LC Network Reflection and Transmission Coefficients

This blog post is inspired by a previous blog post in which I looked at the reflections and re-reflections on a transmission line given an LC tuner circuit placed at the transmitter.

In other words, the combined LC tuner and transmitter were considered to be a single "lumped-element" network.

I was wondering how these reflections would change if the LC tuner was moved further away from the transmitter and connected to it with a second transmission line (rather than being connected directly to the transmitter at the transmitter's output).  And I realized, although I knew how to calculate the Reflection Coefficients at the LC network's two ports, I did not know how to calculate the Transmission Coefficients through the network (either from the input port to the output port, or from the output port to the input port).

A quick googling of the internet did not reveal anything, so I did some pencil and paper noodling and came up with a way to calculate these coefficients for the general case of  a lossless, two-port network consisting of any combination of lumped-element inductors, capacitors, and transformers. 

(Note:  this technique should be applicable to lossy two-port networks, too).

And thus this post.

First, a review the Reflection and Transmission Coefficients when two transmission lines of different characteristic impedances are connected in series:

In the system above energy must be conserved.  If the system is lossless (ideal transmission lines with no loss), then the power of the incident wave will equal the sum of the powers of the forward and the reflected waves:

P(incident) = P(forward) + P(reflected)

Therefore, for the diagram, above:

(|V(incident)|^2 ) / Za = (|V(forward)|^2) / Zb + (|V(reflected)|^2 ) / Za

But suppose I insert a lumped-element, lossless, two-port network between the two transmission lines?  How do I calculate the new reflection and transmission coefficients if this network is now in-line?

The Reflection Coefficient should be obvious -- it is just the normal equation for a Reflection Coefficient but using the network's Zin (with the network terminated with Zb) in place of the second transmission line's Zo.

The Transmission Coefficient is not so obvious.  But considering that the incident wave sees the network's Zin at the network's input, I make the assumption that the transmitted wave is first transformed using the standard "Transmission Coefficient" equation using the network's Zin.

But more is required than just this calculation, -- a quick check of the results will show that energy (calculated as power) is not conserved.

Another factor is needed, and that is the voltage attenuation (or gain) from the network's input port to its output port when the output port is terminated in the characteristic impedance of the second transmission line.

Including this second factor will result in voltage values in which energy is conserved.

The formulas for calculating the Reflection and Transmission Coefficients for general 2-port, lossless, lumped-element networks are shown in the figure, below:

Next, an example applying these equations...

1.  Example, LC Network:

Let's insert a lossless LC network between the two transmission lines.  We can calculate the resultant Reflection and Transmission Coefficients as shown in the figure, below:

2.  Results:

I will assign L = 1.378 uH and C = 137.8 pF.  At 10 MHz their impedances are:

 Zl =  + j86.58 ohms

Zc = -j115.6 ohms

And I will define Za = 50 ohms and Zb = 25 ohms.

Using the equations in the LC network figure, above, let's do the following calculations:

Calculating Γ₁₁:

To calculate Γ₁₁ (the Reflection Coefficient looking into the network's input,) we first need to calculate Zin of the LC Network.  Using the equation in the figure, above, the result is: 

Zin =  23.88 + j81.41 ohms (using MATLAB).

Therefore, using the equation for Γ₁₁ and substituting in the values for Zin and Za, the result is:

  Γ₁₁ = 0.389 + j0.674.

Calculating T₂₁:

To calculate T₂₁ I first calculate T₁ using Zin and Za.  The result is:

 

T₁ = 1.389 + j0.674

Then I calculate the voltage gain, A_V21, with the network terminated with Zb.  The result is::

A_V21 =  0.0207 - j0.2872

Next, let's calculate voltages...

Set Vincident = 1 volt and Calculate Vf and Vr:

Vr =   Γ₁₁ * Vincident =  0.389 + j0.674 volts

Vf = Γ₁₁ * A_V21 * Vincident =    0.222 - j0.385 volts

And note that the magnitudes of these voltages are:

|Vincident| = 1 volt

|Vr| = 0.778 volts 

|Vf| = 0.445 volts

Check if Energy is Conserved:

Let's verify that power, and thus energy, is conserved:

Pincident = (1^2) / 50 = 20 mW.

