Switching characteristics of phase modulators

Klaus Schiinemann, Frank Sporleder, and Laszlo Szabo

Indexing term: Modulators

Abstract: The influence of the p-i-n -diode switching transients on the performance of digital phase modulatorsis studied. To this end two models are introduced: a diode model which allows one to calculate the transientimpedances, and a black-box model of a phase modulator which can be incorporated in network analysisprograms in order to simulate system performance. Based on these models, guidlines are given for anoptimum choice of various design parameters as for example the Mayer width of the p-i-/i-diode, theswitching-current ratio, the switching-current waveforms and the modulator bandwidth.

1 Introduction

In digital radio-link systems the p.c jn. is done in a reflection-type phase modulator with p-i-n-diode. A few years ago, itwas shown that the transient response of the modulatorinfluences the overall transmission characteristics of thecommunication system to a large extent.1 The possibletransient reflection coefficients have been classified. Onthe basis of two extreme solutions the influence of thesetransients on the system performance has been estimated. Itis the aim of this paper to extend those investigations in tworespects: (a) A realistic diode model will be developed,which allows one to analytically calculate the transientmicrowave impedances of the p-i-n-diode. (b) This diodemodel is incorporated in the black-box model of the phasemodulator, which is suitable to be directly used in existingnetwork-analysis programs for simulation of the systemperformance. The compound model can then be utilisedto establish guidelines for the optimum design of bothdiode and embedding network parameters.

2 Formulation of the problem

The switching transient of the reflection-type phasemodulator is defined as the trace of the input reflectioncoefficient in the Smith chart between the two steadystates. (For a transmission-type modulator the reflectionhas to be replaced by the transmission coefficient. Theinvestigations can then easily be derived from thosepresented here.) Two extreme traces can be defined.1 Theywill be referred to as the a.m. and p.m. traces (see Fig. 1).In the case of the a.m. trace, only amplitude variations areproduced, while the phase changes abruptly during thetransients, whereas in case of the p.m. trace only phasevariations are produced while the amplitude stays constant.

The trace of the reflection coefficient is of importancefor the system performance. Owing to the finite switchingtimes of the modulator, which may be caused by the finiteswitching times of the p-i-n-diode and by bandwidth limi-tations of the modulator network (diode embedding), someextra frequency components are generated in the spectrumof the pulse-phase-modulated carrier wave, partly leading to

Paper T412 M, first received 5th February and in revised form 11thJuly 1979Dr. Schiinemann is with the Institut fur Hochfrequenztechnik derTechnischen Universitat, Postfach 3329, D-3300 Braunschweig, WestGermany. Dr. Sporleder is with the Forschungsinstitut der DeutschenBunderspost, Postfach 5000, D-6100 Darmstadt, West Germany. Dr.Szabo was formerly with the Institut fur Hochfrequenztechnik derTechnischen Universitat Braunschweig, West Germany, and is nowwith Standard Elektrik Lorenz AG-ITT, D-7000 Stuttgart, WestGermany.

a degradation of the demodulated signal. These extracomponents are generated by an a.m. trace as well as by ap.m. trace; they are more numerous, however, in the case ofa p.m. trace. For brevity these components will be denotedby 'extra noise components' in the following.

Fig. 1 Definition of two extreme cases for the transient reflectioncoefficient of an 180° phase modulator

rltr2 . . . reflection coefficients at steady state

In the demodulator the extra noise components aredetected and superimposed on the signal. This is due tounsymmetrical amplitude characteristics and to nonlinearphase characteristics in the transmission system. Hence, thesignal/noise ratio degrades. As extra noise components aremainly due to p.m. traces of the transient reflection coef-ficients, a.m. traces should be approximated as closely aspossible. The effect of the switching transients on theoverall transmission characteristic of the communicationsystem has been theoretically studied in Reference 1.Assuming idealised switching transients with constantswitching speed between the two states, it has been shownthat both the signal/noise ratio (error rate) and the patternjitter degrade more the closer a p.m. trace is approached.

