Get our free email newsletter

Transmission Line Reflections at a Discontinuity

Foundations

In this article, we consider the effects of a discontinuity along the transmission line which occurs when the transmission line characteristic impedance changes from ZC1 to ZC2, as shown in Figure 1.

Figure 1: Discontinuity along the transmission line
Figure 1: Discontinuity along the transmission line

 

- Partner Content -

Magnetic Field Conversions

This download provides a table of magnetic field conversion factors of microvolts-per meter, gauss, picotesla, microampere-per-meter, weber per-square meter, and gamma with values of either 0 db or 1 as a starting point.

Let’s consider a voltage wave, vi1, traveling on transmission line 1 incident onto the junction, as shown in Figure 1(a). Upon its arrival at the junction the reflected wave, vr1, and the transmitted wave, vt2, are created.

The reflected voltage is related to the incident voltage by

1702_EC_rev_eq1(1)

where Γ12 is the reflection coefficient for the wave incident from the left onto the boundary, given by

1702_EC_rev_eq2(2)

- From Our Sponsors -

The transmitted voltage is related to the incident voltage by

1702_EC_rev_eq3(3)

where T12 is the transmission coefficient from the left to the right, given by

1702_EC_rev_eq4(4)

Similarly, for the wave incident on the boundary from the right, shown in Figure 1(b), we have

1702_EC_rev_eq5(5)

1702_EC_rev_eq6(6)

In this case, the reflection coefficient for the wave incident from the right is given by

1702_EC_rev_eq7(7)

and the transmission coefficient for the wave incident from the right is

1702_EC_rev_eq8(8)

Verification

The experimental setup and the circuit model for reflection measurements is shown in Figure 2. Its circuit model is shown in Figure 3.

Figure 2: Experimental setup for the discontinuity reflection measurements
Figure 2: Experimental setup for the discontinuity reflection measurements

 

Figure 3: Circuit for the reflection measurements
Figure 3: Circuit for the reflection measurements

 

A 10 Vpp (open-circuit voltage) pulse signal was sent from the function generator along the 6 ft-long RG58 coaxial cable (ZC1 = 50 Ω) connected to the 6 ft-long RG59 coaxial cable (ZC2 = 75 Ω) and terminated with an open circuit. The rise time of the waveform is tr = 2.5 ns. The voltages at the source (VS), the discontinuity (VD) and at the load (VL) were measured using the oscilloscope probes.

When the switch closes, the initial voltage wave is created at location z = 0. The value of this voltage is

1702_EC_rev_eq9

After about t =T= 9 ns (one-way travel time along RG 58) this waveform arrives at the discontinuity where it gets reflected and transmitted. The reflection coefficient (from the left) at the discontinuity is

1702_EC_rev_eq10

The reflected voltage at the discontinuity is

1702_EC_rev_eq11

The total voltage at the discontinuity at t = T is thus

1702_EC_rev_eq12

The reflected voltage wave propagates back towards the source and arrives there about at t = 2T later. The total voltage at the source becomes (after t = 2T+2tr).

1702_EC_rev_eq13

The incident wave that arrived at the discontinuity at t = T is also transmitted. The voltage transmission coefficient for the wave incident from the left is

1702_EC_rev_eq14

The transmitted voltage at z = 6ft is

1702_EC_rev_eq15

which, of course, is the same as the sum of the incident and transmitted voltages at t = T. These voltages are shown in Figure 4(a).

Figure 4: Measurement results
Figure 4: Measurement results

 

The transmitted voltage wave travels towards the load were it gets reflected with a load reflection coefficient equal to one (open load). The total voltage at the load is, therefore, at t = 2T is (shown in Figure 4(b)):

1702_EC_rev_eq16

The reflected voltage at the load (6V) travels back towards the discontinuity where it gets reflected and transmitted. Let’s look at the reflected voltage first. The reflection coefficient (from the right) at the discontinuity is

1702_EC_rev_eq17

The reflected voltage at the discontinuity is

1702_EC_rev_eq18

This reflected voltage wave propagates towards the load where it gets reflected. The total voltage at the load, at t = 4T,

1702_EC_rev_eq19

This is shown in Figure 4(c). The incident wave that arrived at the discontinuity (from the right) at t = 3T  is also transmitted.

The voltage transmission coefficient for the wave incident from the right is

1702_EC_rev_eq20

The transmitted voltage (from right to left) at z = 6ft is

1702_EC_rev_eq21

Resulting in a total voltage at the discontinuity of

1702_EC_rev_eq22

This is shown in Figure 4(d). Finally, the transmitted voltage of 4.8 V arrives at the source at t = 4T, resulting in the source voltage rising to (shown in Figure 4(d)).

1702_EC_rev_eq23

This process continues until the steady state value (of 10 V at all locations) is reached.

author_adamczyk-bogdan

Dr. Bogdan Adamczyk is a professor and the director of the EMC Center at Grand Valley State University (GVSU). He is also the founder and principal educator of EMC Educational Services LLC (www.emcspectrum.com) which specializes in EMC courses for the industry. Prof. Adamczyk is the author of the upcoming book “Foundations of Electromagnetic Compatibility with Practical Applications” (Wiley, 2017). He can be reached at profbogdan@emcspectrum.com

 

Related Articles

Digital Sponsors

Become a Sponsor

Discover new products, review technical whitepapers, read the latest compliance news, trending engineering news, and weekly recall alerts.

Get our email updates

What's New

- From Our Sponsors -

Sign up for the In Compliance Email Newsletter

Discover new products, review technical whitepapers, read the latest compliance news, trending engineering news, and weekly recall alerts.