L-Network Design Equations for Transforming a Complex Load to a Real Zo

This blog post summarizes L-Network equations I derived for transforming any complex load impedance (as long as its real part is not negative), to a real impedance (i.e. no imaginary component) when looking into the input port of the L-Network.

The derivation of these equations is described in a much longer blog post back in 2015 that can be found here:

https://k6jca.blogspot.com/2015/04/revisiting-l-network-equations-and.html

These derivations can also be found in my article, "Correcting a Common L-Network Misconception," published in the March/April issue of QEX.

In addition, I also present an alternate expression of these equations in terms of Q.

Note that the equations described in this post are for lossless L-Networks.  Design equations for L-Networks with lossy components are significantly more complex, and are described in the following blog post https://k6jca.blogspot.com/2018/09/l-networks-new-equations-for-better.html

And a final note:  I have recently posted L-Network equations for transforming any impedance to any other impedance, not just to a non-complex (i.e. real-only) impedance (as described in this blog post).  You can find these equations here:  https://k6jca.blogspot.com/2023/08/l-network-equations-for-any-impedance.html

Introduction:

There are a total of eight possible lossless L-Network configurations, consisting of four Parallel-Series L-Network configurations and four Series-Parallel L-Networks configurations, as shown in the figure, below:

The range of impedances that each L-Network can match (to Zo) are shown as the gray regions on the Smith Charts, below:

Let us first look at the design equations for a lossless Parallel-Series L-Network...

Design Equations for a Lossless Parallel-Series L-Network:

A Parallel-Series L-network can transform Zload = Rload + jXload to a target-resistance Zo if the following condition is satisfied:

Rload ≤ Zo

Assuming this condition is satisfied, the design equations for a Parallel-Series L-network result in two sets of B, X pairs (B is the susceptance of the "parallel" (i.e. shunt) component, and X is the reactance of the series component).

Each B, X pair represents a unique Parallel-Series L-network implementation.  Thus, the impedance transformation can be implemented with either of the two Parallel-Series L-Networks represented by these two B, X pairs.

The figure below describes the equations for calculating the two B, X pairs for a Parallel-Series L-Network: 

These same equations can also be expressed in terms of Q.  However, unlike the usual L-Network equations using Q that only transform a non-complex (e.g. resistive) load to a real, non-complex impedance, e.g. Zo (see here, here, or here), these new equations can be used to transform any complex impedance to Zo.

In order to restate the above equations in terms of Q, first recognize that for a Parallel-Series L-Network we can derive Q to be:

Q = (Zo/Rload - 1)^0.5 = ((Zo - Rload)/Rload)^0.5

We can therefore substitute Q into the original Parallel-Series susceptance and reactance equations, above, wherever the quantity “((Zo - Rload)/Rload)^0.5” occurs in each equation.  This quantity will need to be teased out of the reactance equations, but it is there.  Just recognize that (Rload*(Zo-Rload))^0.5 is the same as Road*((Zo-Rload)/Rload)^0.5. 

The resulting equations are shown, below:

Design equations for a Lossless Series-Parallel L-Network:

A Series-Parallel L-network can transform Zload = Rload + jXload to a target-resistance Zo if the following condition is satisfied:

Gload ≤ 1/Zo

where Gload (and Bload) can be calculated from Rload and Xload per the final equations in the following derivation:

Assuming the condition that Gload ≤ 1/Zo is satisfied, the design equations for a Series-Parallel L-network result in two sets of B, X pairs (B is the susceptance of the "parallel" (i.e. shunt) component, and X is the reactance of the series component).

Each B, X pair represents a unique Series-Parallel L-network implementation.  Thus, the impedance transformation can be implemented with either of the two Series-Parallel L-Networks represented by these two B, X pairs.

The figure below describes the equations for calculating the two B, X pairs for a Series-Parallel L-Network:

Like the Parallel-Series equations earlier, these same equations can also be expressed in terms of Q. 

