Microwave Theory

In this section the important microwave engineering concepts and nomenclature are summarized with particular attention given to microwave transmission line matching techniques.

The range of the electromagnetic spectrum from 300 MHz to 300 GHz is commonly referred to as the microwave range. For applications with wavelengths from 1 meter to 1 millimeter, low frequency circuit analysis techniques can not be used; we must use transmission-line theory. In transmission-line theory, the voltage and current along a transmission line can vary in magnitude and phase as a function of position.

Many different types of microwave transmission lines have been developed over the years. In an evolutionary sequence from rigid rectangular and circular waveguide, to flexible coaxial cable, to planar stripline to microstrip line, microwave transmission lines have been reduced in size and complexity. The microstrip transmission line is the technology employed in the current hyperthermia applicator studied.

For fields having a sinusoidal time dependence and steady-state conditions, a field analysis of a terminated lossless transmission line results in the following relations:

Figure 1 Diagram of lossless transmission line with load showing incident, reflected-transmitted waves.

If an incident wave of the form , where is the phase constant or wave number given by , is incident from the -z direction then the total voltage on the line can be written as a sum of incident and reflected waves:

.

The total current on the line is

Where is the characteristic impedance of the microstrip line, that is, the impedance the transmission line would have if it were infinitely long or ideally terminated. The incident wave has been written in phasor notation and the common time dependence factor has not been written.

The amplitude of the reflected voltage wave normalized to the amplitude of the incident voltage wave is known as the voltage reflection coefficient, G

where is the load impedance.

The total voltage and current waves on the line can then be written in terms of the reflection coefficient as

From the previous equations we see that the voltage and current on the line are a superposition of an incident and reflected wave. If the system is static, i.e. if and are not changing in time, the superposition of waves will also be static. This static superposition of waves on the line is called a standing wave.

Because of the complicated shape of this standing wave, the voltage will vary with position along the line, from some minimum value to some maximum value. The ratio of to is one way to quantify the mismatch of the line. This mismatch is called the standing wave ratio (SWR) or voltage standing wave ratio (VSWR) and can be expressed as:

The SWR is a real number such that 1£ SWR £ ¥ and with a perfect match SWR = 1. By definition, impedance, characteristic or otherwise, is the ratio of the voltage to the current a particular point on the line. The standing waves cause the impedance to fluctuate as a function of distance from the load. The variation in impedance along the transmission line caused by the line/load mismatch can be written.

Where is the distance from the load. If we substitute the expression for G in terms of the impedances, the generalized input impedance of the load plus transmission line simplifies to:

With this equation the impedance anywhere along the line can be calculated if the load impedance and characteristic impedance are known.

In the most basic sense, then, if the load impedance equals the line impedance, the reflection coefficient is zero and the load is said to be matched to the line. All of the microwave impedance matching techniques can be reduced to this simple idea: minimize the reflection of the incident wave to as nearly zero as possible.

When the load is mismatched to the line and thus there is a reflection of the incident wave at the load, the power delivered to the load is reduced. This loss is called return loss (RL) and is equal (in dB) to

This ends the summary of the relevant general microwave engineering concepts. Some relations specific to microstrip will now be discussed before moving on to discuss the compensation of microstrip discontinuities.

The geometry of a typical microstrip line can be seen in figure 4.

Figure 2 Side view of microstrip showing actual and effective geometry.

Starting with a two-layer PCB the top layer is chemically etched away to leave copper traces of width W, separated from the groundplane by a dielectric substrate of some thickness d and relative permittivity .

Because of the anisotropic dielectric geometry, the microstrip line cannot support a true TEM wave for the following reasons: a microstrip line has most of its electric field concentrated in the region between the line and the groundplane; a small fraction propagates in the air above. Because the speed of light is different in air and dielectric the boundary-value conditions at the air-dielectric interface can not be met with a pure TEM wave and the exact fields constitute a hybrid TM-TE wave. Because the dielectric substrate is electrically very thin , for this application, the fields are quasi-TEM. Because the fields are quasi-TEM, good approximations for the phase velocity, propagation constant, and characteristic impedance can be obtained from the static solution.

The phase velocity in microstrip line is given by

.

and the propagation constant is given by

,

where is the effective dielectric constant and is given by

.

The effective dielectric constant is the dielectric constant of an equivalent homogenous medium that replaces the air and dielectric layers.

The characteristic impedance of a microstrip line can be calculated, given the width W and substrate thickness d with the result

.

