Propagation, Termination and Discharge Labwork for Coaxial Transmission Lines



Motivation: To gain practical experience with pulses, reflections, impedances and discharges on transmission lines.

Method: Measure propagation delay, cable impedance, cable attenuation and cable transient characteristics for 100 ft and 500 ft cables of RG-58/AU.


Equipment and Supplies Used


The equipment I used is commonly found in electronic technician training and workspaces.
Scope, Function Generator and Spool of Cable
Snapshot of Lab Equipment



Distinction Between Interior and Exterior Cable Properties


The purpose of a coaxial cable transmission line is to provide a protected environment to send a signal from one end to the other. The interior of this cable will have a characteristic L' (inductance per unit length), C' (capacitance per unit length), R' (series resistance per unit length) and G' (leakage conductance per unit length). With decent cables, R' and G' are small, and the cable is characterized by a dynamic impedance Z = sqrt(L'/C') and velocity v = 1/sqrt(L'*C').

My lowest priced cable carefully does not indicate the manufacturer. Using Belden's specs as a reference, they publish numbers for their RG-58/AU as

  • L' = 80.1 nH/ft
  • C' = 28.5 pF/ft
  • R' = 15.5 mOhms/ft
  • dt = 1.54 ns/ft = 0.66 c

    Using these numbers, we have a characteristic impedance of 53 Ohms, nominally called out as 50 Ohms. As most terminators are not 53 Ohms, small reflections are normal on coax.

    Using the 500 ft spool, with both coax ends open, I expect 14.25 nF between center pin and shell, and I measure 12.88 nF. Placing a shorting plug on the far end of the cable, I measure 86 uH inductance between the center pin and shell on the near end. I expected 40 uH, and attribute the factor of two to the effective path length being 1000 ft, being there and back, so to speak.

    The two ends of the cable have a substantial exterior inductance, which is strongly influenced by cable layout. On a spool, we get substantial mutual inductance terms between turns. For the 500 ft cable, the inductance from shell to shell is 8.71 mH. The inductance from center pin to center pin is 8.75 mH. This inductive decoupling between cable ends helps limit ground currents at high frequency, while allowing destructive ground loop currents at low frequencies (such as 60 Hz).

    Using the DVM, the resistance between center pin to center pin is measured at 5.8 Ohms (nominally 20 gauge should be about 5 Ohms), and the resistance between shell to shell is measured at 5.9 Ohms. With a shorting plug on the far end, the center pin to shell resistance at the near end is 11.5 Ohms.


    Measurement of Propagation Delay and Attenuation



    The propagation delay is measured using the function generator main output feeding one end of the cable and Ch1 of the scope, the other end of the cable feeding a 50 Ohm terminated Ch2 of the scope, and external triggering of the scope using the function generator sync output. This is shown schematically in the figure below.

    Propagation Delay Measurement Hookup
    Propagation Delay Measurement Hookup



    For the 500 ft spool, we see a delay of 750 ns corresponding to a speed of 667 ft/us.
    We can calculate the dB attentuation per 1000'.
    Loss/1000ft = -(1000ft/500ft)*20*log_10(4.6V/5.9V) = 4.3 dB/1000'.

    500 ft Propagation Delay Measurement
    500 ft Spool Propagation Delay Measurement



    The negative-going reflection on Ch1 beginning at 1750 ns (seventh large division) indicates actual cable impedance is more than 50 Ohms.

    Now we repeat the same measurements using the 100' cable

    For the 100 ft spool, we see a delay of 150 ns corresponding to a speed of 667 ft/us.
    We can calculate the dB attentuation per 1000'.
    Loss/1000ft = -(1000ft/100ft)*20*log_10(5.28V/5.68V) = 6.3 dB/1000'.

    100 ft Propagation Delay Measurement
    100 ft Spool Propagation Delay Measurement


    The attenuation numbers are higher than I like. I am not happy with the number from the 500' spool, and I believe the 100' spool is overestimated. Cable, in general, has higher losses at higher frequencies due to skin effect and dielectric losses. For the waveforms above, we are in the 20 MHz regime, and would expect about 2dB/1000' based upon tables at http://www.bcar.us/Coax%20Cable%20Specifications.htm. A better technique seems to be the reflected open pulse technique used a bit below, which give results more in accordance with the reference above.


    Reflections from Open 500' Cable



    In this section, we remove the terminator from Ch2. For the 500' spool, we have a nice reflection.

    500 ft Open Cable Reflection
    500 ft Spool Open Cable Reflection



    Note the change in scale on Ch2. During reflection, we get double the expected voltage on the output cable. In this case, we put in a 6V pulse, which attenutates to 85% (5V), but gets doubled at the open during reflection to 10V. The reflected wave propagates back to the input where we see the twice attenuated 4.4V pulse on Ch1. Measuring attenuation using the reflected pulse (taking into account 1000' travel) gives an attenuation of 2.7 dB/1000', much more in accordance with expectations.

