Transmission Line Design:

Quarter-Wave

&

Single-Stub

Applied Electromagnetic Waves ECE 3317

Professor Chen

Julia London

Spring 2014

London 2

Table of Contents

I. Abstract........................................................................................................................................2

II. Introduction.................................................................................................................................4

III. Analysis and Design..................................................................................................................4

Quarter-Wave Transformer..........................................................................................................4

Figure 1 – QWT Initial Design.................................................................................................5

Table 1 – Initial Givens for Transmission Lines......................................................................5

Figure 2 – QWT Final Design..................................................................................................6

Short-Stub Transmission Line.....................................................................................................7

Figure 3 – SST Initial Design...................................................................................................7

Figure 4 – Solution for d..........................................................................................................9

Figure 5 – SST Final Design..................................................................................................10

IV. Results.....................................................................................................................................11

Quarter-Wave Transmission Line..............................................................................................11

Figure 6 – QWT SWR vs. ff 0

.................................................................................................12

Figure 7 – QWT SWR vs. ff 0

Extended.................................................................................13

Table 2 – QWT Varying Bandwidth......................................................................................14

Single-Stub Transmission Line..................................................................................................14

Figure 8 – SST SWR vs. ff 0

...................................................................................................15

Figure 9 – SST SWR vs. ff 0

Extended...................................................................................16

Table 3 – SST Varying Bandwidth........................................................................................17

V. Conclusions...............................................................................................................................18

VI. References...............................................................................................................................19

London 3

I. Abstract

A problem in designing an effective quarter-wave transmission line and a single-stub

transmission line was presented towards the middle of the semester. After designing said

transmission lines the second task was to observe the change in their standing wave ratios due to

the change in their normalized frequencies. Once this was observed, the percent bandwidths of

both types of transmission lines were to be recorded and compared, along with their standing

wave ratio relationships.

Using MATLAB, several derived equations and allowing the single-stub transmission

line to have a short circuit stub, both graphs were plotted and recorded in increments of 1[MHz].

Through observation, it was noted that the quarter-wave transmission line had a more consistent

change in SWR and bandwidth than the single-stub transmission line. It was also noted that the

single-stub had a higher chance of reflecting an entire signal back to the generator than actually

allowing the load to absorb the signal. Although the bandwidth for the quarter-wave transmission

line at first appeared to have been too large for the analysis, through further research it was

confirmed that the bandwidth was a reasonable amount for the stated initial frequency.

London 4

II. Introduction

Transmission lines are used to transmit the alternating currents of radio frequencies. Such

signals operate on such high enough frequencies that they must be taken into account; otherwise

the transmission would be lost. Transmission lines are also used in everyday life from

distributing cable television to network connection for high-speed computers. Different

transmission lines range from coaxial cables to optical fibers, but the transmission line that will

be discussed is the losses transmission line in two forms, quarter-wave (QWT) and single-stub

(SST). With these two types of transmission lines a graph will be generated to see how the

change in frequency will relate to their relative Standing Wave Ratios (SWR). The standing

wave ratio is the ratio between the highest amplitude of the single versus its lowest amplitude.

This ratio is used to determine the efficiency of a transmission line with SWR being equal to 1 as

the most idea transmission line, and SWR being equal to infinity being the most inefficient. Once

the SWR graphs have been generated their relative bandwidths will be noted and compared.

III. Analysis and Design

Quarter-Wave Transformer

Below in Figure 1 is the set up for a quarter wave transmission line.

London 5

Figure 1 – QWT Initial Design

Before being able to generate a SWR vs. the normalized frequency graph some initial conditions

must be built into the transmission line.

Below in Table 1 is the initial conditions given for the project.

Table 1 – Initial Givens for Transmission Lines

Givens

Z0 75 [Ω]

ZL 75 [Ω]

Z0 s 75 [Ω]

c 3x108 [m/s]

Knowing this we can calculate the initial length, l, lambda, λ0, Z0T, stub-length,ls, and stub

impedance. Because this is a quarter-wave transmission line the stub impedance is unnecessary

thus making the stub length and its impedance equal to zero. Next the initial lambda can be

found by using Equation 1

λ0=c / f 0, (1)

37.5 [Ω]

London 6

With f0 as an initial frequency and c equal to the speed of light. For this project f0 and c will be

equal, simply to aid in the ease of generating the SWR graph.