In order to restate the above equations in terms of Q, it is useful to derive Q for a Series-Parallel L-Network in terms of the load conductance (Gload) and Zo expressed as conductance (i.e. 1/Zo, as Zo is a real, not complex, impedance).  The resulting Q equation is:

Q = ((1/Zo)/Gload - 1)^0.5 = ((1/Zo - Gload)/Gload)^0.5

We can therefore substitute Q into the original Series-Parallel susceptance and reactance equations, above, wherever the quantity “((1/Zo - Gload)/Gload)^0.5” occurs in each equation (recognize that for the susceptance equations (Gload*(1/Zo-Gload))^0.5 is the same as Goad*((1/Zo-Gload)/Gload)^0.5 ). 

The resulting equations are shown, below:

Identifying  L-Network Type from the Signs of B and X:

For a Parallel-Series L-Network solution, the signs of B and X of each pair will determine which of the four possible Parallel-Series L-Networks each B, X pair represents.

Similarly, given the two pairs of B, X solutions for a Series-Parallel L-Network, the signs of B and X for each pair will determine which of the four possible Series-Parallel L-Networks each B, X pair represents.

The table, below, summarizes network configuration versus the signs of B and X:

Converting  B and X to Inductance and Capacitance Values:

Capacitor and Inductor component values are calculated from B and X using the formulas in the table, below:

An Example: Transforming Zload = 20 + j40 ohms to 50 ohms:

Let us find the L-Networks that will transform a load impedance Zload = 20 + j40 ohms (at 10 MHz) to be a resistive value Zo, where Zo = 50 ohms:

In other words:

Target Impedance: Zo = 50 ohms

Zload = 20 + j40 ohms

Frequency = 10 MHz

By inspection of Zload:

Rload = 20 ohms

Xload = 40 ohms

and we can calculate:

1/Zo = 0.02 mhos

Gload = 0.01 mhos

Bload = -0.02 mhos

Next, we must verify which L-network topologies (Parallel-Series or Series-Parallel) can transform Zload to Zo:

1.  Is Rload ≤ Zo?  Yes, 20 is less than 50, and so Zload can be transformed to Zo using a Parallel-Series L-network.

2.  And is Gload ≤ 1/Zo?  Yes, 0.01 is less than 0.02, and so Zload can be transformed to Zo using a Series-Parallel L-network. 

Because both selection criteria are satisfied, Zload = 20 + j40 ohms can be transformed to Zo using any one of four L-networks (i.e. either one of the two Parallel-Series networks or one of the two Series-Parallel L-networks).

The EXCEL spreadsheet, below, shows the B, X pairs and associated configuration for these four L-networks and calculates the inductor and capacitor component values:

Below are these four L-Networks, shown in schematic form:

Note that only two of the four possible Parallel-Series L-Networks and two of the four possible Series-Parallel L-Networks can perform this particular impedance transformation.

Therefore, there remain four L-Networks that can not transform Zload = 20 + j40 ohms to 50 ohms.  These networks are:

LpLs, CpLs, LsLp, and CsLp

Final Comments:

If you examine the Series-Parallel L-Network equations for X and B and the quantities under their square root signs, you will see that a matching solution exists not only when G < 1/Zo, but also when G = Zo.

However, when G = Zo, there is no difference between the two sets of Series-Parallel equations for X and B, and X becomes zero, leaving single shunt susceptance, B, as the matching network, rather than a two-element L-Network.

A similar situation exists for the Parallel-Series L-Network equations.  If Rload = Zo, the two sets of Parallel-Series equations become identical, with B equal to zero and only a single series reactance, X, as the matching network, rather than a two-element L-Network.

Standard Caveat:

As always, I might have made a mistake in my equations, assumptions, drawings, or interpretations.  If you see anything you believe to be in error or if anything is confusing, please feel free to contact me or comment below.

And so I should add -- this information is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.