Transmission Lines
Part 1 of this article series on the world of real schematics ended with a look at wiring. Picking up where that left off, this article looks at real schematics from a transmission line perspective. George shows how transmission line characteristics can be represented by breaking them into schematic elements.
Our excursion into the world of real schematics ended last month (December, Circuit Cellar 341) with wiring. We saw that as long as an AC source connects to a load by conductors shorter than one eighth of the source’s signal wavelength λ, the schematic diagram Figure 1a with lumped component values is a fair rendering of the circuit. The source wavelength λ in meters is approximately c/f where c is the speed of light in meters/second and f is the signal’s frequency in Hertz. The speed of light is slightly less than 300 × 106 meters/second—or 299.792458 × 106 m/s to be precise. In engineering and many sciences 300 × 106 m/s is considered a good approximation.
All around us you can see power cords, coaxial TV and computer cables, twin leads, telephone wires and many other wired interfaces. Once their length comes close to one-eighth of the shortest wavelength they carry, they begin to take on transmission line characteristics. A schematic rendering of such a circuit is shown by Figure 1b. Keep in mind, though, that λ/8 is an empirical value at which the transmission line aspects could begin to be observed. It’s not a precise wavelength at which the circuit characteristics suddenly change.
You can describe a transmission line as consisting of an infinite number of infinitesimal resistors, inductors and capacitors spread along its entire length as depicted by Figure 1b. Letter G stands for conductance, which is the reciprocal of resistance or 1/R. The unit of conductance is Siemens expressed by symbols S, Ω-1, ℧ or the now obsolete “mho.”
The sum of resistances R in Figure 1b represents the serial resistance of the conductor and the sum of Gs the leakage between the conductors. Because most transmission lines are made of quality materials, the resistance and conductance elements are usually negligible with respect to the circuit characteristics. It is customary to call such transmission lines lossless and ignore the resistive elements as is illustrated by Figure 1c. Because its elements are infinitely small, the cable characteristics are specified by their sums per one meter of length.
BUCKET BRIGADE
Figure 2, albeit simplified, explains the principle of a transmission line operation. You can compare a transmission line to a bucket brigade. Just follow along with Figure 2. Initially, when the switch SW is open (Figure 2a), there is no voltage along the line and across the load. Closing the switch connects a 5 VDC source to the line and capacitor C1 begins to charge through inductor L1 up to the source voltage level (Figure 2b). During this process some energy is stored in L1, some begins to trickle to C2 and if the switch SW remains closed, the second element L2-C2 begins to charge (Figure 2c). This is followed by L3-C3 and so forth until the wave reaches the load (Figure 2d). Because during the charging process no current flows through the load, the impedance loading the source is determined by the values of the inductance and capacitance only. This constitutes the characteristic impedance of the transmission line:
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From the above quation you can see that the characteristic impedance of a transmission line is purely resistive. In case we needed to consider the line losses, that equation can be expanded by adding Rs and Gs:
where ω is the angular frequency equal to 2πf, and f is the frequency.
The characteristic impedance of the majority of transmission lines, for example coaxial cables is 50 Ω. This somewhat arbitrary value has been chosen based on several observations. First, as shown in Part 1 of this article series, a large resistance is susceptible to the parasitic capacitance, while a low resistance is mainly affected by its parasitic inductance. What we want, therefore, is a characteristic impedance to be somewhere in between, such that filters, amplifiers and other components can be reasonably sized and built.
Coincidentally, a coaxial cable with air dielectric designed for maximum power transfer happens to have the characteristic impedance of 30 Ω. An air dielectric coax designed for the minimum loss has its characteristic impedance 77 Ω, so the 50 Ω cable impedance is a good compromise. Furthermore, a coaxial cable with a Teflon dielectric optimized for the least loss has 50 Ω characteristic impedance and the impedance of a monopole antenna is 50 Ω too. Cables with 75 Ω impedance and air dielectric are manufactured for applications where losses must be minimized due to long cable runs—such as for cable TV distribution.
