The Quarter-wave Transformer: Transient and Steady-State Reflections

This post is the second post of two analyzing transient and steady-state reflections in systems with transmission lines and impedance matching devices.

This analysis is done using the system's Impulse Response and convolving that with a Stimulus that is bounded in time (that is, the stimulus has a defined start time and a defined stop time).

This method of using the Convolution Integral allows us to examine both the system's transient response (when the stimulus switches from absent to present or from present to absent) and  its steady-state response.  For more background on this technique, please refer to my first post in this series:  Antenna Tuners: Transient and Steady-state Analysis.

The Quarter-wave Transformer:

The quarter-wave impedance transformer is a two-port impedance-matching device consisting of a quarter wavelength of transmission line whose characteristic impedance is the square-root of the product of the impedances attached at its two ports.

For example, in the drawing above, Transmission Line 2 will match the load (Rload) to the characteristic impedance of Transmission Line 1 (Za) if Transmission Line 2's characteristic impedance (Zb) is the square-root of the product of Za and Rload:

Zb = (Za*Rload)^0.5

Calculating the System's Impulse Response:

To analyze the transient-state of the reflections within the matching system using the Convolution Integral, I first need to calculate the system's impulse response (i.e. its response to a single impulse).

To simplify the analysis, I'll assume that the transmission lines are lossless and behave like perfect delay lines.  Also, I will assume that the Source is matched to Transmission Line 1.

Therefore, an impulse traveling on one transmission line, when it arrives at the second transmission line of a different impedance, will be divided into two parts at the impedance discontinuity.  One part is reflected backwards in the opposite direction of travel and the other part continues forward.  Their amplitudes as they travel along the transmission lines will only be a function of the various Reflection and Transmission Coefficients within the antenna system and not due to transmission line loss.

The Reflection and Transmission Coefficients that a single impulse sees as it travels through the antenna system are solely the function of the impedances immediately at either side of an impedance discontinuity, and they are calculated using the standard formulas:

R = (Zb - Za)/(Zb + Za)

T = 2Zb/(Zb + Za)

where Zb is the impedance that the impulse is traveling to, and Za the impedance it is traveling from.

The drawing below shows the calculated Reflection and Transmission Coefficients at each impedance discontinuity, assuming the load is 200 ohms, the characteristic impedance of Transmission Line 1 is 50 ohms, and the characteristic impedance of the Quarter-wave Transformer (Transmission Line 2) is 100 ohms:

If I shoot a single impulse from the source into Transmission Line 1, it will move from left to right until it reaches the discontinuity at Transmission Line 2.  At that point, a portion of the impulse will continue onwards, towards the load, while a second portion is reflected back to the source.  

(Note that I've defined the source to be matched to Transmission Line 1, so there is no re-reflection back towards the load when this part of the impulse arrives back at the source).

The first portion, when it reaches the load, will again see an impedance discontinuity and it will again be divided into two parts.  One part is dissipated by the load's resistance, and the other part is reflected back towards Transmission Line 1.

When this reflection from the load reaches Transmission Line 1, it is again divided into to parts.  One part reflects back to the load, and one part passes through the impedance discontinuity and continues back to the source.

And thus it goes -- there will be an infinite number of reflections and re-reflections on Transmission Line 2.

Below is a diagram showing the amplitudes of the impulse (in terms of Reflection and Transmission coefficients) as it is divided into two parts whenever it reaches an impedance discontinuity:

For example, let Vi(t) be a single impulse in time.  Let's define its arrival (from the source) at the impedance discontinuity between Transmission Line 1 and Transmission Line 2 to be at time t = 0.  

From the lattice diagram above I can calculate impulse responses for Vr1, Vf2, Vr2, and Vld.  

For example, Vr1 (the voltage seen at the interface between the two transmission line that is traveling back towards the source) can be expressed mathematically as an infinite series.  It is the terms in this series that define the impulse response of Vr1:

Here is the Impulse Response of Vr1 (calculated with MATLAB):

You can see that the higher terms of the series (which is infinite in length) quickly become imperceptible compared to the first few terms.

Similar infinite series can be written using the terms in the Lattice Diagram for the voltages Vf2, Vr2, and Vld, and from these series the Impulse Response of each voltage can be calculated.

Convolving the  System's Impulse Response with a Sinusoidal Stimulus:

I can take the calculated impulse responses of Vr1, Vf2, Vr2, and Vld and, using the Convolution Integral, calculate their output waveforms for any input stimulus waveform.

But because the quarter-wave transformer is an impedance-matching device defined for use at a single frequency where its length is a quarter-wavelength of the stimulus signal, the stimulus I will use will be a sine wave.  (A quarter wave-length thus represents one-quarter cycle of the sine wave).

In steady-state (with a sinusoidal drive), the signal from the source arriving at the interface of the two transmission lines will be either in-phase or 180 degrees out-of-phase with the reflections arriving back from the load, depending upon how many up-and-back trips a reflection has made (they will be out-of-phase if the number of trips is odd, in-phase if the number of trips is even). 

Therefore, we can express Vr2 as the following infinite series.  Note that the phase of the arriving reflection flips between 0 and 180 degrees with each term, depending upon if the number of round-trips it has taken on the second transmission line is even or odd..