Once the initial lambda has been found the initial distance that Z0T will be affecting can

be found with Equation 2,

l= λ0/4. (2)

Next the impedances were calculated. Initially the load has two impedances that are in

parallel, these impedances can be combined. Because the wires connecting the impedances have

the same resistance the impedances can be combined into one by using Equation 3

ZLTotal=1

1ZL

+ 1ZL

, (3)

and Z0T can be found by using Equation 4

Z0T=√Z0∗Z LTotal. (4)

The final layout for the quarter wave transmission line can be seen in Figure 2

Figure 2 – QWT Final Design

Now that the initial conditions have been found the changing aspects of the transmission

line, due to the change in frequency, can be found.

To find how frequency relates to SWR Equation 5

SWR=1+|Γ|1−|Γ|

, (5)

is used, where Γ is found using Equation 6

London 7

Γ=Z¿−Z0

( Z¿+Z0 ) , (6)

In Equation 6, Z0 is the transmission line’s original impedance, as shown in Figure 1, and

Z¿ is the change in impedance as the current heads towards ZLoad Total. Z¿ can be found using

Equation 7

Z¿=Z0 T

Z LTotal+ j Z0 T tan( 2 πlλchanging

λ0)(Z0 T+ j ZL Total tan( 2 πl

λchangingλ0))

, (7)

In this Z0T is the constant found from Equation 4. Equation 7 shows how the normalized

frequency will change the SWR, since Equation 1 shows us that the λ0 directly relates to the

frequency. λ0 is also multiplied within tan to normalized the frequency, as seen in Equation 8

f normalized=f changing

f 0 . (8)

These formulae were input into MATLAB in order to generate a SWR vs. f normalized graph.

Short-Stub Transmission Line

Below in Figure 3 is the initial set up for a short-stub transmission line.

London 8

Figure 3 – SST Initial Design

The known values for Figure 3 are the same as that in Figure 1 so Table 1 was referred to

for this portion of the calculations as well. Again, with the information from Table 1, several

unknowns can be found for Figure 2. These unknowns include the distance the stub is away from

the load, d , the stub length, ls, and the impedance of the stub, jβ stub.

Again, as was done for Figure 1 the load impedances can be combined using Equation 3.

To simplify the problem λ0, used for the quarter-wave transmission, line will be used for these

calculations. Knowing the new load impedance d can be found using a smith chart,

corresponding to whether the stub is an open or short circuit. For the following calculations the

stub is a short circuit.

Figure 4 shows the solution for d with the stub as a short circuit. ZLN Represents the

normalized load impedance, Y LN represents the normalized load admittance and d represents the d

for Figure 3.

Gin=1

London 9

Figure 4 – Solution for d

To find the d using the smith chart the normalized impedance of ZLTotal must be found and

indicated on the chart, this can be found using Equation 9

ZNormalized=ZLTotal

Z0. (9)

From there the distance Y LNis from the open circuit side of the smith chart, plus to where our

original circle hits Gin, starting from the open circuit side of the smith chart equals the distance d.

With the d found the initial Z¿ can be found using Equation 10

London 10

Z¿=Z0∗Z LTotal+ j Z0 tan( 2πd

λ0 )(Z0+ j ZLTotal tan( 2 πd

λ0 )), (10)

and with the initial Z¿ the initial ls can be found using Equation 11

ls=cot−1 (ℑ {1/Z¿ }∗Z0 ) λ0

2 π. (11)

Now that all the initial conditions have been found several equations were derived in

order to relate the change in the normalized frequency to SWR, and the final set up for Figure 3

can be seen in Figure 5

Figure 5 – SST Final Design

From Equations 5 and 6 it can be noted that Z¿will affect SWR and from Equation 10 it

can be noted that Z¿ is directly affected by the changing frequency. With this knowledge

Equation 12 can be derived

Z¿=Z0

ZL Total+ j Z0 tan( 2 πdλChanging

λ0)(Z0+ j Z LTotal tan( 2 πd

λChangingλ0))

. (12)

Because the impedance of the stub is, at f 0, supposed to cancel out the imaginary portion of the

admittance of Z¿ so that Z¿ matches Z0 Equation 13 can be derived from Equation 12 to solve

for the stub’s admittance

London 11

jβ stub=− j

Z0 s tan( 2πλChanging

λ0 ls). (13)

As seen in Figure 3 it can be noted that the stub impedance and Z¿ are in parallel, thus to find the

new Z¿, that can be used in Equation 6, Equation 14 was derived from Equation 3

Z¿ new=1

jβ+ 1Z¿

. (14)

In MATLAB Z¿ new can be input into Equation 6 which will be used in Equation 5 to generate the

different SWRs for each frequency.

