apply parallel-series networks if Rload < Rs
  • Apply series-parallel networks if Rload > Rs
(Note that some authors use Zo in lieu of Rs as the target resistance.)

(click on image to enlarge)

But suppose the load is complex (as it often is)...the constraints really should be:
  • Apply parallel-series networks if Rload < Rs 
  • Apply series-parallel networks if Gload < 1/Rs
Where Gload is the conductance term of Yload, Zload's equivalent admittance ( Yload = Gload + jBload). 




Note that the first constraint has not changed.  But the second constraint is now in terms of load conductance, not load resistance.

These new constraints also allow us to define network usage slightly differently:
  • If Rload > Rs:  Can only use series-parallel networks (LsCp or CsLp).
  • If Gload > 1/Rs:  Can only use parallel-series networks (LpCs or CpLs).
  • If Rload < Rs and Gload < 1/Rs:  Can use either series-parallel or parallel-series networks (LsLp, LpLs, CsLp, or CpLs if Xload is negative;  CsCp, CpCs, LpCs, or LsCp if Xload is positive).

Note, too, that Gload is defined to be:

Gload = (Rload) / (Rload2 + Xload2)

Using this equation, we can do a quick "sanity" check.  If our load is resistive, then Xload  = 0, and the equation above reduces to Gload = 1/Rload.  If we then substitute 1/Rload for the Gload term in our constraint equation, Gload < 1/Rs, and rearrange, we end up with the original constraint: Rload  > Rs.  So, for the subset of loads that are solely resistive, our new, more general constraint, is completely compatible with the original constraint.

These new constraints are the result of a set of L-network design equations that can be used with complex loads.  These equations are not limited to resistive loads (unlike design equations utilizing Q).  And they are universally applicable to all eight L-network configurations:



Let's now look at these design equations and their derivation.  And in doing so, we will see how we derive the new constraints...


L-Network Design Equations

Parallel-Series Network Equations:

Let's first derive the equations for the parallel-series L-network.  We can draw a general form of this network with the series-element a reactance X and the parallel element a susceptance B.


The load is complex: Zload = Rload + jXload And we want to transform it to be Rs.

The impedance looking into the network, Zin, is:

Zin = 1 / (jB + (1 / (Rload + j(Xload + X))))

Let's expand this further...

Zin = (Rload + j(Xload + X)) / (1 + jBRload - B(Xload + X))

Our goal is to select B and X such that Zin = Rs.  In other words:

Rs = (Rload + j(Xload + X)) / (1 + jBRload - B(Xload + X))

We can expand this equation and then separate out the equation's real terms and the imaginary terms.  If we do, we get the following:

Equating the Real terms:

Rs - Rload = BRsXload + BRsX

Equating the Imaginary terms:

X = BRsRload - Xload

We can take that last equation, plug it into the X term of the previous equation, and solve for B.  The resulting equation will be in quadratic form and therefore has two solutions.  Each of these two solutions for B can then be plugged into the equation for X (the last equation above).  The final results are two B, X pairs:


Solve for B and X, then convert the resulting reactances and susceptances into actual inductors and capacitors.  Note:
  • If B is negative, the component will be a shunt inductor
  • If B is positive, the component will be a shunt capacitor
  • If X is negative, the component will be a series capacitor
  • If X is positive, the component will be a series inductor
Thus, the signs of B and X will determine what kind of network a particular B,X pair creates.  The choices are:
  • shunt-inductor, series-inductor (LpLs)
  • shunt-inductor, series-capacitor (LpCs)
  • shunt-capacitor, series-capacitor (CpCs)
  • shunt-capacitor, series-inductor (CpLs)

Deriving the constraint  Rload < Rs:

Note the term Rs-Rload under the square root sign.  This term must be positive for the answer to exist.

And thus, setting this term to be greater than 0 and then rearranging it, we have our constraint for parallel-series networks: 



Let's now look at series-parallel networks...

Series-Parallel Network Equations:

In a fashion similar to our analysis of parallel-series networks, we can draw a general form of the series-parallel network, with the series-element a reactance X and the parallel element a susceptance B.