Experimental investigations on realised phase modulatorshave shown that, in the case of a p-z'-n-diode as switchingdevice, an a.m. trace is approached.2'4 This real trace is,however, by no means ideal, because the switching speed isnot constant. Furthermore, the turn-on trace is not congru-ent with the turn-off trace. Whereas the former shows aninductive component and a switching speed which decays

MICROWA VES, OPTICS AND ACOUSTICS, SEPTEMBER 1979, Vol. 3, No. 5 197

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with time, the latter is capacitive and its switching speedincreases with time. There are, thus two questions whichmust still be answered: (i) To what extent can the traces beinfluenced by diode or by network parameters inorder to approximate the ideal a.m. trace? (ii) How large isthe influence of the real trace on the system performance?

In order to solve these problems a simple model for cal-culating the transient microwave impedances of p-/-«-diodeswill be derived next.

3 The p-/-n-diode transient impedance

p-i-n-diodes for microwave applications show Mayer widthsof only a few micrometres. Although this is much shorterthan a diffusion length in the intrinsic middle layer, recom-binations of charge carriers cannot be neglected. This is dueto an additional charge storage in the highly doped contactlayers, where lifetimes are orders of magnitude shorter thanin the /-layer. The contact-layer storage charges play animportant role in the case of high forward-current densities(larger than 19 A/cm2) and for narrow /-layer widths, as hasbeen shown in Reference 5. They have to be taken intoaccount for the calculation of the transient impedances.

The following calculations are closely related to asimplified charge-control model of the p-/-«-diode, whichhas been presented in Reference 5. The characteristicassumptions, which we shall make, are illustrated in Fig. 2.

a rn

p=n

recombinations in the contact layers into account, the/-layer lifetime T,- is replaced by an effective lifetime

T =

1 +eANcw

2 AJ T,

(1)

In eqn. 1 the subscript / refers to the Mayer, the subscriptc to the contact layers. A is the cross-section of the diode,w the Mayer width, e the magnitude of the electron charge,Dc the diffusion constant, and Nc the doping concentrationof the contact layers. Qi denotes the stored charge in theMayer.

The concept of an effective lifetime in the Mayer hasbeen introduced in Reference 6. An inspection of eqn. 1shows that T decreases with increasing forward currentdensity (i.e. with increasing Mayer storage charge Qt) andwith decreasing Mayer width w. By this the contact layerrecombinations are modelled.

3.1 Turn-on transient

We assume in the following that the diode current isimpressed. Qi and the forward current IF are related to oneanother by the continuity equation

dQi.Qi~—I =dt T

(2)

because Qt nearly equals the total stored charge.s Hence(2,(0 is known from the solution of eqn. 2 as is the conduc-tivity of the Mayer. Provided that Qt(t) only changessmoothly during one period of the r.f., the small-signalmicrowave impedance of the /-layer reads

**(') =wJ

(3)

ju is the mobility of the charge carriers. The factor 2 takesinto account that both electrons and holes contribute tothe Mayer conductivity.

In order to take conductivity modulation into account,an inductance Lt has to be added to Rt. It can be takenfrom the results of Reference 7, where Z,,- has been relatedto R{ via

!<&) = ^ (4)

Fig. 2 Doping profile (solid line) and distribution of the storedcharge (dashed line) of a real (a) and idealised (b) p-i-n-diode

NA> ND . . concentration of acceptors, donorsp, n . . . concentration of holes, electronsTn> Tp> Ti • • • carrier lifetimes in the various regionsT . . . effective carrier lifetime in the /-layer

The upper part shows the doping profile of a p-n-n+ diodeand the distribution of the stored charge carriers: a uniformcharge plasma in the weakly doped middle layer andminority carrier diffusion tails in the contact layers. Thecarrier lifetimes Tn and rp are typically two orders ofmagnitude shorter than r,-. This 1-dimensional dopingprofile will be replaced by the ideal box profile shown inthe lower part of Fig. 2. The middle layer is assumed to beintrinsic. The potential barriers at the p-i- and ^-/-junctionsconfine the stored charge to the /-layer. In order to take the

D[ is the diffusion constant in the Mayer.The inductance must be assumed to be in shunt to the

resistance as is sketched in Fig. 3. Because the impedancesof the junctions may be neglected with respect to theimpedance of the Mayer at microwave frequencies, theturn-on transient impedance is completely given by eqns. 3and 4. The model is, of course, only a rough approximationof reality; it is, however, well suited to be incorporated innetwork-analysis programs. Moreover, it describes theexperimental results of References 1—4 and 8.

