Impedance Matching

VOLTAGE

3. WAVE PROPAGATION & MANIPULATION

3.2 Impedance Matching

I.MI CROWAVE ENGINEERING FUNDAMENTALS AND SPECTROMETER DESIGN 29 nulls the incident wave E

max /

^{Emin~ 00,} and a perfectly absorbing termination reflects no wave, creating no nodes, so that

Emax / Emin =

1.0.

Complete reflection represents an infinite impedance to wave propagation, whereas the perfect absorber 'matches' the transmission line by absorbing all the incident power. The ratio of the wave maximum and minimum, which has been defined in the preceding paragraph and is called the Voltage Standing Wave Ratio (VSWR), therefore provides a measure of the match between a microwave circuit device and the transmission line source of electromagnetic radiation. Impedance is therefore only defined with respect to the transmission line; one does not need to be concerned with the absolute value of the impedance. The important design factor is the ratio Z / 20 , where Z is the impedance of the device and 20 the impedance of the transmission line feed.

relative orientations on the DC and oscillatory magnetic fields (Bleaney &

Stevens, 1953), and rectangular waveguide that operates in the TE mode is the most frequently used transmission line. A second advantage of rectangular guide is that departures from linearity or uniformity of cross section appear only as a reactance, which can easily be compensated by the addition of another reactance (Marcuvitz, 1951).

UNIVERSAL TRANSFORMER

-'1-+

-b

DOUBLE SHUNT

-41

3,\

/81-4-J

Figure 14. Impedance matching transformers. The 'universal' transformer consists of an adjustable short whose position along the line can be adjusted to a point at which the conductance equals the characteristic admittance. In practice, the device that best approximates a universal transformer is the slide screw tuner, whose variable reactance can be positioned at some point along the line where a 'sweet spot' repeats itself at some multiple nAI2. Other practical devices cope with positional limitations along the line by adding one or more reactances (double and triple slug tuners, right).

Devices for impedance matching correct for a non-optimal reactance by introducing a variable reactance. A theoretical 'universal' transformer is merely a shunt on a transmission line (Figure 14). There are only two adjustable components on such a transformer: the length of the shunt, and the distance of the shunt from the mismatched load. The distance of the shunt from the mismatched load is selected so that the conductance equals the characteristic admittance (i.e. the resistance equals 20 ) , At this point, therefore, all one has to do is introduce a pure reactance equal and opposite to the reactance of the line, and one can eliminate the reactive component of the impedance; sinceR

=

20 at this point, the loaded line appears matched to the source. The length of the shunt controls the reactance of the transformer, which should be equal and opposite the line's reactance at the point of its attachment.

In EMR spectrometers, all of the components are matched in the sense that the bridge circuit is laid out to the optimal

Zo;

this is true even of home-built spectrometers, which tend to be tinker-toyed together from discrete

I.MICROWAVE ENGINEERING FUNDAMENTALS AND SPECTROMETER DESIGN 31 components that have connectors and are optimized for 50

n

circuits. The only 'unknown' is the sample resonator, whose impedance will vary according to what sample material occupies the field lines. A cavity resonator tends to be coupled to a rectangular guide via an aperture and a variable post (the so-called iris), but other devices, such as the E-H tuner, the slide-screw tuner, and the double-stub tuner, may be used to match non-standard sample resonators (laboratory experiments for demonstrating the use of these devices are compiled in Reich et al., 1957). E-H tuners are specified in some spectrometers that employ a helical sample resonator (Webb, 1962), and a slide-screw tuner was initially specified for matching the Broker ENB250 TM\\o ENDOR cavity when the coil was in place."

3.2.1 Apertures and Irides

A transmission line has a characteristic resistance, capacitance, and inductance per unit length, and it is common to find theoretical treatments of transmission line behavior expressed in terms of an infinite or semi-infinite repetition of an RLC unit. The effective inductance and capacitance are parameters that depend on the geometry of the line and its dominant propagating mode. For example, in a rectangular guide the capacitance and inductance are determined by the dimensions of the narrow and broad walls, respectively . The analogy, based on the orientation of the respective field lines, to conventional capacitors and inductors is illustrated in Figure 15.

Detailed analyses of the network properties of waveguide are found in texts by Slater (1942), Marcuvitz (1951), and Collin (1960).

17One would not ordinarily use two or more tuning devices in this manner, but each tuning device has only a finite range of impedances over which it can be varied. The slide screw supplies a greater range than the iris alone for matching the coil-containing ENDOR cavity; a similar strategy is used for an ESE-ENDOR cavity, in which the matching range of a Gordon Coupler is extended by adding an iris.

f ^{11 11 111}

- - B

-Figure 15. The analogy between the rectangular waveguide cross-sectional parameters and distributed parameter devices (capacitor and inductor).

A common method of correcting a mismatch is to insert a compensating inductance or capacitance into the circuit. One can therefore insert a very narrow diaphragm¹⁸ through the broad or narrow wall to introduce

THIN THICK

PARTITION PARTITION

0 0

= ^~

or

T ^o

T

^{T 0}

0 0

:~: o

Figure 16. Waveguide obstructions and their equivalent circuit representations. Thin obstructions (left) and thick obstructions (right) are indicated.

capacitance or inductance (Figure 16). As appertains coupling to a cavity or

18Narrow diaphragms in waveguide are taken, in a practical sense, to be of thickness equal to orless than1/32",see Marcuvitz, 1951.