Again, Zload = Rload + jXload And again we want to transform the load impedance to be Rs.

The impedance looking into the network, Zin, is:

Zin = jX +  1 / (jB + (1 / (Rload + jXload))

Again, I'll set Zin equal to our target resistance, Rs.  Expanding the above equation and then separating out the resulting real and imaginary terms, we get:

Real terms:

B(XRload - XloadRs) = Rload - Rs

Imaginary terms:

X = (BRsRload - Xload) / (1 - BXload)

The real-term equation can also be solved for X:

X = 1/B + (XloadRs)/Rload - Rs/(BRload)

Through X, we can now equate this last equation to the previous one and solve for B.  Again, the results for B are in the form of a quadratic equation whose solutions are two roots, as shown in the equation below.  These two roots, when calculated, are then substituted into the second equation below (which is our equation for X, from above):


The results are again two pairs of susceptances and reactances that can then be converted into actual component values.  Note:
  • If B is negative, the component will be a shunt inductor
  • If B is positive, the component will be a shunt capacitor
  • If X is negative, the component will be a series capacitor
  • If X is positive, the component will be a series inductor
Thus, the signs of B and X will determine what kind of network a particular B,X pair creates.  The choices are:
  • series-inductor, shunt-inductor (LsLp)
  • series-inductor, shunt-capacitor (LsCp)
  • series-capacitor, shunt-capacitor (CsCp)
  • series-capacitor, shunt-inductor (CsLp)

Deriving the constraint Gload < 1/Rs:

Note the term under the second square root sign in the equation for  B

Rload2 + Xload2 - RsRload

This term must be positive for the equations' results to exist.

And thus, setting this equation to be greater than 0 and then rearranging, we have our new constraint for parallel-series networks:


That's it for deriving design equations and the constraints defining when they can be used.  For the sake of brevity I've left out quite a few steps.  Here's what my derivations actually look like...



Sanity Check!

Let's do a quick test to verify if the new constraint for series-parallel networks is really true.  Let Zload = 25 + j50 and our target resistance Rs = 50 ohms.

Clearly, Rload (25 ohms) is less than Rs (50 ohms), so if the constraint "Rload > Rs" is to be believed, we shouldn't use a series-parallel network.  Yet, here's a series-parallel solution (LsCp) plotted on a Smith Chart:

(click on image to enlarge)


This same result can be verified with an on-line Impedance Matching Network Designer.


Other Fun Facts:

Given the constraint Rload < Rs, parallel-series networks can be used to match impedances in the yellow area of the Smith Chart below.

(click on image to enlarge)


And given the new constraint Gload < 1/Rs, series-parallel networks can be used to match impedances in the yellow area of the Smith Chart below.

(click on image to enlarge)

Note the similarity in form between the two charts above.  You do not get this symmetry if the constraint for series-parallel networks is limited to Rload > Rs. 

This similarity in form is interesting given that the equations for series-parallel networks are so much more complex than the equations for parallel-series networks.  Which got me wondering -- is there a way to express the series-parallel equations with the same simplicity as the parallel-series equations?

I found that if we express the network in terms of admittances, then yes, there is.  Let's use the following diagram:


This drawing is equivalent to the series-parallel diagram earlier in the post, except now the load is expressed as an admittance.

And rather than writing an equation for Zin, the impedance looking into the network, I'll instead write an equation for Yin, the admittance looking into the network.

The equation for Yin is:

Yin = 1 / (jX + 1/(Gload + j(Bload + B)))

If I expand this equation and then equate the imaginary terms on either side of the "equals" sign and then do the same with the real terms in the equation, I get the following two equations:

Real terms:

Yin - Gload = YinX(Bload + B)

Imaginary terms: 

B = YinXGload - Bload

If I plug this last equation into the previous and solve for X, we'll again get a quadratic equation.  The equations representing the two roots of this quadratic equation can then be plugged into the X term in the equation above for B.  The results will be two X,B pairs:

(Note the similarity in form with the parallel-series equations derived earlier!)