If all microstrip based circuits consisted of a proper width straight feedline terminating in a load, there would not be much need to worry about compensating for discontinuities. Even in this ideal case, the transition from microwave source to microstrip line and from the microstrip to load can be the source of large reflections. Typical microstrip discontinuities are junctions, bends, step changes in width and the coaxial cable to microstrip junction. If these discontinuities are not compensated, they introduce parasitic reactances that can lead to phase and amplitude errors, input and output mismatch, and possibly spurious coupling. The strength of a particular discontinuity is frequency dependent, where the higher the frequency, the larger is the discontinuity. The following typical discontinuities and their compensation are discussed in descending order of importance.

Impedance Mismatches

Quarter-Wave Transformer

A general mismatch in impedance between two points on a transmission line can be compensated with a quarter-wave transformer. The quarter-wave transformer is a very useful matching technique that also illustrates the properties of standing waves on a mismatched line. First, an impedance-based explanation of how a quarter-wave transformer works will be described; then a more intuitive explanation that is analogous to destructive interference in thin films will be discussed. A quarter wave transformer in microstrip is shown in fig 5.

Figure 3 Diagram of quarter wave impedance transformer showing multiple reflections.

In a quarter-wave transformer, we want to match a load resistance to the characteristic feedline impedance through a short length of transmission line of unknown length and impedance. . The input impedance looking into the matching section of line is given by;

If we choose the length of the line = then , divide through by and take the limit as to achieve

For a perfect transition with no reflections at the interface between microstrip and load, G =0 so and this gives us a characteristic impedance as

which is the geometric mean of the load and source impedances. With this geometry, there will be no standing waves on the feedline although there will be standing waves on the matching section. Why was the value of chosen? In fact, any odd multiple (2n + 1) of will also work.

The astute reader may recognize these conditions as similar to those found in destructive interference in thin films. In thin films, if light is incident on mediums with progressively higher index of refraction, it will undergo a 180 degree phase change at both interfaces. For there to be destructive interference, the path length difference must be . The microstrip quarter-wave transformer works in exactly the same way. When the line length is precisely the reflected wave from the load destructively interferes with the wave reflected from the interface and they cancel each other out. It should be noted that this method can only match a real load. If the load has an appreciable imaginary component, it must be matched differently. It can be transformed into a purely real load, at a single frequency, by adding an appropriate length of feedline.

Junctions

A junction between two dissimilar width sections also introduces a large discontinuity. A standard T-junction power divider is shown in figure 6.

Figure 4 Diagram of T-junction power divider.

In this diagram, the input power is delivered to the intersection on a microstrip of width and impedance . The line then branches into two arms with power, width and impedance given by and respectively. The design equations for this divider are

This simplest type of matched T-junction is the lossless 3dB power divider. It can be seen from the equations above that if the power will split evenly into the arms of the T with each arm having half the original power. It is interesting to note that the impedances of the two arms act just like resistors wired in parallel. To match the impedances of the arms of the T to the impedance of the base, the arms must have twice the impedance of the base.

Another typical microstrip discontinuity results from a simple bend in the line. Figure 7 shows some typical bend discontinuities and the required compensation techniques.

Figure 5 Different bend discontinuities in microstrip and their compensations.

The increased conductor area in the region of the bend produces a parasitic discontinuity capacitance. This effect can be eliminated by making a smooth swept bend where there is no change in the conductor area. The radius has to be r³ 3W, which takes up a large amount of space, space that is always at a premium. A more space-effective compensation method is to miter the right angle bend.

Source-microstrip transition

To launch a wave, on the microstrip transmission line the microwave signal is brought from the generator on a coaxial cable which connects to an on-board PCB mounted jack which is soldered directly to the groundplane and feedlines. To minimize reflections in this process the generator, coaxial cable and jack all have characteristic impedances of 50 ohms. The actual transferring of the wave from the jack to the microstrip is the main source of reflections in this process. To minimize these reflections the microstrip line impedance must match the impedance of the jack.

The compensation methods for a step change in width and the parasitic reactance of a T-junction are shown in figure 8.

Figure 6 Second order discontinuities and their compensation techniques.

These discontinuities are second order, only becoming significant at frequencies above 3 GHz. . For this reason, these methods of compensation were not employed in this research.

We have reviewed basic transmission line theory, explaining the terms used to describe microstrip circuits and the techniques used to match different elements in a circuit. In the next section, we discuss how to apply this theory

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