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AC Electricity : Transmission Lines

Fifth Part of AC Electricity for Industrial Instrumentation

A two-conductor cable conveying an electrical signal whose period is short compared to the propagation time along the cable’s length is known as a transmission line. In low-frequency and/or physically small circuits, the effects of signal propagation go unnoticed because they are so brief. In circuits where the time delay of signal propagation is significant compared to the period (pulse width) of the signals, however, the effects can be detrimental to circuit function.

When a pulse signal is applied to the beginning of a transmission line, the reactive elements of that cable (i.e. capacitance between the conductors, inductance along the cable length) begin to store energy. This translates to a current draw from the source of the pulse, as though the pulse source were driving a (momentarily) resistive load. If the transmission line happened to be infinitely long, it would behave exactly like a resistor from the perspective of the signal source; i.e. it would never stop “charging.”

During the time when a transmission line is absorbing energy from a power source – whether this is indefinitely for a transmission line of infinite length, or momentarily for a transmission line of finite length – the current it draws will be in direct proportion to the voltage applied by the source. In other words, a transmission line behaves like a resistor, at least for a time. The amount of “resistance” presented by a transmission line is called its characteristic impedance, or surge impedance, symbolized in equations as Z0.

A transmission line’s characteristic impedance is a function of its conductor geometry (wire diameter, spacing) and the permittivity of the dielectric separating those conductors. If the line’s design is altered to increase its bulk capacitance and/or decrease its bulk inductance (e.g. decreasing the distance between conductors), the characteristic impedance will decrease. Conversely, if the transmission line is altered such that its bulk capacitance decreases and/or its bulk inductance increases, the characteristic impedance will increase. It should be noted that the length of the transmission line has absolutely no bearing on characteristic impedance. A 10-meter length of RG- 58/U coaxial cable will have the exact same characteristic impedance as a 10,000 kilometer length of RG-58/U coaxial cable (50 ohms, in both cases). The only difference is the length of time the cable will behave like a resistor to an applied voltage.

The following sequence illustrates the propagation of a voltage pulse forward and back (reflected) on an open-ended transmission line:

 

Transmission_Lines_Fig_056.JPG

 

The end result is a transmission line exhibiting the full source voltage, but no current. This is exactly what we would expect in an open circuit. However, during the time it took for the pulse to travel down the line’s length and back, it drew current from the source equal to the source voltage divided by the cable’s characteristic impedance (Isurge = Vsource / Z0 ). For a short amount of time, the two-conductor transmission line acted as a load to the voltage source rather than an open circuit.

An experiment performed with a square-wave signal generator and oscilloscope1 connected to one end of a long wire pair cable (open on the far end) shows the effect of the reflected signal:

Transmission_Lines_Fig_057.JPG

The waveform steps up for a short time, then steps up further to full source voltage. The first step represents the voltage at the source during the time the pulse traveled along the cable’s length, when the cable’s characteristic impedance acted as a load to the signal generator (making its output voltage “sag” to a value less than its full potential). The next step represents the reflected pulse’s return to the signal generator, when the cable’s capacitance is fully charged and is no longer drawing current from the signal generator (making its output voltage “rise”). A two-step “fall” appears at the trailing edge of the pulse, when the signal generator reverses polarity and sends an opposing pulse down the cable.

The duration of the first and last “steps” on the waveform represents the time taken by the signal to propagate down the length of the cable and return to the source. This oscilloscope’s timebase was set to 0.5 microseconds per division for this experiment, indicating a pulse round-trip travel time of approximately 0.2 microseconds. Assuming a velocity factor of 0.7 (70% the speed of light), the round-trip distance calculates to be approximately 42 meters, making the cable 21 meters in length.

The following sequence illustrates the propagation of a voltage pulse forward and back (reflected) on a shorted-end transmission line:

Transmission_Lines_Fig_058.JPG
The end result is a transmission line exhibiting the full current of the source (Imax = Vsource / Rwire ), but no voltage. This is exactly what we would expect in a short circuit. However, during the time it took for the pulse to travel down the line’s length and back, it drew current from the source equal to the source voltage divided by the cable’s characteristic impedance (Isurge = Vsource / Z0). For a short amount of time, the two-conductor transmission line acted as a moderate load to the voltage source rather than a direct short.