You can also describe the transmission line by a waveguide model. The electromagnetic wave moving through a transmission medium comprises a magnetic field related to the wave current and an electric field related to the wave voltage. The wave travels through the medium at velocity ν close to the speed of light, reduced by the medium’s permittivity ε and permeability μ. Consequently, the wave velocity is:
and the two models are related such that we can write:
In cables where the wire pairs are separated—for example by Teflon—the wave travels at roughly 70% of the speed of light in vacuum. The equation above shows that signal propagation within a wire transmission line is slower than through vacuum, determined by the transmission line’s velocity factor ν. For instance, Cat-5e cable’s velocity factor ν = 64%. An electromagnetic wave traveling in vacuum needs 3.34 ns to travel one meter, but it takes 5.21 ns for it to go the same distance in Cat-5e cable. In other words, 1 m of Cat 5e cable delays a signal by 1.88 ns. This phenomenon is employed in signal processing for constructing delay lines. At very high frequencies transmission lines are often replaced by waveguides due to their significantly lower loss, typically around 0.2%. Waveguides, essentially metal pipes using air dielectric, are another topic.
When a traveling electromagnetic wave is confronted by an abrupt change in impedance, a part of its energy continues on its way, while the remainder is reflected back to the source. That’s why impedance matching is so important for RF power transfer. The origins of power transmission physics date back to 1840, when Moritz von Jacobi postulated that maximum power transfer from a source to a load is obtained when their impedances match. But even if maximum power transfer is not your concern, the incidental and reflected waves combine, creating standing waves that can and often do destroy the signal integrity. The degree of impedance matching is expressed by Standing Wave Ratio (SWR), calculated as:
or
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Achieving SWR = 1 is perfect, but SWR less than 2 is still considered good match. Voltage Standing Wave Ratio (VSWR) is frequently used instead of SWR because it is easier to measure. High SWR means that the reflected energy turns into heat which must be dissipated by the cable.
STUBS AND BALUNS
Stubs are fractions of transmission lines used for impedance transformation, matching and as resonant circuits. A λ/4 long stub, shorted at its output has high impedance input. Conversely, the input impedance is low when the λ/4 stub is open-ended. A λ/2 stub works the other way around. Consequently, a stub impedance depends on its length. This phenomenon is the basis for construction of baluns—devices for matching impedances of balanced with unbalanced transmission lines [1] among other uses. Figure 3 illustrates how a stub can be used to match impedance of a load to the impedance of a transmission line.
This technique is useful, for example, for matching a RF amplifier output to an antenna. You calculate the stub length Lstub and its location dstub based on the frequency, source impedance and the load characteristics. But those calculations are outside the scope of this article. Transmission lines called microstrips are created on PCBs. They behave just as wire transmission lines but their design requires special skills.
All transmission lines, when properly terminated by their characteristic impedance, appear as infinitely long, no matter what their actual length might be. This applies to cables as well as PCB traces. High frequency digital signals with short rise time—clocks included—have bandwidth often extending into high Gigahertz. Just a few centimeters of a PCB trace can become a transmission line. To protect data integrity, it is imperative that such traces are properly terminated. Moreover, remember that signals on a PCB trace propagate at a speed of about 150 mm/ns (6”/ns)! This delay must be considered during the PCB layout when several signals are required to arrive at the same time.
In this two-part series, we discussed the behavior of passive components when exposed to alternating voltages and currents. The Circuit Cellar article materials webpage provides other sources for those interested in finer details. Next month we’ll take a closer look at infrared sensors.
For detailed article references and additional resources go to:
www.circuitcellar.com/article-materials
Reference [1] as marked in the article can be found there.
PUBLISHED IN CIRCUIT CELLAR MAGAZINE • JANUARY 2019 #342 – Get a PDF of the issue
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George Novacek was a retired president of an aerospace company. He was a professional engineer with degrees in Automation and Cybernetics. George’s dissertation project was a design of a portable ECG (electrocardiograph) with wireless interface. George has contributed articles to Circuit Cellar since 1999, penning over 120 articles over the years. George passed away in January 2019. But we are grateful to be able to share with you several articles he left with us to be published.