    91 Ohm Termination on 500' cable



    We now use a 91 Ohm (nominally 93 Ohm) terminator. We see the echo substantially reduced. The standard reflection coefficient formula is Gamma = (R - Z_0)/(R + Z_0). For a 91 Ohm, with a 6V input pulse, with 85% transmission there and back, we expect 6V*0.85*0.85*(91-50)/(91+50) = 1.26V. This return echo seems a bit weak, but in the right range.

    500 ft 91 Ohm Cable Reflection
    500 ft 91 Ohm Cable Reflection




    75 Ohm Termination on 500' cable



    We now use a 75 Ohm terminator. We see the echo is almost gone. We expect 6V*0.85*0.85*(75-50)/(75+50) = 0.87V. Again, we are in the right range.

    500 ft 75 Ohm Cable Reflection
    500 ft 75 Ohm Cable Reflection


    15 Ohm Termination on 500' cable



    We now use a 15 Ohm terminator. Notice that the echo has gone negative. We expect 6V*0.85*0.85*(15-50)/(15+50) = -2.33V. This is in the correct range.

    500 ft 15 Ohm Cable Reflection
    500 ft 15 Ohm Cable Reflection


    Zero Ohm Termination on 500' cable



    We now use a 0 Ohm terminator. Notice complete reflection with inverted sign.

    500 ft 0 Ohm Cable Reflection
    500 ft 0 Ohm Cable Reflection


    Function Generator, no transmission line cable



    We now use hook the function generator directly to the scope for reference. We see rise and fall times on the order of 50 ns, and unloaded output level of 10V.

    Function Generator Direct to Scope
    Function Generator Direct to Scope




    Reflections on the 100' Line



    The propagation delay on the 500' samples was longer than the pulse duration, leaving pulse and echo distinct. With the 100' cable, we will have reflections superimposing on the transmitted signal, with constructive and destructive interference, depending upon the terminating resistance compared to the line impedance.

    Open 100' Line Showing Constructive Interference with Echo
    Open 100' Line Showing Constructive Interference with Echo

    Our input pulse is 500 ns long. During the transmission, the echo arrives at 450ns, taking the Ch1 voltage up almost double. At 650ns, the input voltage drops, yet the continuing reflection keeps the input voltage above zero. The echo finally ends near the end of this trace at 950 ns.

    Gallery of Reflections on the 100' Line



    The next cluster of images illustrate different levels of overlap between pulses and their open echoes.

    Open 100' Line Showing Echo Frequency Doubling
    Open 100' Line Showing Echo Frequency Doubling



    Open 100' Line Showing Echo Merging
    Open 100' Line Showing Echo Merging



    Open 100' Line Showing Echo Merging Pulse Duration Doubling
    Open 100' Line Showing Echo Merging Pulse Duration Doubling



    Open 100' Line Showing Echo Mesa Shapes
    Open 100' Line Showing Echo Mesa Shapes



    Open 100' Line Showing Individual Pulses and Reflections
    Open 100' Line Showing Individual Pulses and Reflections


    100' Line with 91 Ohm Termination





    Open 100' Line with 91 Ohm Termination
    Open 100' Line with 91 Ohm Termination


    100' Line with 75 Ohm Termination





    Open 100' Line with 75 Ohm Termination
    Open 100' Line with 75 Ohm Termination


    100' Line with 15 Ohm Termination





    Open 100' Line with 15 Ohm Termination
    Open 100' Line with 15 Ohm Termination


    100' Line with Zero Ohm Termination





    Open 100' Line with Zero Ohm Termination
    Open 100' Line with Zero Ohm Termination


    Capacitive Loading Gallery



    These next images show the 100' line terminated with a 100nF capacitor. These spikes are reminiscent of fangs, or attic roofing nails. Dynamically, the capacitor looks like a short. This response will be set upon the RC response of the capacitor and signal source. The initial spike on Ch 1 is the initially applied voltage divided between the function generator internal impedance and the line impedance. The echo off the capacitor is of opposite polarity, and pretty much cancels the applied pulse, resulting in a spike of applied signal level at twice the cable length duration. Now, under quasi-steady conditions, the line (and load capacitor) are being charged with a time constant tau=50 Ohms * 100nF = 5us. This corresponds to a rise time 2.2*tau = 11 us. Note we charge to twice the pulse height. The initial pulse is at half the open circuit voltage, due to the 50 Ohm internal resistance of the function generator being divided into the 50 Ohm cable impedance. During the long charge time, the internal resistance of the function generator becomes insignificant for final voltage. Likewise, the cable impedance becomes insignificant during the RC charging time constant calculation, and just the function generator R seems to matter.