Two switching-current waveforms will be considered: aconstant current IF = IF0 and a more realistic waveform

IF = (5)

In eqn. 5 the time constant j3 is a measure of the finite slopeat the beginning of the switching pulse. The solution of the

198 MICRO WA VES, OPTICS AND ACOUSTICS, SEPTEMBER 1979, Vol. 3, No. 5

charge-continuity eqn. 2 reads for a time-independentcurrent

Qt = IFoT(l-e-t/T)

and for the current waveform of eqn. 5

Qt = 1 - ,-t/T

(6a)

(6b)

The turn-on time ton is defined with respect to the storedcharge reaching 90% of its steady-state value. In the case ofa constant current one obtains

ton = 2-3 T (7)

This result shows the relative importance of the turn-ontransient, since a reasonable value for the effective lifetimeis r — Ins.

charge gradients are established at the junctions by thereverse current, whose slope is given by

tan a = (9)

The border concentrations reach nh the intrinsic concen-tration, at

IFOT w2

to = 00a)

if the charge distribution is triangular. This is valid providedthat

w<2 (11)

1V//////A

Ri

1

Fig. 3 Storage charge Qi during turn on and equivalent circuit ofthe i-layer

3.2 Turn-off transient

The treatment of the turn-off transient closely follows thediode model of Reference 5. We assume that a reversecurrent —IR is applied to the p-/-«-diode at t — 0, whichrecovers the Mayer storage charge QIO^IFO7- Recombi-nations may now be neglected, because the carrier lifetimein the /-layer is much larger than the turn-off time. Thereare only recombinations at the beginning of this phase,when the contact layer storage charge is extracted. Thiseffect is, however, insignificant, because there are no longerany replacements of charge carriers in the contact layersfrom the /-layer. Hence the charge-continuity equationreads

dQt/dt = - / * , = IF0T (8)

The turn-off transient can be subdivided into two intervals,a plasma phase and a space-charge phase, which will betreated separately.

3.2.1 The plasma phase: For t < 0 the charge plasma is uni-formly distributed throughout the Mayer. This does nothold any longer under reverse-current conditions. Followingthe framework of Reference 5, the charge distribution willbe modelled by a broken-line approximation for t > 0. Two

Fig. 4 Linearised charge carrier distribution during the plasmaphase of the turn-off transient and a.c. resistance components

a Definition of t{, t2

b Before f,c After r,

MICRO WA VES, OPTICS AND ACOUSTICS, SEPTEMBER 1979, Vol. 3, No. 5 199

The inequality condition (eqn. 11) is fulfilled in most prac-tical cases. Otherwise the charge distribution can be approxi-mated by a trapezoid and t2 is given by

= wIn the following, the calculation of the plasma resistancewill be derived for the case when the inequality condition(eqn. 11) holds. The other case can then be treated similarlyby combining the former calculations and the results inReference 5. The small-signal plasma resistance can beevaluated by regarding the charge distribution in the Mayer(see Fig. 4). Prior to t = tx there are two contributions tothe overall resistance Rp: a part R x which is characterisedby a linear charge distribution, and another one R2 for aconstant charge. Rx is given by

dx(12)1 Jo 2neAp{x)

with the hole concentration

p(x) = (tan ct)x + pr + rii = (x — z) tan a + p0 — nt

(13)p0 and pr are defined in Fig. 4. The initial hole concen-tration p0 is

Po = Aew(14)

z can be calculated by equalising the recovered charge toIR t yielding

05)

3.2.2 The space-charge phase: When time exceeds t2 twospace-charge zones of width s develop at the junctions andgrow toward the centre of the /-layer. The plasma chargeQi is completely recovered at the turn-off time tOff, whichcan be calculated from the continuity eqn. 8. The /-layerimpedance is now composed of a decaying plasma resist-ance, which is due to the shrinking plasma zone in themiddle of the /-layer, in series with the impedance of thespace-charge zones. The latter can be modelled by capacit-ances Cs which are shunted in space-charge resistances Rs.As has been sketched in Fig. 5, the capacitances are due tothe displacement current flowing across the space-chargezones. The electric field strength E increases when s widens.

