Notes on Antenna Tuners: the T-Network

Previously I'd looked into L-networks (see this post: K6JCA, L-Networks).  I thought I'd now look into T-networks and how they compare to L-networks.  (I'll use Smith charts from time to time.  If you're rusty, here's a quick tutorial:  K6JCA, Smith Chart Tutorial).

L-networks were easily analyzed because they can be defined with a set of equations that, for a given topology, generate a unique set of component values for a given Zload.  And I was able to write an EXCEL spreadsheet to perform these calculations.

In contrast, a T-network with its three variables (the two series components in the arms and the shunt component) will have a range of settings that generate a 1:1 SWR, not a single unique setting.  What I wanted to know: which setting would be the best?

So I decided to use a "brute-force" approach and analyze a T-network by stepping through all of the possible component values (within a range that I'd select) while searching for those combinations that generated results that met a criteria I've selected, such as lowest loss with an SWR equal-to or better-than a target SWR (e.g 1.2:1).

I have no idea how to do this sort of iterative analysis in EXCEL, so I decided to pick up a copy of Matlab and see if I could create an analysis tool using it.  Fortunately, Matlab is available in an inexpensive Home version: Matlab Home Version  ($149 as of 23 April 2015).  I also used their Smith Chart routine, available in their RF Toolbox  (an additional $45).

Here's the GUI I created for my analysis tool:

(click on image to enlarge)

This tool allows me to do the following:

1.  Enter in a Zload and manually adjust the network component values (via sliders or typed-in values) for a match.  The program automatically calculates SWR and network power loss.

2.  For the same Zload, press the "Autotune" button and have the program automatically calculate the component values to satisfy user-entered criteria (such as minimal loss).

3.  For a given SWR, step around a Smith chart's "circle of constant SWR" in discrete steps, calculating the component values which satisfy the user-defined criteria and, after stepping around the entire circle, plot the component and power loss values and also store them in a .CSV file.

But before I get into T-networks, let's briefly look at L-network component values from my previous post...

Specifically, let's look at the maximum component values required to match to any load with an SWR of 10:1 at 3.5 MHz:

Note that the maximum values for the lowpass L-networks (LsCp/CpLs) are much less than the values required for the highpass L-networks.

Here's a plot of the component values for the lowpass versus the highpass.  You can really see how the highpass component values skyrocket at certain reflection-coefficient angles.

(click on image to enlarge)

Well, I'm not likely to design a tuner with variable capacitor whose maximum value is 95,000 pF and with a 238 uH variable inductor.  Instead, let's suppose I limit the max capacitance value to, say, 3000 pF and the max inductance to 12 uH:

(click on image to enlarge)

Clearly I could design a low-pass L-network that covers the entire range of 10:1 reflection coefficients.  The highpass L-networks, on the other hand, has some gaps because the values are no longer large enough.

So if I have 3000 pF variable caps, the lowpass L-network is clearly the way to go.

But even 3000 pF is a lot of capacitance, if it's variable.  Yes, I could have a set of capacitors whose values increase in powers of 2 and which could be switched in parallel with each other via relays.  But controlling the relays, although very do-able, becomes more complex than just turning a couple of knobs.

Suppose instead I just use 500 pF variable transmitting caps?  How effective would these L-networks be?

Here's a plot of how both types of networks would handle the range of 10:1 reflection coefficients at 3.5 MHz.

(click on image to enlarge)

Note that now the highpass CsLp/LpCs networks match a wider range of reflection coefficient angles than do the lowpass LsCp/CpLs networks (although there's also a very wide range that cannot be matched by either). 

So let's take the fact that the highpass network with 500 pF caps has a wider reflection-coefficient "angular" range than the lowpass network and see if we can improve upon it.  What would happen if we add a second series capacitor (also 500 pF) and make the highpass L into a highpass T network?  Let's look...

T-Networks:

We can think of a T-network as being two L-networks connected back to back:

(Yes, I know.  I'm stating the obvious).

As we saw above, the high-pass L-network required some incredibly large component values if it were to match all possible Zload combinations for, say, an SWR of 10:1 at 3.5 MHz.  But even though a high-pass T-network can be thought of as being two high-pass L-networks back-to-back, it doesn't have the same high component values -- they are much more reasonable.

In other words, changing a highpass L-network to a T network brings the peak component values down to something  much more reasonable.