    100 nF capacitor load 8.88kHz rep rate
    100 nF capacitor load 8.88kHz rep rate



    100 nF capacitor load 28.8kHz rep rate
    100 nF capacitor load 28.8kHz rep rate



    100 nF capacitor load 28.8kHz rep rate Expanded Sweep
    100 nF capacitor load 28.8kHz rep rate expanded sweep



    100 nF capacitor load 500 ns
    100 nF capacitor load 500 ns

    Notice the spike pulse duration (300 ns) is twice the cable propagation delay (150 ns).

    100 nF capacitor load almost shorted pulses
    100 nF capacitor load almost shorted pulses


    Dual Cable End Feed Gallery



    Tie the two ends of the cable together using a BNC tee to create a storage ring. Feed this ring with a function generator. We now have a 25 Ohm connection, initially at zero, charging eventually to our pulse voltage.

    These next images show the pulse generator feeding both ends of the coax line tied together with a BNC tee, no resistive termination and feed from the function generator to the middle of the BNC tee. We see multiple reflections on the increasing and decreasing voltage.

    The first step is consistent with a 11V source feeding through 50 Ohms (the function generator) driving two parallel 50 Ohm loads at zero volts initially. At the end of one cable propagation delay, we are now at 1/3 our applied voltage. At this point, for reasons I don't yet fully understand, but suspect relate to viewing the loop discharging low voltage rather than the function generator applying positive voltage, we now achieve 2/3 of the remaining voltage on each cable transit. The increasing roundness of the echo pulses reflects the higher attenuation of the higher frequency components of the pulses.

    100' line being dual fed
    100' line being dual fed



    100' line being dual fed 100 us/div
    100' line being dual fed 100 us/div



    100' line being dual fed longer pulses
    100' line being dual fed longer pulses



    Reference Waveform Direct to Scope
    Reference Waveform Direct to Scope



    100' line being dual fed longer still pulses
    100' line being dual fed longer still pulses



    500' Spool Loop double fed
    500' Spool Loop double fed

    Feeding the 500' spool, we see the initially step is larger than the 1/3 of the 100' cable. Numerically, this is consistent with a cable impedance of 62 ohms, rather than 50. I see the remaining steps being consistent with achieving 2/3 of the remaining voltage on each transit.

    Reference Pulse for 500' Loop
    Reference Pulse for 500' Loop

    To better understand the step response, I greatly reduced the pulse duration to more clearly see the reflection behavior on the storage ring. The reference input voltage unloaded by the cable is shown below.

    Reference Pulse Direct to Scope
    Reference Pulse Direct to Scope

    Our input voltage peak is 11V.

    500' double fed line with reflection
    500' double fed line with reflection

    Our loaded peak is 4V, consistent with 11V in series with 50 Ohms feeding parallel 57 ohm cables. When this input is place upon the coil, we launch clockwise and counter-clockwise pulses in the storage ring. After one cable delay, we see the 4V pulse again, with a bit of edge rounding due to high frequency losses. The clockwise cable has an internal 50 Ohm impedance, and sees two parallel paths, being the loop continuation and the feed from the function generator. This results in a 1:3 attenuation for the clockwise cable. In a similar fashion, the counter clockwise cable also sees a 1:3 attenuation. For the longer duration pulse, the plateaus we would see would be 1/3 (initial), +1/3 (first circulation), +1/9 (second circulation), +1/27 (third circulation) and so on.


    Discharge of Charged 500' Cable



    The coax cable is charged to 3V with a circuit consisting of a 9V battery in series with a 1M resistor, with Ch1 and Ch2 monitoring the two ends of the cable. Given the scope impedance of 1M, this results in a divider resulting in 3V across the cable and scope inputs.

    Resistor is Manually Connected to Ch1 Thus Discharging the Line
    Resistor is Manually Connected to Ch1 Thus Discharging the Line

    In the illustration above, a 15 Ohms resistor is shown being connected to Ch 1. In the following galleries, the specified resistor are connected to Ch1 or Ch2 as called out in the legend.

    91 Ohm Ch2 250 ns/div
    91 Ohm Ch2 250 ns/div



    91 Ohm Ch2 500 ns/div
    91 Ohm Ch2 500 ns/div showing multiple RC decay echoes

    We see that upon connecting the 91 Ohm load on Ch 2, that a reduced voltage wave propagate toward Ch 1, reaching Ch 1 750ns later. This low voltage wave reflects off the 1M load on Ch 1, and reflects back toward Ch 2. Ch 2 voltage does not begin to decline until after two cable propagation time lengths. During this time, the voltage at Ch 2 reflects a 3V source in series with 50 Ohms (line impedance) driving a 91 Ohm load, namely 1.9 to 2 V depending upon which trace we look at.