\ \

p AA

_ kT1 eIR

2DJIFOT +

— wIRz(16)

kT/e is the temperature voltage.The resistance R2 of the constant charge region is

R-, =w — 2z

2neAp0(17)

Eqns. 16 and 17 hold for a trapezoidal distribution at t = t2

as well. In the case of a triangular distribution, however,they are only valid prior to t = tx, because R2 has decreasedto zero at this moment. tx can be calculated from z = w/2:

fj = w2/(8A)

The border concentration pr is now given by

(18)

wPr = «0 - - t a n a —

2 e^vvyielding

tx <t<t2 (19)

kT—— IneIR \2DiIF0T

2DiIF0T + 2eADiniw - 2DJR (t —

2eADiniw- - tx) - w2IR/2

(20)

r dQ,

k

Fig. 5 Linearised charge-carrier distribution p and electric fieldE during the space-charge phase of the turn-off transient andac-impedance components (a); recovered charge dQl by a wideningds of the space charge zone (b)

We will assume for a moment that s(t) be known. Theplasma resistance ^ 4 can then be calculated to be

kT ,— meIR 2eADini

(21)

in the case of a triangluar initial distribution. Otherwise thespace-charge phase must be divided into two intervals.During the first one {t2 < t < ^3) the trapezoid narrows toa triangle, in the second (t23 < t < tOff) the plasma chargein the triangle is recovered. t23 is given by

(22)

200 MICROWA VES, OPTICS AND ACOUSTICS, SEPTEMBER 1979, Vol. 3, No. 5

The total plasma resistance Rp reads in this case

R = nR5 +R6 t7<t<t23

l 2 / ? 4 t23 < t < toff

with/?4 given by eqn. 21 and

eIRlR\ eAvmt

vv/2 —s—

(24)

(25)

The space-charge resistance Rs can approximately be cal-culated by assuming that the charge carriers of either signdrift with their saturation velocity vs. The hole concen-tration in the left space charge zone is then given by

p, =eAv*

yielding

R. =svs

(26)

(27)

The electric-field strength in the space-charge zonesincreases with time. This effect can approximately be takeninto account by shunting Rs with a capacitance

C, = eA/s (28)

with e the dielectric constant.In order to complete the calculations of the Mayer

impedance during the space-charge phase, the space-chargewidth s(t) must be determined. To this end we equate therecovered charge dQ{ of Fig. 5b to 5 IRdt. This yields thedifferential equation

Two current waveforms will be regarded as in the case ofthe turn-on transient: a constant current IR0 and a morerealistic waveform

In =IRO 0 < t < t2

Xoff(30)

This takes into account that the current actually decreases,when the space-charge zones are built up. The solution ofeqn. 29 is, in the case of a constant current,

toff = h - I n II - (32b)

These solutions are valid provided that the charge distri-bution at t = t2 is triangular. In the case of a trapezoidaldistribution one obtains, for a constant current,

tn<t<toff (33a)

The turn-off time is, similarly, given by eqn. 32a.The differential equation for a current, which decays

against time according to eqn. 30, can be taken from Refer-ence 5. It reads

, s(t2) = 0,

h < t < t23

w

~ s | ds = Di

The constant a is given by

a = w —

, s(t23) = a/2,

wlR0

(34)

(35)

and the moment t23, at which the trapezoid has beenreduced to a triangle, by

- t2-pin 1 +—-[ J3J3 \WIFO] &IRQ

= 0 (29) The solution of eqn. 34 is

S =

W

2IF0T '23

VV/c

Equating s to vv/2 yields the turn-off time

w -2D,(t-t2)

yielding a turn-off time [s(toff) = vv/2]

ho?totf — (32a)' j R O

and, in the case of a time-dependent current,

f(31*)

(33b)

( 3 7 )

s(t) being known, the impedance during the space chargephase is completely determined. The next step in investi-gating the switching characteristics of the phase modulatoris to establish a black-box model of the reflection modu-lator.