So, rather than a highpass L-network with a 95,000 pF variable capacitor and a 238 uH variable inductor, let's say I have two much smaller 500 pF variable capacitors and a 10 uH variable inductor and I connect them as a C-L-C highpass T-network.  With these values I can match the entire Smith chart "circle of constant SWR" for a 10:1 SWR at 3.5 MHz.

Here's how the component values change as we rotate around the constant-SWR circle:

(click on image to enlarge)

With the T-Network, we can now cover the entire range of SWR 10:1 reflection coefficients with reasonably sized components (L-networks require higher component values). 

But there's a tradeoff...

Power loss in the T-network (compared to the L-network) has increased!

Here is how network power loss (as a percentage of total power) varies for these same settings.  I've included the loss for the highpass L-networks for comparison (blue curve -- but again note that for the L-networks to meet this curve the maximum component values need to be very large).

(click on image to enlarge)

So, although the T-network now lets us match all of the load values with reasonable component values, the tradeoff is the significantly increased power dissipation in the T-network (which peaks between 20 and 25% in the graph above).

My Matlab analysis tool generated the T-network plot above by adjusting components to provide minimum loss along with a match..

But it's not really practical (if it's even possible) to actually tune a tuner for minimum loss.  And we can't just tune for minimum SWR, because a T-Network will tune to a 1:1 SWR over a range of different combinations of the C-L-C components, some of which (as we will later see) can cause significant amounts of power to be wasted within the tuner.

So -- given that tuning for minimum SWR is insufficient, let's add a qualifier...let's tune for minimum SWR with minimum inductance.  

How does efficiency look if we include this minimum-inductance requirement?

(click on image to enlarge)

There's a bit more loss when tuning for minimal inductance, but it's not too bad.

It's important to note: 

Minimum inductance does not necessarily correlate with minimum loss.  

Some authors claim that when a T-network has been tuned for a 1:1 SWR with minimum inductance, this point also corresponds to minimum loss.  This is not true, and you can prove it using W9CF's on-line T-Network Tuner Simulator.

For example, enter into the W9CF simulator a load of 100 + j0 ohms, frequency 3.5 MHz, and set the Cmax values for each capacitor to 2000pF.  If you press the simulator's "Autotune" button, the algorithm will tune for minimum inductance.  The result is a network with an inductance of 3.2 uH and a power-loss of 2.2%.

But if I use the simulator's controls to manually adjust the component values, I can find a 1:1 match with an inductor of 3.7 uH.  And power-loss is now 1.3%, which is obviously lower than the "minimum-inductance" power loss of 2.2%.

You can see the Smith Chart plots of these two networks in the image below:

(click on image to enlarge)

Fortunately, although the hand-tuned network has less loss, it's not less by much, and so tuning for "minimum-inductance" seems to be a good compromise.

Let's look further into power loss.  Is there a way to improve upon the T-network's rather dismal power-loss performance?

Yes, there is.  Let's add two switches that will let us "devolve" from the T-network to one of the two L-network configurations by shorting out either the Cin or Cout capacitor.

(This is not a new idea.  See:  http://www.g3ynh.info/atu/mfj989c.html)

So I'll run my Matlab analyzer to include shorting-out either Cin or Cout (or both), if doing so improves network performance (the analyzer will "tune" for minimum inductance that a meets an SWR of 1.2 or better).

Here are plots of the component values.  Note that the red triangles signify whenever either of the two capacitors is shorted to devolve the T-network into an L-network.

(click on image to enlarge)

Here's the new loss curve (with the previous one for comparison).  Note that we've improved power loss for almost 40% of the SWR circle.

(click on image to enlarge)

You're probably wondering what's happening at 215 degrees where both the Cin and Cout bypass-switches are switched on.  As it happens, at this point the load lies on (or very close to) the Smith chart's "G = 0.02 mhos" circle.  And so the load can be matched with a single shunt inductor, as shown below:

(click on image to enlarge)

For comparison, let's also look at power loss for an SWR circle of 4:1 and compare it to our 10:1 SWR results.

(click on image to enlarge)

Note that the power-loss, although significantly better at the lower SWR, has a smaller angular-range of improvement when devolving to an L-network, compared to the 10:1 SWR.