1. MICROWAVE ENGINEERING FUNDAMENTALS AND SPECTROMETER DESIGN 33 another section of waveguide, an aperture may be viewed in much the same way as a thin diaphragm. Either circular holes or slots may be used, but the former couples via the electric field, whereas the latter couple via the magnetic field. One variant of the latter, the cruciform coupler, is useful because it allows one to impose field orientation constraints. It is important to recognize, however, that coupling apertures must be of a size so that they do not resonate.

Finite thickness obstructions scatter an incident electromagnetic wave and, through the analogy between wave scattering and impedance, introduce both series and shunt reactances. A post perpendicular to the electric field lines acts as a capacitive shunt, whereas a parallel post acts as an inductive shunt. The equivalent circuits of capacitive and inductive obstructions are indicated in Figure 16. Transformers are readily fabricated from fractional waveguide obstructions and optimally 'tuned' to the mismatch, coated with a thin layer of solder, slid into the optimal position, and heating the waveguide to make the solder joint. Bandpass filters often feature a step-like cross sectional profile of A / 4 transformer sections (see Matthaei, Young, & Jones ,

1980, Chapter 3).

3.2.2 The Slide-Screw Tuner

A vertical post inserted through the broad wall of a waveguide behaves as though it were an inductor, but if the post does not completely pass through the waveguide, there is a capacitance that arises between the post and the wall. A post whose position is variable may be used to introduce a reactance in the form of a series inductance and capacitance, which may then be used as a means to provide a nulling reactance so that a load may be matched . The slide-screw tuner is such a device that works on the principle of the universal transformer described in the introductory paragraphs of this section.

The adjustable post is mounted on a carriage so that it may be moved along the guide in the direction of propagation, and it follows that there are two adjustable parameters that optimize the impedance match . The optimal point along the transmission line, where R

=

Zo, repeats itself at half wavelength intervals, and therefore the slide screw tuner device can be installed at some arbitrary point near the load(i.e .the tuner is a device that may be purchased as a waveguide section to be 'dropped in'). Since most applications of tuners correspond to resonator tuning, the slide screw is situated between the source and close to the resonator. The screw is withdrawn from the waveguide, and the resonator tuned as well as possible by using the iris, if one is available (presumably, one is monitoring coupling via the 'dip' in the source's output). The screw is then inserted into the waveguide until it affects the tuning parameters, and then its position along

the waveguide is adjusted in order to maximize the effect . Once the optimal position is found, the screw is inserted to optimize coupling.

3.2.3 The Double-Stub Tuner

The universal transformer that was described in the introduction to this section requires that one find a 'sweet spot' along the transmission line where R

=

Zoo An adjustable short can, in principle, be placed anywhere along the line, but in practice this becomes problematic because one simply cannot move a waveguide or coaxial junction arbitrarily along the main transmission line. The double-stub tuners obviate the problem by introducing a second variable short that compensates for the immobility of the tuning plungers.

Placement of the stubs affects the performance of the tuner. For two stub device, the placement should be at some odd multiple of1J8;the three stub tuner performs optimally when the distance separating the stubs is1J4.The stub closest to the load to be matched provides an adjustable admittance so that the conductivity of the load matches that of the line; the second stub (furthest from the load) then provides an adjustable admittance to the transmission line.

3.2.4 Waveguide Tees and the E-H Tuner

A stub tuner is realized as a junction in a coaxial or waveguide transmission line (Figure 14), but because of the TEo1 mode pattern in rectangular guide, the equivalent circuit of shunts varies according to which wall bears the junction. An E-plane junction puts all three branches in the electric plane, and therefore the circuit elements lie in series. By contrast, an H-plane junction puts all three branches in the magnetic plane; any circuit element (e.g. a load) attached to a given terminal will appear in parallel.

Waveguide tees are therefore analogous to parallel and series junctions at low frequency and may be used to divide power among the other two ports . Power division is equal if the ports are identically matched.

I.MI CROWAVE ENGINEERING FUNDAMENTALS AND SPECTROMETER DESIGN 35

1

4

Figure 17. The hybrid tee. Branch 3 appears in series with the main waveguide section denoted by ports Iand2;branch4appears in parallel.An E-Htuner features adjustable shorts in both of the branches and therefore puts reactance in series and parallel to the main circuit that is connected to ports I and 2. Because of the symmetry and distribution of power delivered to the device , it is also use as a means of creating bridge circuits (see Sections 2.5 and 4)

A hybrid tee features both E-plane and H-plane junctions on the main guide, and the symmetry of the device renders certain pairs(i.e. 1-2,3-4; see Figure 17) decoupled. If signals enter the tee from port 3, it will be divided equally among ports I and 2 if these two ports are matched loads. A signal appears at port 4 only if there is a mismatch between ports I and 2, and this is the bridge method of detection that was used in many early autodyne EMR spectrometers. Ports 1 and 2 are connected to the sample cavity and a standard 50

n

load. With the sample off-resonance, the sample cavity is tuned to match the 50

n

load, thereby nulling the signal at port 4. As the magnetic field is swept through the resonance condition the impedance of the sample resonator changes, creates a mismatch, and diverts power to port 4. As impedance-matching devices, tees enable one to introduce reactive elements in series or parallel to the load because ports 3 and 4 are decoupled.

A sliding short is situated in both the E- and H-plane at a common junction, and operates on the same principle as the double-stub tuner. This tuning device is more compact than the double-stub (i.e, series) tuner and may be used at frequencies up to 300 GHz. In this case, port 1 is the input and supplies power to port 2(i.e. the load to be matched) only ifthe ports 3 and 4 are mismatched, which is the case because they are series and parallel

shunts. Since the desired match condition(i.e. source to load) requires that no power be reflected back from port 2 to port 1, ports 3 and 4 are matched from the standpoint of port 2.