An experiment performed with the same signal generator and oscilloscope connected to one end of the same long wire pair cable (shorted on the far end) shows the effect of the reflected signal:

Transmission_Lines_Fig_059.JPG

 Here, the waveform steps up for a short time, then steps down toward zero. As before, the first step represents the voltage at the source during the time the pulse traveled along the cable’s length, when the cable’s characteristic impedance acted as a load to the signal generator (making its output voltage “sag” to a value less than its full potential). The step down represents the (inverted) reflected pulse’s return to the signal generator, nearly canceling the incident voltage and causing the signal to fall toward zero. A similar pattern appears appears at the trailing edge of the pulse, when the signal generator reverses polarity and sends an opposing pulse down the cable.

Note the duration of the pulse on this waveform, compared to the first and last “steps” on the open-circuited waveform. This pulse width represents the time taken by the signal to propagate down the length of the cable and return to the source. This oscilloscope’s timebase remained at 0.5 microseconds per division for this experiment as well, indicating the same pulse round-trip travel time of approximately 0.2 microseconds. This stands to reason, as the cable length was not altered between tests; only the type of termination (short versus open).

Proper “termination” of a transmission line consists of connecting a resistance to the end(s) of the line so that the pulse “sees” the exact same amount of impedance at the end as it did while propagating along the line’s length. The purpose of the termination resistor is to completely absorb the pulse’s energy so that none of it will be reflected back to the source.

Finally, we see the oscilloscope plot of the voltage signal (measured at the source end of the cable) with a  resistor of the (nearly) correct value connected to the far end of the cable:

Transmission_Lines_Fig_060.JPG

The pulse looks much more like the square wave it should be, now that the cable has been properly terminated2. With the termination resistor in place, a transmission line always presents the same impedance to the source, no matter what the signal level or the time of signal application. Another way to think of this is from the perspective of cable length. With the proper size of termination resistor in place, the cable appears infinitely long from the perspective of the power source because it never reflects any signals back to the source and it always consumes power from the source. A transmission line’s characteristic impedance will be constant throughout its length so long as its conductor geometry and dielectric properties are consistent throughout its length. Abrupt changes in either of these parameters, however, will create a discontinuity in the cable capable of producing signal reflections. This is why transmission lines must never be sharply bent, crimped, pinched, twisted, or otherwise deformed.

The speed at which an electrical signal propagates down a transmission line is never as fast as the speed of light in a vacuum. A value called the velocity factor expresses the propagation velocity as a ratio to light, and its value is always less than one:

 

Velocity factor = v / c

 

Where,

v = Propagation velocity of signal traveling along the transmission line

c = Velocity of light in a vacuum (3.0 × 108 meters per second)

Velocity factor is a function of dielectric constant, but not conductor geometry. A greater permittivity value results in a slower velocity (lesser velocity factor).

Data communication cables for digital instruments behave as transmission lines, and must be terminated at both ends to prevent signal reflections. Reflected signals (or “echoes”) may cause errors in received data in a communications network, which is why proper termination can be so important. For point-to-point networks (networks formed by exactly two electronic devices, one at either end of a single cable), the proper termination resistance is often designed into the transmission and receiving circuitry, and so no external resistors need be connected. For “multi-drop” networks where multiple electronic devices tap into the same electrical cable, excessive signal loading would occur if each and every device had its own built-in termination resistance, and so the devices are built with no internal termination, and the installer must place two termination resistors in the network (one at each far end of the cable).

The probe for a guided-wave radar (GWR) level transmitter is another example of a transmission line, one where the vapor/liquid interface creates a discontinuity: there will be an abrupt change in characteristic impedance between the transmission line in vapor space versus the transmission line submerged in a liquid due to the differing dielectric permittivities of the two substances. This sudden change in characteristic impedance sends a reflected signal back to the transmitter. The time delay measured between the signal’s transmission and the signal’s reception by the transmitter represents the vapor space distance, or ullage.

Details on the theory and function of radar level measurement will be discussed later in future posts.

1The signal generator was set to a frequency of approximately 240 kHz with a Th´evenin resistance of 118 ohms to closely match the cable’s characteristic impedance of 120 ohms. The signal amplitude was just over 6 volts peak-to- peak.

2The termination shown here is imperfect, as evidenced by the irregular amplitude of the square wave. The cable used for this experiment was a length of twin-lead speaker cable, with a characteristic impedance of approximately 120 ohms. I used a 120 ohm (+/- 5%) resistor to terminate the cable, which apparently was not close enough to eliminate all reflections.

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