Here are the component values at an SWR of 4:1:

(click on image to enlarge)

We've been looking at performance with the maximum capacitance set to 500 pF.  What happens if we make Cmax 2000 pF?

Here are the plots of the component values (SWR is still 4:1).  Note that we devolve to the more efficient L-networks much more often.

(click on image to enlarge)

And network power dissipation has also improved significantly:

(click on image to enlarge)

Let's now keep Cmax at 2000 pF but change the SWR back to 10:1.

Component plots:

(click on image to enlarge)

And power-loss plot:

(click on image to enlarge)

Better!

So...

The greater the maximum values of Cin and Cout, the more often we can "devolve" to the more efficient L-network by shorting out either Cin or Cout.

Now let's take a slight digression.  A number of T-tuners use tapped-inductors (tap selected with a switch) rather than continuously-variable inductors, such as roller inductors.

Let's take a look at T-network power loss if inductance were fixed, not variable.

I'll use as an example the EZ-Tuner, a T-network tuner designed by W8ZR and which appeared in QST in 2002.

W8ZR recommends that, for 80 meters, one of these three values of inductance should be used: 10, 4.6, or 3.1 uH.  His design uses two variable caps with a Cmax of about 500 pF.

Let's again assume an SWR of 10:1 at 3.5MHz and look at matching with these fixed values of inductance.  (I don't know what the Q is for the EZ-Tuner components, but let's assume they're the same numbers I've been using:  capacitor Q is 2000 and inductor Q is 100).

Here's the calculated power loss for the three different inductor values:

(click on image to enlarge)

Note that, although the 10 uH inductor allows the network to be matched over the entire range of "10:1 SWR" loads, internal power loss can almost reach 50 percent (assuming inductor Q is 100)!  

(Of course, from the plots above, the user should use the 3.1 uH inductor if the Gamma angle is between about 30 and 180 degrees, the 4.6 uH inductor from 180 degrees to about 290 degrees, and the 10 uH inductor everywhere else. But if the user isn't paying attention to the inductance they've selected, a large amount of power could be lost in the tuner!)

If inductor Q were changed from 100 to 250, power-loss (for 10uH) is roughly halved.  Much better, but still not great.

(click on image to enlarge)

So, the points of this digression are several.  If using a high-pass (C-L-C) T-network tuner:

1.  You may find find a match yet have terrible power loss within the tuner if you choose the wrong value of inductance to match with.

2.  Rule of thumb -- use the smallest value of inductance possible to provide the match.

3.  If the Q's are equivalent, go with a continuously-variable inductor rather than a tapped-inductor (but note that a roller inductor might have significantly worse Q).

OK, end of the digression.  Back to T-networks...

Note that the value of the 2000 pF caps used in the analyses above is not that far from 3000 pF.  What's significant about 3000 pF?  Well, if Cmax is 3000pf, we can skip T-networks entirely and instead use a simple lowpass L-network:

Here are its component values (we saw these at the beginning of this post, too).

(click on image to enlarge)

Here's its power dissipation, compared against a T-network (with shorting-switches and with 3000 pF, not 2000 pF, caps).

(click on image to enlarge)

Not too different, but...

1.  The lowpass L-network only requires one 3000 pF variable cap, not two.

2.  The lowpass L-network will tune to a unique solution.  This is very important.  There's no worry about tuning to a match with a value of inductance that causes a significant amount of power to be lost within the tuner.

So,in my opinion, given a choice between highpass T and low-pass L networks, choose the lowpass L network, if you can manage to get enough capacitance.

Here are the lowpass L-network's maximum component values required to match loads with 10:1 SWRs on 160, 80, and 10 meters:

 

Conclusions:

If designing an antenna tuner, my first choice would be a lowpass L-network that's switchable between LsCp and CpLs configurations.  Assuming I could get the appropriate component values, this network minimizes loss and tunes to a single, unique, matching solution.

My second choice would be a highpass T-network:

• I'd want its capacitors to be as large as possible.  And high-Q, of course.
• I'd want either a continuously-variable inductor or, if a tapped-inductor, one with many taps.  And again, maximizing inductor Q (especially the inductor's Q) is very important!  (Note that roller-inductors can have low Q.  Check out the discussion here:  http://www.w8ji.com/antenna_tuners.htm )
• There should be switches to selectively short-out Cin or Cout, thus "devolving" the T-network to a more efficient L-network whenever possible.