To allow MATLAB to not over flow with information the changing frequency ranges

from zero to 600 [MHz], which will be plotted in intervals of 1 [MHz] Once the SWR graphs

have been created the bandwidths of both the quarter-wave and single-stub transmission lines

will be calculated using Equation 15

Bandwidth=|f 2− f 1|

f 0100 %. (15)

f 2 and f 1are the frequencies, one immediately after each other, where the Standing Wave Ratio

equals 2.

IV. Results

Quarter-Wave Transmission Line

Figure 5 displays the SWR vs. the normalized frequency for the quarter wave

transmission line.

London 12

Figure 6 – QWT SWR vs. ff 0

As the frequency neared f 0 the SWR gradually decayed from 2 to 1. This is because as

the frequency increased, still being divided by f 0, it eventually reaches its match frequency,

making Z¿ match up with Z0 once more. As seen in Equations 5 and 6 once Z¿ andZ0 match SWR

equals one, meaning that the amplitude of the wave being reflected back equals zero of that

which enters the load. This is known to be an ideal transmission line, in that all of the power

being sent is acquired by the load considering none of it will be reflected back to the generator.

Another observation is that as the frequency becomes greater or less that the f 0 the SWR

increases from 1 to 2. This occurs because as the frequency increases the change in Z¿ takes the

form of a sin graph, due the tan portion of Equation 7. As the frequency increases Z¿ increases,

London 13

lowering the SWR, until the frequency reaches f 0 of which as the frequency continues to

increase Z¿ decreases, increasing SWR.

If this trend continues the SWR will continue to repeat itself as seen in Figure 7 below

Figure 7 – QWT SWR vs. ff 0

Extended

Because the SWR graph is periodic any two points where the frequency generates the SWR to

read 2 can be used in Equation 15. As a result the bandwidth for the quarter-wave transmission

line was 200%.

Upon later inspection this bandwidth appeared to be almost too large for this system, but

through research and similar tests it was noted that it was because f 0 and c were equal why the

bandwidth was so large. During this research it was noted that if f 0 increased the bandwidth

London 14

would decrease and if f 0 would decrease the bandwidth would increase. The trend of increasing

f 0 can be seen in Table 2 below.

Table 2 – QWT Varying Bandwidth

Varying Bandwidth

f 0 Bandwidth

400 [MHz] 150 [%]

500 [MHz] 120 [%]

600 [MHz] 100 [%]

700 [MHz] 85.714 [%]

800 [MHz] 75 [%]

For this transmission line lowering f 0 below the value of c would result in an unfeasible

bandwidth considering that “the theoretical limit for percent bandwidth is 200%” (Wikipedia 4).

After considering these observations the bandwidth calculations were determined to be valid.

Single-Stub Transmission Line

Figure 8 displays the zoomed in version of the SWR vs. the normalized frequency graph

for the single-stub transmission line.

London 15

Figure 8 – SST SWR vs. ff 0

For a reasonable graph to be shown boundaries had to be set for the SWR. This is

because of how a zero frequency affects Z¿ and jβ stub. Imputing zero for λChanging into Equations

12 and 13 cancels out tan portion of the equation resulting in jβ stubto equal infinity and Z¿ to

equal ZLTotal. With an incredibly large imaginary impedance, Z¿ new becomes very small resulting

in Γ being equal to one, thus generating an infinite SWR. An infinite SWR is the exact opposite

of a zero SWR; it means that all the power transmitted to a load has been reflected back. This

makes sense since the stub is a short circuit, making it so that most if not all the current going

through the transmission line will go through the short rather than the load. This is not the best

result considering an ideal reflection is when no power has been returned to the generator, and all

of it has been absorbed by the load.

