RADIOFREQUENCY CIRCUIT ELEMENTS

Transmission lines

Before proceeding to the subject of com­munications circuits, it is necessary to 'deal briefly with the interesting subject of transmission lines. Transmissidn lines play a central role in radiofrequency circuits, where they are used to pipe signals around from one place to another within a circuit, and often to an antenna system. Transmission lines provide one of the most important exceptions to the general principles that a signal source ideally should have a source impedance small compared with the impedance of the load being driven and that the load should present an input impedance large compared with the source impedance driving it. The equivalent rule for transmission lines is that the load (and possibly the source) should present an impedance equal to the characteristic impedance of the line. The line is then "matched." Transmission lines for signals of moderate frequency (up to 1000MHz, say) come in two major types: parallel conductors and coaxial line. The former is typified by the inexpensive molded 300 ohm "twin lead" used to bring the signal from a television antenna to the receiver, and the latter is widely used in short lengths with BNC fittings to carry signals between instruments. In the domain of ultra-high-frequency circuitry there are "stripline" techniques that involve parallel conductor transmission lines as part of the actual circuit, and at the higher "microwave" frequencies (up­wards of 2GHz, say) conventional lumped circuit elements and transmission lines are replaced by cavity and waveguide techniques, respectively. Except at these extremes of frequency, the familiar coaxial cable is probably the best choice for most radiofrequency applications.  Compared with parallel conductor line, a properly matched coax line has the advantage of being totally shielded, i.e., there is no radiation or pickup of external signals.

Characteristic impedance and matching

A transmission line, whatever its form, has a "characteristic impedance" Z₀, meaning that a wave moving along the line has a ratio of voltage to current equal to Z₀.  For a lossless line, Z₀ is resistive and equal to the square root of L/C, where L is the inductance per unit length and C is the capacitance per unit length. Typical coaxial lines have impedance in the range of 50 to 100 ohms, whereas parallel conductor lines have impedances in the range of 300 to 1000 ohms.When used with high-frequency (or short-rise-time) signals, it is important to "match" the load to the characteristic impedance of the line. The important facts are the following: (a) A transmission line terminated with a load equal to its characteristic impedance (resistance) will transfer an applied pulse to the termination without reflection. In that case all the power in the signal is transferred to the load. (b) The impedance looking into such a terminated line, at any frequency, is equal to its characteristic impedance (Fig. 1).

Figure 1

This is surprising at first, since at low frequencies you tend to think of a length of coax as a small capacitive load, generally a pretty high (capacitive) impedance. Also, at low frequencies (wavelength » length of cable) there is no need to match the line's impedance, provided you can handle the capacitance (typically 30pF per foot). If the cable is terminated with a resistor, on the other hand, it magically becomes a pure resistance at all frequencies.

Mismatched transmission lines

A mismatched transmission line has some interesting, and occasionally useful, properties. A line terminated in a short circuit produces a reflected wave of opposite polarity, with the delay time of the reflected wave determined by the electrical length of the line (the speed of wave propagation in coax lines is about two-thirds the speed of light, because of the solid dielectric spacing material). You can see the reason for this, since the short circuit enforces a point of zero voltage at the end; the cable produces this obligatory condition by creating a wave of opposite phase at the short. In similar manner, an open-circuited cable (boundary condition of zero current at the end) produces a noninverted reflection of amplitude equal to the applied signal.This property of a shorted cable is sometimes exploited to generate a short pulse from a step waveform. The step input is applied to the cable input through a resistance equal to Z₀, with the other end of the cable shorted. The waveform at the input is a pulse of width equal to the round-trip travel time, since the reflected step cancels the input (Fig. 2).

Figure 2. Pulse generation with shorted transmission line (inverted reflection).

Cables terminated with a resistance R unequal to Z₀ also produce reflections, although of lesser amplitude. The reflected wave is inverted if R < Z₀ and uninverted if R > Z₀. The ratio of reflected wave amplitude to incident wave amplitude is given by

Ar/Ai  = (R - Z₀)/(R + Z₀)

Transmission lines in the frequency domain

Looked at in the frequency domain, a transmission line matched at the far end looks like a load of impedance Z₀, i.e., a pure resistance if line losses are neglected. That makes sense, because it just swallows any wave you apply, all the power going into the matching resistor. This is true independent of cable length or wavelength. It is when you deal with mismatched lines that things begin to get interesting in the frequency domain. Since, for a given line length, the reflected wave arrives back at the input with a. phase (relative to the applied signal) that depends on applied frequency, the impedance seen looking into the input depends on the mismatch and on the electrical length of the transmission line, in wavelengths. As an example, a line that is an odd number of quarter wavelengths long ter­minated in an impedance Zload at the far end presents an input impedance Zin = Z0^2/Zload. If the load is resistive, the in-put will look resistive. On the other hand, a line that is an integral number of half wavelengths long presents an input imped­ance equal to its terminating impedance (Fig. 3).

Figure3

The presence of reflected signals on a transmission line is not necessarily bad. For operation at a single frequency, a mis­matched line can be driven (through a line tuner) in such a way as to match its resultant input impedance, often with only negligibly greater line losses (due to higher voltages and currents for the same forward power) than with a matched load. But a mismatched line has different properties at different frequencies (the famous "Smith chart" can be used to determine transmission line impedances and "standing-wave ratio," or SWR, a measure of the amplitude of reflected waves), making it undesirable for broadband or multifrequency use. In general, strive to terminate a transmission line in its characteristic impedance, at least at the receiving end.

Stubs, baluns, and transformers

There are some interesting applications of transmission lines that exploit the properties of mismatched sections or generally use sections of line in an unconventional way.  The simplest is the quarter-wave matching section, which exploits the relationship Zin = Z0^2/Zload. This can be rearranged to read Z0= (ZinZload)^1/2. In other words, a quarter-wave section can be used to match any two impedances by choosing the characteristic impedance of the matching section appropriately. In a similar manner, a short length of transmission line (a "stub") can be used to "tune" a mismatched load by simply putting the stub across or in series with the mismatched line, choosing the stub length and termination (open or shorted) and its position along the mismatched line correctly. In this sort of application the stub is really functioning as a circuit element, not a transmission line. At very short wavelengths the use of sections of transmission line as circuit elements is common (Fig.4). Sections of transmission line (or a transformer made with several interconnected windings) can be used to construct a "balun," a device for matching an unbalanced line (coax) to a balanced load (e.g., an antenna). There are simple configurations for making fixed-impedance transformations at the same time (1:1 and 4:1 are common).

A. quater-wave matching section

B. matching stubs

Figure 4

Perhaps the nicest circuit element made from transmission line is the broadband transmission line transformer. These gadgets consist simply of a few turns of miniature coax or twisted pair wound on a ferrite core, suitably interconnected. They avoid the high-frequency limitations of conventional transformers (caused by the resonant combination of "parasitic" winding capacitance and inductance) because the coils are arranged so that the winding capacitance and inductance form a transmission line, free of resonances. They can provide various impedance transformations  with  excellent  broadband performance (e.g., less than 1dB loss from 0.1 MHz to 500MHz), a property not shared by transformers constructed from simple coupled inductors. Transmission line transformers are available as packaged modules. Figure 5 shows a few examples of baluns and a transmission-line transformer.

A.   tuned balun

B.   tuned balunr

C.    4:1 unbalanced transmission-line transtormer

Figure 5. Transmission-line transformers.