Читать книгу Handbook of Microwave Component Measurements - Joel P. Dunsmore - Страница 41

The simplest and most ubiquitous microwave components are transmission lines. These can be found in a variety of forms and applications, and they provide the essential glue that connects the components of a microwave system. RF and microwave cables are often the first exposure an engineer has to microwave components and transmission systems, the most widespread example being a coaxial cable used for cable television (CATV, aka Community Antenna TeleVison).

The key characteristics of coaxial cables are their impedance and loss. The characteristics of coaxial cables are often defined in terms of their equivalent distributed parameters (Magnusson 2001), as shown in Figure 1.11, described by the telegraphers' equation

(1.73)

(1.74)

where v(z), i(z) are the voltage and current along the transmission line, and r, l, g, c are the resistance, inductance, conductance, and capacitance per unit length.

Figure 1.11 A transmission line modeled as distributed elements.

For a lossless cable, the impedance can be computed as simply

(1.75)

but it becomes more complicated when loss is introduced, becoming

(1.76)

In many applications, the conductance of the cable is negligible, particularly at low frequencies, so that the only loss element is the resistance per unit length, yielding

(1.77)

Inspection of Eq. (1.77) shows that the impedance of a cable must increase as the frequency goes down toward DC. Figure 1.12 demonstrates this with a calculation the impedance of a nominal 75 Ω cable, with a 0.0001 Ω mm⁻¹ loss and capacitance of 0.07 pF mm⁻¹ (typical for RG 6 CATV coax). In this case, the impedance deviates from the expected value at 300 kHz by over 10 Ω; and by 1 Ω at 1 MHz.

Figure 1.12 Impedance of a real transmission line at low frequency.

This low‐frequency response of impedance for any real transmission line is often unexpected by those unfamiliar with Eq. (1.77), and it is sometimes assumed that this is a result of measurement error. However, all real transmission lines must show such a low frequency characteristic, and verification methods must take into account this effect.

An “airline” coax consists of a cable with an air dielectric, sometimes supported by dielectric beads at either end or sometimes supported only by the center conductor of the adjacent connectors, as shown in Figure 1.13. This type of cable has virtually no conductance, so series resistive loss is the only loss element. The small white ring on the airline sometimes used to prevent sagging at the male end of the pin so that it may be more easily mated.

Figure 1.13 An airline coaxial transmission line.

In some special applications, such as using measurements of a transmission line loaded with some material to determine the properties of the material, none of the elements of the telegraphers' equation can be ignored.

At higher frequencies, the loss of a cable is increased due to skin effect, which can be shown to increase as the square root of frequency (Collin 1966).

(1.78)

Thus, the insertion loss of an airline coaxial cable depends only upon the resistance per unit length of the cable, and so the insertion loss (in dB) per unit length, as a function of frequency, can be directly computed as

(1.79)

where R_a and R_b are the inner and outer conductor radius and r contains the square root of frequency. Thus, all the attributes can be lumped into a simple single loss‐term, A. Figure 1.14 shows the loss of a 10 cm airline as well as the idealized loss, as described in Eq. (1.79), where good agreement to theory is seen. However, the introduction of dielectric loading of the coaxial line will add some additional loss due to the loss tangent of the dielectric. This additional loss often presents itself as an equivalent conductance per unit length, and this loss is often more significant that the skin‐effect loss. Because of dielectric loss, the computed loss of (1.79) fails to fit many cables. The equation can be generalized to account for differing losses by modifying the exponent to obtain

(1.80)

where the loss is expressed in dB, and A and b are the loss factor and loss exponent. From the measured loss at two frequencies, it is possible to find the loss factor and loss exponent directly, although better results can be obtained by using a least‐squares fit to many frequency points. Figure 1.14 shows the loss of a 15 cm section of 0.141 in. semi‐rigid coaxial cable. The values for the loss at one‐fourth and three‐fourths of the frequency span are recorded. From these two losses, the loss factor and exponent are computed as

Figure 1.14 Loss of a 15 cm airline and a 15 cm semi‐rigid Teflon‐loaded coaxial line.

Taking the log of both sides, this can be turned in to a linear system as

(1.81)

This system of linear equations can be solved for the loss factor A and the loss exponent b.

The computed loss for all frequencies from Eq. (1.80) is also shown, with remarkably good agreement to the measured values over a wide range. Ripples in the measured response are likely due to small calibration errors, as discussed in Chapter 5.

The insertion phase of a cable can likewise be computed; in practice, a linear approximation is typically sufficient, but the phase of a cable will vary with frequency beyond the linear slope due to loss as well.

The velocity of propagation for a lossless transmission line is

(1.84)

The impedance of a lossy cable must be complex from Eq. (1.77), and thus the phase response must deviate from a pure linear phase response, due to the phase velocity changing with frequency at lower frequencies. A special case for airlines, which have no dielectric loss, is

(1.85)

For cables in general, the dielectric loss will cause a deviation in the velocity of propagation similar to that seen for loss. So far the discussion has focused on ideal low‐loss cables, but in practice cables have defects that cause the impedance of the cable to vary along the cable. If these defects are occasional, they cause little concern and are typically overlooked unless they are so large as to cause a noticeable discrete reflection (more of that in Chapter 5). However, during cable manufacturing it is typical that the processing equipment contains elements such as spooling machines or other circular equipment (e.g. pulleys, spindles). If these have any defects in the circularity, or even a discrete flaw like a dimple, it can cause minute but periodic changes in the impedance of the cable. A flaw that causes even a one‐tenth Ω deviation of impedance periodically over a long cable can cause substantial system problems called structural return loss (SRL), as shown in Figure 1.15. These periodic defects add up all at one frequency and can cause very narrow (as low as 100 kHz BW) very high return loss peaks, and thereby cause insertion loss dropouts at these same frequencies. In practice, the SRL test is the most difficult for low‐loss, long‐length cables such as those used in the CATV industry. Figure 1.16 shows a simulation of a SRL response caused by a 15 mm long, 0.1 Ω impedance variation, every 30 cm, and another −0.1 Ω variation every 2.7 m, each on the same 300 m coaxial cable with an insertion loss typical for main‐line CATV cables. In the figure, two SRL effects are shown; a smaller effect every 50 MHz or so, due to the 2.7 m periodic variation, and a much higher effect every 500 MHz or so due to the 30 cm impedance variation. The higher impedance variation occurs more often, and so the periodic error will have a greater cumulative effect resulting in a nearly full reflection, as shown in the figure.

Figure 1.15 A model of a coax line with periodic impedance disturbances.

Figure 1.16 The return loss of a line with structural return loss.