Technical memorandumDESIGN FORMULAS FOR IMPEDANCEMATCHING USING A HERMITE LINE

Indexing term: Microwave circuits and networks

Abstract: Impedance matching using a Hermite lineis studied. The problem formulation leads to a non-linear differential equation which, after solving withappropriate boundary conditions, yields a formulasuitable for impedance matching on microwave inte-grated circuits (MIC). Numerical examples are con-sidered. The results show tremendous reduction inmatching component size which could be achieved ifHermite lines are used instead of uniform transmis-sion lines. This is a considerable advantage in theminiaturisation of solid-state circuits.

j = N/(— 1). Let us write Z,- as

(2)

z(L)=z2

ri '' z,(x)

Fig. 1 Generalised nonuniform line with symbols

Introduction

It is common practice to use uniform transmission lines forimpedance matching on microwave integrated circuits(MICs). This sometimes leads to application of shuntopen- or short-circuited stubs when complex loads areinvolved. Apart from the problem of providing good openand short circuits at microwave frequencies, the uniformlines provide matching elements whose sizes are not suit-able for circuit miniaturisation. In addition, the high drainline impedance of 110 Q for optimum noise figure designof a distributed amplifier poses considerable problem ofmatching to external 50-Q load when uniform transmis-sion lines are employed [1]. These and many more attractthis study.

A Hermite line has been defined as a nonuniform linewhose characteristic impedance Z(x) varies with the linelength x in the form Z(x) = a exp [(bx)2] [2]. This defini-tion was derived in part earlier by Swamy and Bhatta-charyya [3] who, using suitable transformations, derivedHermite's differential equation from a general equation ofa nonuniform line. In the work of Kobayashi et al., anequivalent circuit of a Hermite line is presented [2]. Theirequivalent circuit does not render itself to easy applicationin impedance matching. This is particularly so when opti-misation processes are required to miniaturise matchingelements.

In this paper, a design formula suitable for impedancematching on microwave integrated circuits, using Hermitelines, is presented. Numerical examples are considered forthe case of an arbitrary complex load and also the specificproblem of the high drain line impedance of the distri-buted amplifier mentioned above.

Governing equation

Consider the nonuniform line terminated by a generallycomplex load impedance ZT as shown in Fig. 1. It can beshown that the variation of input impedance Z, with linelength x is [4]

(1)

where fi is the phase constant, Z(x) is the line characteristicimpedance which varies with the line length x and

Then, differentiating eqn. 2 with respect to x, using eqn. 1and simplifying, gives

ii+r Jdx2 \_Z( x) dx ]dx

j / \ dx / • lZ{x)

Squaring eqn. 2 and using the result in eqn. 3 gives

= 0 (3)

dx dx J dxdy 2

d + P y (4)

By the definition of a Hermite line, the characteristicimpedance Z(x) is

Z(x) = a exp [(6x)2] (5)

where a and b are constants which, using the boundaryvalues, give

a = Z(0) = Zi

'-if-©]1/2

(6)

where L is the length of the matching line. Using eqn. 5 ineqn. 4 gives,

dx(7)

To study the use of Hermite lines in impedance matching,the solution of eqn. 7 is required subject to the boundaryconditions

= 1

and

dy\ JfiZTdx)x=0 Zx

(8)

Solution of the problem

The method of Frobenius was employed to solve eqn. 7 forwhich the recurrence relation is,

(2b2r(9)

IEE PROCEEDINGS, Vol. 133, Pt. H, No. 4, AUGUST 1986 319

The solution of eqn. 7 is a rapidly converging series, which,on approximation and with the boundary conditions ineqn. 8, can be written as

2x2

y= l -P2x (4b2 + P2)

24

jpZT[ (2b2 + P2) 2~|(10)

The relationship of the input impedance with x is obtainedby employing eqn. 10 in eqn. 2 to give

Z(x)

2x2

1 - P2x (4b

i -

24

(2b2 + 02)

•]

(4b2+i2) ,-Ffl-^-](11)

For a matching length L, the characteristic impedance atx = L is Z2 and when employed in eqn. 11 we have

24

l -(2b2 + p2)

•>}l -

(4b2 + /?2) r2"

§tf-(2b2 - p2)

(12)

Writing Z ^ = Klt. ZJZ2 = K2, 2nL/Xg = K3,Z2jZx = /C4 and noting eqn. 6, eqn. 12 can be rewritten as

jK3[l-fln + - In (X4)

+ ^ Ci In {K4) + Ml ~ 1]

Now

=rT+jxT

(13)

(14)

where r, is the normalised real input impedance, and rT

and x r are the normalised real and imaginary parts of thecomplex load ZT defined as

X-r =Z,

(15)

Equating the real and imaginary parts of eqn. 13 andsolving gives the design formula

[1 - In (X4) - {K\\rT + K3[l - f In (X4)

r, = (16)

Numerical examples

(a) Let us consider matching a complex load(30 +y"45)Q to a 50 Q system at a frequency of 19 GHz,using a Hermite line with Z(0) = 20 Q and Z(L) = 40 Q,on fused silica whose relative permittivity is 3.78. Usingeqn. 15, rt = 1.25, rT = 1.5 and xT = 2.25. K4 = 2 andIn (iC4) = 0.693. Eqn. 16 then becomes

2.015 + 1.211K3-4.4O6K2

Eqn. 17 is solved graphically (see Fig. 2) for 50 Q matching2 5 r

2.0

1.5

1.0

0.5

0.2 0.4 0.6 0.8 1.0

Fig. 2 Graphical solution for the numerical examples

(i.e. r, = 1.25) to give a required matching length of800.1 nm. In comparison, a combination of 2338.9 ^mlength of series line and 1158.2 ^m length of shunt open-ended stub will be required to achieve the same matchingcondition if uniform 50 fi lines are employed.

(b) For the distributed amplifier problem we have, usingeqn. 16,

/*,• =1.689 - 2.75K2,

1 - 0.327K2 + 0.042/Ct(18)

320

Solving eqn. 18 (see Fig. 2), we have the required matchinglength of 880 //m at 12 GHz. The problem of matching thehigh drain line impedance of distributed amplifiers toexternal 50 Q loads using uniform transmission lines hasalready been highlighted [1].

Conclusions

A formula suitable for impedance matching on a micro-wave integrated circuit, using a Hermite line, has been pre-sented. Numerical examples are carried out to show howsmall associated matching lengths could be and to high-light the drastic reduction in sizes of matching elementswhen Hermite lines are employed to replace uniform linematching elements. This is a considerable advantage incircuit miniaturisation.

In conclusion, it is hoped that this formula will be veryuseful in microwave integrated circuits design.

22nd October 1985 S.O. AJOSE

Department of Electrical EngineeringUniversity of LagosLagosNigeria 4726H

IEE PROCEEDINGS, Vol. 133, Pt. H, No. 4, AUGUST 1986

References

1 Report on Research Activity in the Department of Electronics, ChelseaCollege, University of London, 1980-83, pp. 6

2 KOBAYASHI, K., NEMOTO, Y, and SATO, R.: 'Equivalent rep-resentation of nonuniform transmission lines based on the extendedKuroda's identity', IEEE Trans., 1982, MTT-30, pp. 140-146

3 SWAMY, M.N.S., and BHATTACHARYYA, B.O.: 'Hermite lines',Proc. IEEE, 1966, 54, pp. 1577-1578

4 AHMED, M.J.: 'Impedance transformation equation for exponential,cosine-squared, and parabolic tapered transmission lines', IEEE Trans.,1981, MTT-29, pp. 67-68

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