Recall that our target impedance for Zin was the resistance Rs.  Yin is just the reciprocal of Zin and so its equivalent target is the conductance 1/Rs -- when solving these equations, this is the value you'd substitute into Yin.

And again, for the solutions to exist, the term under the square root sign must be positive.  So the equations' constraint is:


Not surprisingly, this is the same constraint that we found for the earlier series-parallel network analysis.  But note how much simpler the design equations are.

If you'd like to verify that both sets of equations produce the same results, keep in mind that:

Gload = Rload / (Rload2 + Xload2)
and
Bload - Xload / (Rload2 + Xload2)

Using a spreadsheet, I did a quick test using Zload = 25 + j50 and a target resistance, Rs, of 50 ohms:

(click on image to enlarge)

As you can see, the two different sets of series-parallel equations return the same results.


Even More Fun Facts

Given the new constraint for series-parallel networks (Gload < 1/Rs), these equations allow the entire Smith Chart to now be analytically "matched" to a target impedance with just a pair of networks.


If physically realized, a simple toggle-switch can select the appropriate network for the appropriate Smith Chart region.  For example, a switch-selectable LsCp -- CpLs network pair can span the Smith Chart, as can a switch-selectable CsLp -- LpCs pair of networks:  


Here are the regions covered by the CsLp -- LpCs pair.  One network spans the yellow region, the other spans the non-shaded region.

(click on image to enlarge)

And here are the regions covered by the LsCp -- CpLs pair:

(click on image to enlarge)

If I'd used the original constraints (specifically Rload > Rs for series-parallel networks) to select networks for, say, a computer-driven analysis of matching networks as the load is rotated around a Smith Chart's "circle of constant SWR" (as I did in this post: L-Networks ), there would be a large swath of the circle where I would be missing networks.

For example, suppose I wanted to analyze component values required for a CsLp -- LpCs switch-selectable network.  I won't have calculated matching networks for the entire bottom part of the chart because, if I use the original constraints, neither network will create a match for impedances within that area! 

(Click on image to enlarge)

(Note the same issue exists for the LsCp/CpLs duo, but in this case the top part of the Smith Chart cannot be matched by the CpLs configuration.)

Of course, in actual life this isn't the case. If I built the networks, I would find that they covered the entire constant-swr circle.  The fact that an analysis missed part of the circle is solely due to the incorrect constraint used for the analysis.  And if we instead used the correct constraint, Gload < 1/Rs, our analysis would be fine.


References:

The following websites have similar derivations for the L-network design equations.

http://www.engr.uky.edu/~gedney/courses/ee523/notes/Set4_Matching%20Networks.pdf 

http://whites.sdsmt.edu/classes/ee481/notes/481Lecture7.pdf

http://www.ece.msstate.edu/~donohoe/ece4333notes5.pdf

http://www.ittc.ku.edu/~jstiles/723/handouts/section_5_1_Matching_with_Lumped_Elements_package.pdf

Note however, all of these websites define the use of series-parallel L-networks only when Rload > Rs (or Zo in lieu of Rs, per author preference).


My Related Posts:

http://k6jca.blogspot.com/2015/03/a-brief-tutorial-on-smith-charts.html
A quick tutorial on Smith Chart basics

http://k6jca.blogspot.com/2015/03/notes-on-antenna-tuners-l-network-and.htmlA look at L-Networks

http://k6jca.blogspot.com/2015/04/notes-on-antenna-tuners-t-network-part-1.html
A look at highpass T-Networks

More on the W8ZR EZ-Tuner:  http://k6jca.blogspot.com/2015/05/notes-on-antenna-tuners-more-on-w8zr-ez.html  (Note that this tuner is also discussed in the highpass T-Network post).

The Nye Viking MB-V-A tuner and the Rohde Coupler: http://k6jca.blogspot.com/2015/05/notes-on-antenna-tuners-nye-viking-mb-v.html

The Elecraft KAT-500:  http://k6jca.blogspot.com/2015/05/notes-on-antenna-tuners-elecraft-kat500.html






Standard Caveat: 
I could have easily made a mistake anywhere in this blog post.  If something looks wrong, is unclear, or doesn't make sense, please feel free to contact me!

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