Pr = (|Vr|^2) / 50 = 12.1 mW

Pf =  (|Vf|^2) / 25 = 7.9 mW

Pf + Pr = 7.9 + 12.1 mW = 20 mW = Pincident.  

Energy is conserved!

Verifying the results using Simulink:

To verify my results, I'll use the following  Simulink model: 

(For more information regarding its Directional Coupler models, please refer to this post: http://k6jca.blogspot.com/2021/05/antenna-tuners-lumped-element-tuner.html).

The Simulation waveforms are below (note that the simulation voltages are spec'd as peak values, not RMS):

First, note that Vincident = 1 V (peak).  The simulated Vf equals 0.444 volts (peak), which is essentially identical to the value calculated, above.

Below, Vr equals 0.777 volts peak, which is essentially the same as the result calculated, above.

So simulated results match calculated results!

(Note that these are the steady-state values.  You can see that there is a short initial transient in both Vr and Vf when the sine-wave source is first gated on.  The length and amplitude of these transients are related to the network type (e.g. L-network, PI, T, etc.), the component values, and impedances seen by each network port.)

Other Notes:

1.  The technique I present, above, has not been rigorously proven mathematically, so take it with a grain of salt.  Never the less, in the few cases where I have applied it, the calculated results match the simulated results.

2.  For Reflection and Transmission Coefficients from the network's Port 2 to its Port 1 (in other words, in the opposite direction), use the same technique.  Note that the impedance (Zin) looking into the network's output (Port 2) equals Zc || (Zl + Za), where Za is the impedance of the transmission line connected to Port 1, and A_V12 (the voltage gain from Port 2 to Port 1) = Za / (Za + Zl).

3.  If Za = Zb = Zo, then T21 equals S21 and T12 equals S12.

Other Transmission-Line Posts:

http://k6jca.blogspot.com/2021/02/antenna-tuners-transient-and-steady.html.  This post analyzes the transient and steady-state response of a simple impedance matching system consisting of a wide-band transformer.  I calculate the system's impulse response and find the time-domain response by convolving this impulse-response with a stimulus signal.

http://k6jca.blogspot.com/2021/02/the-quarter-wave-transformer-transient.html.   This post analyzes the transient and steady-state response of a Quarter-Wave Transformer impedance matching device.  I calculate the system's impulse response and find the time-domain response by convolving this impulse-response with a stimulus signal.

http://k6jca.blogspot.com/2021/03/useful-swr-voltage-and-power-equations.html.  This post lists (in an easily accessible location that I can find!) some equations that I find useful

http://k6jca.blogspot.com/2021/05/antenna-tuners-lumped-element-tuner.html.  This post analyzes the transient and steady-state reflections of a lumped-element tuner (i.e. the common antenna tuner).  I describe a method for making these calculations, and I note that the tuner's match is independent of the source impedance.

http://k6jca.blogspot.com/2021/05/lc-network-reflection-and-transmission.html.  This post describes how to calculate the "Transmission Coefficient" through a lumped-element network (and also its Reflection Coefficient) if it were inserted into a transmission line.  

http://k6jca.blogspot.com/2021/09/does-source-impedance-affect-swr.html.  This post shows mathematically that source impedance does not affect a transmission line's SWR.  This conclusion is then demonstrated with Simulink simulations.

https://k6jca.blogspot.com/2021/10/revisiting-maxwells-tutorial-concerning.html  This posts revisits Walt Maxwell's 2004  QEX rebuttal of Steven Best's 2001 3-part series on Transmission Line Wave Mechanics.  In this post I show simulation results which support Best's conclusions.

Standard Caveat:

I might have made a mistake in my designs, equations, schematics, models, etc. If anything looks confusing or wrong to you, please feel free to comment below or send me an email.

Also, I will note:

This design and any associated 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.