The final equation in the figure, above, contains an infinite geometric series that converges to a value.  We can calculate this value using the formula below for solving infinite geometric series:

Solving for Vr1's infinite series, we find the following: 

Using the same technique, we can also calculate the steady-state equations for Vf2, Vr2, and Vld, which are:

Given a sine-wave signal Vi(t) of amplitude equal to 1, the steady-solutions for these equations are shown at the bottom of the lattice diagram, below:

(Note:  we can also use the earlier equation for "Apparent Γ₂" to find the equation that calculates the required characteristic impedance of Transmission Line 2, given the load impedance and the characteristic impedance of Transmission Line 1, that will result in an "Apparent Γ₂" equal to 0.  See: Multiple Reflections and the Quarter-wave Transformer.)

Convolution Results:

I've used MATLAB to first calculate the impulse responses of Vr1, Vf1, Vr2, and Vld, and then to convolve these impulse responses with the sinusoidal stimulus.  I've plotted the results, below.

Below is the stimulus, Vi(t).  It is present for 12 cycles and otherwise off:

Below are the MATLAB plots of Vr1's impulse response and resulting output when convolved with the stimulus.  You can see the non-zero transient response at the start and finish of the sine-wave.  And in steady state Vr1 goes to 0. 

For convenience, I've plotted the stimulus and the Vr1 output together, below:

Below are Vf2's impulse response and resulting output when it is convolved with the stimulus.  

Below are Vr2's impulse response and resulting output when it is convolved with the stimulus. 

And finally, below are Vld's impulse response and resulting output when it is convolved with the stimulus. 

Again, for convenience, I've plotted the stimulus and the Vld output together, below:

You can see the quarter-cycle delay between the stimulus and Vld in the plot, above.

The Re-reflection of Vr2:

In steady-state it is clear that some portion of the Vr2 voltage continues through the discontinuity towards the source.  This is why the wave heading back to the source (Vr1) goes to 0 in steady-state (when load is matched to Transmission Line 1) -- this transmitted part of Vr2 cancels the part of Vi that is reflected by the transformer's input discontinuity.

But some part of Vr2 is reflected back to the load.  How much?  And (assuming that the quarter-wave line is matching load to Transmission Line 1) does  Γ₃ change in steady-state (in a fashion similar to how Γ₂ changes to to an "apparent"  Γ₂ of 0)?

We can calculate the steady-state Γ₃ using the same steady-state equations for Vf2 and Vr2 that were derived earlier.

First, let's call this reflection of Vr2 off of the quarter-wave transformer's "input" impedance discontinuity "Vr2Re_Reflected".  This is the portion of Vr2 that is re-reflected back to the load by the impedance discontinuity (the other part continues through the discontinuity to the source)

This re-reflected voltage sums with the forward voltage from the source that is passing through the input discontinuity towards the load.  In other words, Vf2 = Vi(0°)Tf + Vr2Re_Reflected.

Therefore Vr2Re_Reflected = Vf2 - Vi(0°)Tf , and the "apparent" Γ₃ should therefore be the ratio of Vr2Re_Reflected to Vr2.

Using these relationships and recognizing that Vr2 is 180 degrees out of phase with Vi when Vr2 arrives back at the input to the quarter-wave transformer, we can derive the "apparent" Γ₃ as follows:

Therefore, the steady-state "apparent" Γ₃ is the same as the original Γ₃ calculated for a single impulse on the line.

Given a sinusoidal source driving a 1 volt signal onto the first transmission line, the following illustration summarizes the steady-state voltages in the system.  Note that the blue line represents the signal that is just arriving from the source, while the green lines represent the reflections and re-reflections of previous signals from the source.

Conclusions:

1.  As with any other linear system, the Convolution Integral allows us to calculate the time-domain transient and steady-state responses of the system to a stimulus.

2.  Given a sine-wave stimulus that is bounded in time, there are reflections back to the source during the system's transient states.  These reflections decay to 0 in the steady-state, resulting in an "apparent" Γ₂ of 0.

3.  In the steady-state, Vf2 is a combination of the forward signal arriving from the source (scaled by the Transmission factor of the impedance discontinuity between the two transmission lines) and the re-reflection by Γ₃, at the impedance discontinuity between the two transmission lines, of the backwards traveling Vr2 wave (from the load).  The amplitude of this re-reflection equals Vr2 * |Γ₃|.

4.  Assuming that the quarter-wave transformer is matching the load to Transmission Line 1's characteristic impedance, in steady-state Γ₃ remains unchanged from the value calculated for an impulse response, and Γ₃ is independent of the source impedance.  In other words, because there are no signals sent back to the source in steady-state, it does not matter what the source impedance is because there is no signal to be reflected back by a possible source mismatch.

Additional Notes:

1.  Quarter-Wave Transformer Input Impedance:

Although not obvious from the Lattice Diagram, above, the Input Impedance of a Quarter-Wave Transformer is independent of the impedance of the "left-hand" transmission line (or lumped-element source impedance, if there is no transmission line) attached to its input port.

In other words, the Transformer's Input Impedance is only a function of the characteristic impedance Zo of the quarter-wave line and the load impedance attached to the output of the Transformer.

Regarding the diagram, above, this would mean that if the characteristic impedance of the left-hand transmission line (Za) was not 50 ohms, there would be reflections from the input of the transformer back towards the source that do not go to zero in steady-state.

Also, the transient voltage levels of the waves on the transformer's transmission line (Vf1, Vf2, and Vr2) would be different.  

But despite these differences, in steady state the sum of Vf1 and Vf2 (i.e. the overall Vf on the transformer's transmission line moving towards the load), and the level of Vr2 (moving from the load back), would remain unchanged.

And if a "lumped-element" directional coupler (i.e. SWR meter), referenced to 50 ohms, were placed on Transmission Line 1 at the input to the transformer, it would read an SWR of 1:1, irrespective of the Transmission Line 1's characteristic impedance.

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.