The T-network should be tuned so that the desired SWR is reached with minimum amount of inductance, to ensure that the power-loss within the network is near its minimum (note that minimum inductance does not necessarily equate to minimum loss, but it's often close).


Other Thoughts and Comments:

1.  Component Q's:

Please note that the component Q's I used for my calculations are for illustration only.   Inductor Q's might be higher; capacitor Q's might be lower.

For example, at 3.5 MHz, a 2500 pf cap would need to have an ESR of 0.009 ohms (!!!) if it were to have a Q of 2000.  And this resistance could include variable-capacitor wiper contacts, or relay (or switch) contact resistance.  So care must be taken in the tuner design to maintain a high capacitor Q.

Most likely, though, inductor Q will be significantly less than capacitor Q.  In this case, inductor dissipation will dominate the total power dissipation for worse-case loads.  It is only when capacitor Q (for the case of L-networks) approaches inductor Q that the worst-case dissipation in each component become equivalent, as you can see in the figure below, which plots L-network power dissipation for various capacitor Q's, all with the inductor Q fixed at 100.

(click on image to enlarge)

(Note that the plots above are for a highpass L-Network.  A lowpass L-Network performs similarly as capacitor Q is changed).


2.  Current and Peak Voltage in a Lowpass L-Network.

If I were to build a lowpass L-network Antenna Tuner, I was curious what would be the RMS-Current-Through and Peak-Voltage-Across the inductor and capacitor.

Here's a plot for loads of SWR = 10.  Note that Power Into the network is assumed to be 1000 watts.

(click on image to enlarge)

Note that peak voltage, worst-case, is just a bit more than1000 volts.  And RMS current, worst case, is just a bit more than 15 amps.  These values will be lower for loads with lower SWR's.



Resources, articles:

"The E-Z Tuner", James C Garland, W8ZR, QST Apr 2002 p40-43. May 2002 p28-34, Jun 2002 p33-36.

"Understanding the T-tuner (C-L-C) Transmatch" William E Sabin, W0IYH, QEX, Dec. 1997, p13-21.
Suggests that shorting out C1 and C2 will improve efficiency in some situations.

"Save Your Tuner for Two Pence", Tony Preedy, G3LNP, Rad Com, May 2000, p20-25.  Another article which selectively shorts the T-networks capacitors.

"Getting the Most Out of Your T-Network Antenna Tuner", Andrew Griffith, W4ULD, QST, Jan. 1995, p44-47.    http://www.arrl.org/files/file/Technology/tis/info/pdf/9501046.pdf

T-network sites:

W9CF T-network Simulator:  http://home.sandiego.edu/~ekim/e194rfs01/jwmatcher/matcher2.html    (note that the simulator does not always result in lowest power dissipation)

W8JI site:
  http://www.w8ji.com/antenna_tuners.htm (note discussion on roller inductor Q)
  http://www.w8ji.com/loading_inductors.htm

G3YNH site (lots of great information):

  http://www.g3ynh.info/zdocs/z_matcing/part_1.html

  http://www.g3ynh.info/zdocs/z_matcing/refs.html

  http://www.g3ynh.info/atu/mfj989c.html

T-network equations: http://home.earthlink.net/~w6rmk/math/wyedelta.htm  (I'm not a big fan of equations such as these.  They presume resistive source and load.  The first is often unknown, the latter usually reactive.  But some may find them useful).

Network synthesis site: http://home.sandiego.edu/~ekim/e194rfs01/jwmatcher/matcher2.html  Note that T-network solutions require that you also input a Q value.

Crawford Broadcasting: http://www.crawfordbroadcasting.com/Eng_Files/Matching%20Networks%20and%20Phasing.pdf

VK5BR:  http://users.tpg.com.au/users/ldbutler/Approach_Ant_Tuning.htm


My Related Posts:

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

Plotting Smith Chart Data in 3-D:
http://k6jca.blogspot.com/2018/09/plotting-3-d-smith-charts-with-matlab.html

The L-network:
http://k6jca.blogspot.com/2015/03/notes-on-antenna-tuners-l-network-and.html

A correction to the usual L-network design constraints:
http://k6jca.blogspot.com/2015/04/revisiting-l-network-equations-and.html