    After the line has discharged, we then see repeated R/C discharge echoes, as emphasised in the second trace above.

    Next, we change the terminating resistance to 75 Ohms.

    75 Ohm Ch2 500 ns/div
    75 Ohm Ch2 500 ns/div



    75 Ohm Ch1 500 ns/div
    75 Ohm Ch1 500 ns/div

    The output voltage during discharge is 1.8V being 3V*75/125. The discharge is symmetrical between Ch1 and Ch2. Battery placement is not important.

    Next, we terminate with 50 Ohms

    50 Ohm Ch2 500 ns/div
    50 Ohm Ch2 500 ns/div

    The output voltage is consistent with a 3V source in series with 50 Ohms due to line impedance driving a 50 Ohm load. This configuration is known as a transmission line pulse shaping network, and has been remarked upon by Ivor Catts as evidence of bidirection traffic in statically charged lines. We note that in all these examples so far, the terminating end of the cable has maintained voltage for the duration of transit to the far end of the cable and back.

    The next set will work with 15 Ohm and Zero Ohm terminations, resulting in reverse polarity reflections.

    15 Ohm Ch2 500 ns/div
    15 Ohm Ch2 500 ns/div

    The plateau voltage is 3*15/65 = 0.7 V.


    Zero Ohm Ch2 500 ns/div
    Zero Ohm Ch2 500 ns/div

    Variable contact resistance is becoming a problem with manual connection. I understand why Tektronix used reed relays for their pulse line generators.




    Discharge of Charged 100' Cable



    The coax cable is charged to 3V with a circuit consisting of a 9V battery in series with a 1M resistor, with Ch1 and Ch2 monitoring the two ends of the cable. Given the scope impedance of 1M, this results in a divider resulting in 3V across the cable and scope inputs.

    91 Ohm Ch2 100 ns/div
    91 Ohm Ch2 100 ns/div 100'



    75 Ohm Ch2 100 ns/div
    75 Ohm Ch2 100 ns/div 100'



    50 Ohm Ch2 100 ns/div
    50 Ohm Ch2 100 ns/div 100'



    15 Ohm Ch2 100 ns/div
    15 Ohm Ch2 100 ns/div 100'



    Zero Ohm Ch2 100 ns/div
    Zero Ohm Ch2 100 ns/div 100'

    In each of these cases, the output voltage during the plateau was a bit higher than I expected. I attribute this to contact resistance due to manual connection.


    Conclusions

    1. Transmission lines carry bidirectional signals propagating at their characteristic velocity at all times.
    2. The speed for RG-58 coax is 500' in 750 ns or 8"/ns.
    3. The characteristic impedance for RG-58 is a bit above 50 Ohms.
    4. Reflections for impedances above Z_0 maintain polarity. This is consistent with the reflection coefficient Gamma = (R-Z_0)/(R+Z_0).
    5. Reflections for impedances below Z_0 reverse polarity. This is consistent with the reflection coefficient Gamma = (R-Z_0)/(R+Z_0).
    6. The attenuation characteristics of a cable are best measured by using a single channel looking at the reflected pulse off the open back end. This eliminates scope base line and channel match ambiguities.
    7. When a pulse hits an open cable, the voltage at the open is twice the incoming voltage at that open. This is consistent with superposition of reflected and incoming waves, as well as the reflection coefficient Gamma.
    8. When a pulse hits a shorted cable, the voltage at the short is zero. This is consistent with superposition of reflected and incoming waves, as well as the reflection coefficient Gamma.
    9. When an open transmission line is charged to a DC voltage through a large resistor, we can treat the cable as having bidirectional, reflected signals totally reflecting off the ends of the line. The internal, unidirectional voltage component will be half the applied voltage, as the observed voltage is the superposition of the left and right going waves.
    10. When this charged line is discharged into a load, the duration of the initial pulse corresponds to twice the transmission line length regardless of load resistance. This corresponds to clearing propagation of the forward and reflected waves out of the line into the load. Alternatively, we don't get reduced output voltage until the lower voltage pulse caused by the termination connection propagates through the cable, reflects from the back end, and arrives at the termination point. This behavior is the basis for the pulse forming transmission line, and was known at least as early as the 1940's.
    11. The voltage across a load resistor connected to a charged line models as the charged voltage sent through a series impedance (the cable impedance) to the load resistor. V_r = V_c*R/(Z_0 + R). This is consistent with superposition of impressed and reflected waves using the Gamma reflection coefficient with the line voltage being half the cable charging voltage.