Transmission Line Experiments At Low Cost

The GHz-range equipment and components normally required for the basic experiments in transmission line and microwave topics are expensive and often beyond the budgets of small programs. The LC lumped-element transmission line model provides an economical alternative for such experiments. Appropriate choice of inductance and capacitance values for the LC sections makes it possible to establish standing waves several half-wavelengths long on physically small models at operating frequencies well below 1 MHz. At the low operating frequencies, measurement and data collection can be accomplished using general-purpose lab instruments that are readily available in most basic laboratories. The prototype “lines” built by the author and used in a transmission line course are described and the lab exercises and procedures for determining the propagation properties such as standing wave pattern, phase constant, and wavelength are outlined. Typical experimental results are also provided. INTRODUCTION The experiments for the study of the basic characteristics of the propagation along transmission lines are performed at frequencies above 1 Ghz. These experiments usually require the measurement of voltage or the electric field along a test section of the transmission line. From such data, the standing wave pattern, the phase constant, wavelength, and the VSWR of the line can be obtained. To yield these results, the data must cover a span of the line that includes at least one, and preferably several, λ/2 of the standing wave. This requirement can be met for a physically short line if the operational frequency is high. However, since the components and particularly the instrumentation at these frequencies are costly, a lab setup for transmission line experiments can be prohibitively expensive for a small program. For example, a minimum setup, consisting of a signal source, one or more detectors, a detection instrument, a frequency counter, a slotted line section, and a few other components such as attenuators, terminations, and loads can run from $15,000 to over $50,000 per station, depending on the frequency range. Since the basic characteristics of the propagation phenomena are the same for lines operating at different frequencies, an obvious low cost approach would be to utilize lines that operate at a low enough frequency where general-purpose instrumentation could be used for the measurements. Although theoretically possible, this is physically impractical, since the line sections that would be required to obtain the needed data would be very long. For example, the length of the line section required to span even a single λ/2 of the standing waves at an operating frequency of 10kHz (with a propagation velocity equal to the velocity of light in vacuum) is 15 kilometers! P ge 390.1 A practical way of getting around the need for high frequency instrumentation is to use a physical model of the transmission line rather than an actual line. The LC-section lumpedelement model is such an example, in which with appropriate choice of LC values, it is possible to achieve multiples of λ/2 of the standing wave within a practical length at low frequencies. THE LC LUMPED-ELEMENT MODEL The idea of using a lumped-element circuit that experimentally behaves like a transmission line is based on the classical approach to analyzing the wave propagation on transmission lines. In this approach, the line is considered to be composed of an infinite number of sections, each made of discrete lumped elements R, L, C, G (see Fig 1.) Figure 1. Lumped-element equivalent model of a transmission line. By applying the circuit laws to a section and solving the resulting differential equations, one is able to find the voltage and current of the line as functions of time and distance. 1,2 These solutions and some of the other results for time-harmonic operation may be written as v(x,t) = [Ae + Be ]e i(x,t) = (1/Zo)[Ae Be ]e Zo = [(R + jωL)/( G + jωC)] γ = α + j β = [(R + jωL)( G + jωC)] in which A and B are arbitrary constants, to be determined from the input voltage and the load (i.e. the boundary conditions), and Zo and γ are the characteristic impedance and propagation constant of the line. The latter two, for low-loss or loss-less lines (R=G=0) simplify to, Zo = [L/C] γ = 0+jβ = jω[LC] 1/2 . The line constants R, L, C, and G, are distributed properties, uniformly spread along the length of the line. They are, in effect, microscopic properties even though they are given in macroscopic units of per meter, per kilometer, etc. Therefore, the lumped-element model as a circuit description of the line is accurate if the section length for which the lumped components are defined is made infinitesimal. It would be expected, then, that a model (proposed here) that has a finite number of sections made from discrete components should behave, at least in an approximate manner, like a transmission line.