Calculating L-Network values when the components are lossy:
http://k6jca.blogspot.com/2018/09/l-networks-new-equations-for-better.html

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

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

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 Drake MN-4 Tuner:
http://k6jca.blogspot.com/2018/08/notes-on-antenna-tuners-drake-mn-4.html

Measuring a Tuner's "Match-Space":

http://k6jca.blogspot.com/2018/08/notes-on-antenna-tuners-determining.html

Measuring Tuner Power Loss:

http://k6jca.blogspot.com/2018/08/additional-notes-on-measuring-antenna.html

Other interesting sites:

DJ0IP Site: http://www.dj0ip.de/antenna-matchboxes/

CalPoly Tuner:  http://digitalcommons.calpoly.edu/cgi/viewcontent.cgi?article=1191&context=eesp
(L network tuner)

Mathworks broadband networks: http://www.mathworks.com/help/rf/examples/designing-broadband-matching-networks-part-1-antenna.html

Z match:  http://users.tpg.com.au/users/ldbutler/SingleCoilZMatch.htm

W0QE: Antenna Tunera Paper: http://www.w0qe.com/Papers/Antenna_Tuners.pdf

Standard Caveat:

I could have easily made a mistake in the above post.  If anything appears wrong or is confusing, please let me know.  Thanks!

- Jeff, k6jca

Revisiting L-Network Equations and Constraints

For those of you who would rather jump to the new L-Network equations rather than dive into their derivation, below, please go here:

https://k6jca.blogspot.com/2022/06/l-network-design-equations.html

***********************

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 typically based upon a relationship between the load impedance's resistance and the "target" resistance that the load is to be transformed to.

Note that the load itself need only be characterized in terms of its resistance (Rload) to determine if a series-parallel or parallel-series L-network should be selected, even if the load represents a complex impedance (i.e. Zload = Rload + jXload).

These constraints are commonly defined to be:

• Apply parallel-series L-networks if R_load < Rs
• Apply series-parallel L-networks if R_load > Rs

But are these two constraints correct?

Let's examine them...

The four most typical L-networks are shown below, along with their commonly-defined constraints.  So, the constraint for selecting the Series-Parallel LsCp or CsLp L-networks is R_load > Rs, and the constraint for the Parallel-Series CpLs or LpCs L-networks is R_load > Rs.

(click on image to enlarge)

(Note that I am using the ARRL Handbook's convention of labeling the impedance that the load is to be transformed to as 'Rs' (i.e. the Source resistance).  Some authors use Zo in lieu of Rs as the target resistance.)

The two constraints I've defined above are what I call the "classical" L-network constraints, in the sense that they can be found throughout the literature on impedance matching.  The two Smith Charts, below, show the range of impedances associated with each constraint, i.e. the range of impedances that satisfy R_load < Rs and the range of impedances that satisfy R_load > Rs.   

Note that the two Smith Chart regions are mutually exclusive and therefore do not overlap.

The first Smith Chart shows the impedances (in the red area) that meet the R_load < Rs constraint.  And per the "classical" definition of this constraint, these impedances can be matched with a Parallel-Series L-network.

The second Smith Chart shows the impedances (in the yellow area) that meet the R_load > Rs constraint.  And per the "classical" definition of this constraint, these impedances can be matched with a Series-Parallel L-network.  

Note my comment in the figure, above, that the R_load > Rs constraint is too restrictive.  In fact, Series-Parallel L-networks can be used to match impedances outside of the R_load > Rs Smith Chart area, despite statements (that can be found in the literature) that a Series-Parallel L-network can be used only when R_load is greater than Rs.

Below is an example of one such a statement.  (My comments are in red).

Let me stress:  Series-Parallel networks are not limited solely to when R_load > Rs.  

Instead (as I will derive later in  this post), the constraints for selecting Series-Parallel and Parallel-Series L-networks actually are:

• Apply parallel-series networks when R_load < Rs 
• Apply series-parallel networks when G_load < 1/Rs

Where G_load is the conductance term of Y_load, Z_load's equivalent admittance.

Note that Y_load = G_load + jB_load, where G_load = R_load / (R_load² + X_load²) and B_load =  - X_load / (R_load² + X_load²).

In the figure below are two common Parallel-Series L-networks, CpLs and LpCs.  (I will mention two other Parallel-Series L-networks, LpLs and CpCs, later in this post.)