London 16

Another graph was generated to see the change in SWR as frequency increased. Figure 9

displays the SWR vs. the normalized frequency with the last frequency three times the amount of

the original end frequency

Figure 9 – SST SWR vs. ff 0

Extended

From Figure 9 it can be noted that the SWR graph for a single-stub transmission line is

not periodic like the quarter-wave transformer. This is probably because not only is the graph

affected by the tangent of Z¿ it is also affected by the tangent in jβ stub. Because the periods of

these two equations do not divide rationally they simply cannot form a periodic function, thus

not allowing for a consistent bandwidth.

After analyzing the graphs Figure 8 was used to determine the bandwidth of the single-

stub transmission line. To simplify the calculation the two frequencies closest to f 0that equaled 2

London 17

were used in Equation 15. The determined bandwidth of the transmission line was 35.667%, but

this bandwidth was not constant throughout the graph. Several other points were used to

calculate the bandwidth; the results can be seen in Table 3

Table 3 – SST Varying Bandwidth

Bandwidth

Frequencies [MHz]

(f 1 f 2)Bandwidth [%]

259 366 35.667

366 491 41.667

491 492 .333

492 493 .333

493 494 .333

494 548 18

548 549 .333

549 779 76.667

By observing Figure 9 and Table 3 it can be expected that the bandwidth will not become

a constant trend as long as the frequency continues to increase.

Though the data for both the quarter-wave and single-stub transmission lines were not

plotted out in increments of one hertz the data for both graphs seem valid. The outcomes of both

transmission lines were justified by prior knowledge, outside resources and equation analysis.

London 18

V. Conclusions

Overall it can be noted that the quarter-wave transmission line has a much more

predictable outcome when it comes to SWR and bandwidth, than the single-stub transmission

line. The consistency that the quarter-wave transmission line has that the single-stub lacks is due

to it not requiring second imaginary impedance portion, instead the only impedance acting on the

system is the load. As noted in the Single-Stub Transmission Line section of the Results, the

single-stub’s Z¿ new is not only being affect by Z¿ but also jβ stub. Due to the fact that the periods

of these two equations cannot be divided rationally they simply are unable to form a periodic

function, thus making a constant bandwidth impossible.

Another observation was that the quarter-wave transformer has a chance for the entire

signal wave can be absorbed by the load, while for the single-stub has a more likely chance for

the signal to be returned to the generator; this can be seen in Figures 7 and 9. In Figure 7 it can

be noted that SWR periodically hits 1 at multiples of the initial frequency, meaning that the load

has a chance to almost completely absorb the transmitted wave. Although, as seen in Figure 9,

there is a higher chance that the load will not be able to absorbed any portion of the wave but has

a single chance of absorbing almost all of the transmitted wave at the fundamental frequency.

Again considering the stub is a short circuit this conclusion can be validated.

Subsequently it can be noted that a single-stub transmission line would be best suited

when trying to only allow the use of certain frequencies. Due to the quarter-wave transmission

line’s consistent nature it can be used wherever consistency is needed.

London 19

VI. References

[1] Professor Chen. 2014. Transmission Lines (Impedance Matching) Notes 13. Available

http://www0.egr.uh.edu/courses/ECE/ECE3317/SectionChen/Class%20Notes/

[2] Stiles, J. 2014. 5.2 – Single-Stub Tuning [Online]. Available FTP:

http://www.ittc.ku.edu/~jstiles/723/handouts/section_5_2_Single_Stub_Tuning_p

ackage.pdf

[3] Wikipedia. 2014. Transmission Lines [Online]. Available FTP:

http://en.wikipedia.org/wiki/Transmission_line

[4] Wikipedia. 2014. Bandwidth (Signal Processing) [Online]. Available FTP:

http://en.wikipedia.org/wiki/Bandwidth_(signal_processing)

[5] Donohoe, J. Patrick. 2014. Impedance Matching and Transformation [Online]. Available

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

[6] Wikipedia. 2014. Quarter-wave Impedance Transformer [Online]. Available FTP:

http://en.wikipedia.org/wiki/Quarter-wave_impedance_transformer

[7] Chew, C.W..2014. Impedance Matching on Transmission Line ECE 350 Lecture Notes

[Online]. Available FTP: http://wcchew.ece.illinois.edu/chew/ece350/ee350-

10.pdf

[8] Bevelacqua, Peter. 2014. VSWR (Voltage Standing Wave Ratio) [Online]. Available FTP:

http://www.antenna-theory.com/definitions/vswr.php