Viewed on a Smith Chart, the Z_load impedance "space" representing R_load < Rs has not changed from our original discussion, above, and it still looks like:

As you can see, depending upon where Z_load lies on the Smith Chart, a Parallel-Series L-network consisting of either an LpCs or a CpLs network will transform Z_load to Rs.

And in the figure, below, are two common Series-Parallel L-networks, LsCp and CsLp.  (I will mention two other networks, LsLp and CsCp, later in the post).

With the new G_load < 1/ Rs constraint for Series-Parallel networks, the impedances that can be matched with a Series-Parallel network now look like:

In other words, in the above diagram, depending upon where Z_load lies on the Smith Chart, a Series-Parallel L-network consisting of either an LsCp or a CsLp network will transform Z_load to Rs.

The Overlap of R_load < Rs and G_load < 1/ Rs:

If you examine the two Smith Charts, above, you can see that the R_load < Rs region and the G_load < 1/ Rs region are no longer mutually exclusive;  they now partially overlap.  

Load impedances within this 'overlap' area can be matched to Rs with either a Series-Parallel or a Parallel-Series L-network, as the Smith Chart, below, shows (the overlap area is shown in light green).

You can see that in the top 'light green' area, either an LpCs or an LsCp network can transform an impedance to be Rs.

And in the bottom 'light green' area, either a CpLs or a CsLp network can transform an impedance to be Rs. 

For the sake of completeness, let's examine the subset of the region of the R_load < Rs Smith Chart space where it does not overlap the  G_load < 1/ Rs space (i.e. this subset spans the region in which G_load > 1/ Rs).  The matching L-network is still a Parallel-Series network (CpLs or LpCs):

Similarly, let's examine the subset of the region of the G_load < 1/ Rs Smith Chart space where it does not overlap the  R_load < Rs space (i.e. this subset spans the region in which R_load > Rs).  The matching network is still a Series-Parallel network (CsLp or LsCp).  

The table below summarizes which L-network to use depending upon Z_load's characteristics:

 

L-Networks Consisting only of Two Inductors or of Two Capacitors:

Finally, there are four more L-network cases that should be mentioned:  two L-networks that consist solely of two capacitors and two that consist solely of two inductors.  These can be configured as CpCs, CsCp, LpLs, or LsLp networks:

You can see that these four networks can only transform loads in which the  R_load < Rs and the G_load < 1/ Rs regions overlap.  Therefore, they cannot span the entire impedance range represented by the Smith Chart, and so, in my opinion, they are of limited usefulness for applications requiring transformation of a wide range of impedances.

To summarize, these new constraints are universally applicable to all eight L-network configurations:

Let's now look at the Parallel-Series and Series-Parallel L-network design equations and their derivation.  And in doing so, I will point out the common flaw in the Series-Parallel derivation that resulted in the incorrect R_load > Rs constraint and how to derive the correct G_load < 1/ Rs constraint.


Deriving the Parallel-Series Network Equations:

Let's first derive the design equations for the Parallel-Series L-network -- these equations can be used to find the L-network component values for a specific impedance transformation.  

We can draw a general form of this network with the parallel element a susceptance B and the series-element a reactance X.  (Note that both of these elements are assumed to be lossless,  For lossy networks, see this blogpost:  New Equations for Lossy L-Networks.)

The load is complex: Z_load = R_load + jX_load.  And we want to transform it to be a resistance equal to Rs.

The impedance looking into the network, Zin, is:

Zin = 1 / (jB + (1 / (R_load + j(X_load + X))))

Let's expand this further...

Zin = (R_load + j(X_load + X)) / (1 + jBR_load - B(X_load + X))

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

Rs = (R_load + j(X_load + X)) / (1 + jBR_load - B(X_load + 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 - R_load = BRsX_load + BRsX

Equating the Imaginary terms:

X = BRsR_load - X_load

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  R_load < Rs:

Note the term Rs-R_load under the square root sign.  This term must be not be negative for the answer to exist.

In other words, the following condition must be met:

Rs  ≥  R_load 

However, it can be easily shown that if Rs = R_load, the solution is not a two-element L-network but a single series reactance. Therefore, for two-element L-Network solutions, we will only consider the cases in which Rs-R_load is greater than 0.

And thus, setting Rs-R_load 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.  First the classical approach (and I'll point out the error that is often made)...

Deriving the Series-Parallel Network Equations (Method 1, the Classical approach):

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, both of these elements are assumed to be lossless,  For lossy networks, see this blogpost:  New Equations for Lossy L-Networks.)

Again, Z_load = R_load + jX_load.  And again we want to transform the load impedance to be a resistance equal to Rs.

The impedance looking into the network, Zin, is:

Zin = jX +  1 / (jB + (1 / (R_load + jX_load))

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(XR_load - X_loadRs) = R_load - Rs

Imaginary terms:

X = (BRsR_load - X_load) / (1 - BX_load)

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

X = 1/B + (X_loadRs)/R_load - Rs/(BR_load)

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)

The next step is to define under what conditions the above equations give valid solutions, and this is  where this derivation often takes a wrong turn!

The classical approach will typically state that the quantity under the second square root (i.e. R_load² + X_load² - RsR_load) must be positive and that this condition is met when R_load > Rs.

But is the quantity under the second square root positive only when R_load > Rs?

As it turns out, R_load² + X_load² - RsR_load is positive when G_load < 1/Rs, which encompasses a larger range of impedances than those encompassed by the original "classical" constraint of R_load > Rs.

Let's prove this conclusion mathematically...

Deriving the constraint G_load < 1/Rs:

The equation under the second square root sign in the equation for  B is: 

R_load² + X_load² - RsR_load

And has been stated, this equation must be not be negative for the equations' results to exist.

Thus, we have the equation: 

R_load² + X_load² - RsR_load ≥ 0

However, it can be shown that if the above equation equals zero, the matching network is not a two-element L-Network but a single shunt susceptance.  If we ignore this special case, the above equation becomes:

R_load² + X_load² - RsR_load > 0

 Rearranging this equation, we have our new constraint for series-parallel networks:

Note at the complexity of the derivation of the Series-Parallel set of equations compared to the earlier derivation of the Parallel-Series equations.  Given the symmetry between the two "match spaces" when plotted on a Smith Chart, is there were a way to derive the Series-Parallel equations in a less complicated way?

Deriving the Series-Parallel Network Equations (Method 2, the Admittance Approach):

Let's use the following diagram for a Series-Parallel L-network and redefine the input and output impedances to instead be admittances.:

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 one 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 our target is its conductance 1/Rs -- when solving these equations, this is the value you'd substitute into Yin.

Again, for the solutions to exist, the terms under the square root sign must not be negative.  And again, ignoring the special case of Yin = Gload (in which the matching network is a single shunt susceptance, not a two-element L-Network), the equations' constraint becomes:

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:

G_load = R_load / (R_load² + X_load²)

and

B_load =  - X_load / (R_load² + X_load²)

More Interesting Facts

Given the new constraint for series-parallel networks (G_load < 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-yellow 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 R_load > 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, G_load < 1/Rs, our analysis would be fine.


Examples of incorrectly derived Series-Parallel constraints:

The following websites incorrectly derive the constraint for applying Series-Parallel L-networks.  Below are several examples.

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

https://uspas.fnal.gov/materials/10MIT/Lecture11.pdf

(A number of other sites skip the derivations, but still incorrectly state the the Series-Parallel network should be used when Rload > Rs and the Parallel-Series network should be used when Rload < Rs).

My Related Posts:

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

Plotting Smith Chart Data in 3-D:
http://k6jca.blogspot.com/2018/09/plotting-3-d-smith-charts-with-matlab.html

The L-network:
http://k6jca.blogspot.com/2015/03/notes-on-antenna-tuners-l-network-and.html

A correction to the usual L-network design constraints:
http://k6jca.blogspot.com/2015/04/revisiting-l-network-equations-and.html

Calculating L-Network values when the components are lossy:
http://k6jca.blogspot.com/2018/09/l-networks-new-equations-for-better.html

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

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

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 Drake MN-4 Tuner:
http://k6jca.blogspot.com/2018/08/notes-on-antenna-tuners-drake-mn-4.html

Measuring a Tuner's "Match-Space":

http://k6jca.blogspot.com/2018/08/notes-on-antenna-tuners-determining.html

Measuring Tuner Power Loss:

http://k6jca.blogspot.com/2018/08/additional-notes-on-measuring-antenna.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!