4 Network model for a phase modulator

An equivalent circuit for the phase modulator with p-i-n-diode is shown in Fig. 6. The diode symbol represents theinternal p-z-n-diode without parasitics which are assumed to

MICROWA VES, OPTICS AND ACOUSTICS, SEPTEMBER 1979, Vol. 3, No. 5 201

be part of the linear, reciprocal 2-port network. For sim-plicity this network will be assumed to be lossless. It trans-forms the two steady-state impedances of the p-/-w-diode insuch a way that the input reflection coefficients of themodulator are given by rx and r2=—rly respectively.According to Reference 9 the bandwidth of the embeddingusually is so large that a distortionless transformation of thetransient impedances of the p-i-n-diode can be assumed.(The switching times of the modulator would otherwise belimited by the turn-on time of the linear circuitry.) Hencethe network parameters can be determined according toReference 10. These parameters being known the transientinput coefficients are calculated from the transient diodeimpedances by standard methods.

a 'd

A 8 C D VO

Fig. 6 Modulator equivalent circuit

It has been shown in Reference 10 that, in order toproduce an 180° phase shift and the input port of themodulator, the embedding has to match a special averagevalue of the two impedances of the/?-f-«-diode. This averageimpedance is called the 'hyperbolic point impedance'. It isuniquely related to the diode impedances. The modulatorcircuitry is, hence, an impedance-matching network with abandpass characteristic. It will be described here by itsABCD matrix relating the input and output voltages andcurrents to one another via

Vo A jB

jC D

Vd

Id

Reciprocity shows that

D = (l-CB)fA

(38)

(39)

A, B and C are real provided that the two port is lossless, aswill be assumed here. Furthermore, their frequency depend-ence will be neglected, because it is small in comparison tothat of the diode impedances. A, B and C can then becalculated from the matching condition.

The normalised steady-state impedances of the p-i-n-diode will be denoted by zx and z2, respectively. The zX2(t)[z2i(0] is the transient impedance when the diode isswitched from zx(z2) to z2(zx). The hyperbolic middle-point impedance zm is given by

(40)

with

rm = Rez2 J\ z2)

Re zx Im z2 + Re z2 Im zx

ReCzj + z 2 )(41)

Its magnitude \Zm\ has been used as the normalising imped-ance:

zm = Zm/\Zm\, zx = ZJl'ZJ, etc. (42)

Denoting the modulator input impedances by a prime oneobtains

( 4 3 )

(44)

The matching condition reads

Azm +jBjCzm+(l-CB)/A = l

From eqn. 44 two of the three unknowns A, B, C can bedetermined. A third conditions is arbitrarily laid down bysetting

C = l/B

A and B are calculated from eqns. 44 and 45:

A =—x, B =

The input reflection coefficients are now given by

r; = (A+jQz,+jB= T(zt\ i = 1,2

(45)

(46)

(47)

Eqn. 47 can be generalised, yielding the transient inputreflection coefficients

r12(t) = T{za(t)}t r2l(t) = T{z2l{t)} (48)

Now, the reflected voltage waves will be calculated. Toproceed in this the equivalent circuit of the modulator fromFig. 6 is replaced by the network of Fig. 7 showing a volt-age source, an ideal switch, and a 2-port network withtransfer function S instead of the p-/-«-diode, which wasoriginally driven by the input voltage wave. Now the phasemodulation is represented by switching the voltage ampli-tude from — 1 to + 1 or vice versa. In the following, thisblack-box model of the reflection modulator will be com-pleted by relating the transfer function S to the transientimpedances of the p-/-«-diode.

The transient output voltages (i.e. the transient reflectedvoltage waveforms of the modulator) are given by

—r

+ r(49)

co0 means the r.f., r is the stationary reflection coefficientof the modulator in state 1.

VABCD

Fig. 7 Black-box model of a digital-phase modulator

Vo represents the incoming wave, 5 the transfer functions due to thetransient diode impedances, ABCD the diode embedding, and Vmthe outgoing phase modulated wave

202 MICROWAVES, OPTICS AND ACOUSTICS, SEPTEMBER 1979, Vol. 3, No. 5

From eqn. 49 the transfer function S can be calculatedby Fourier transforming the pulse response:

(50)l-oodt

The relation in eqn. 50 completes the black-box model ofthe phase modulator. This model can directly be incorpor-ated in network-analysis programs as, for instance, that ofReference 1, which are capable of simulating the overallperformance of the communication system. Moreover,modelling the transient impedances of the switching diodein a two port with an ambiguous transfer function enablesone to easily draw conclusions on the perfomance of themodulator.

5 Results

The transfer function of the network model of a digitalphase modulator takes two different values, one for theturn-on and another one for the turn-off transient. It ispossible to replace these characteristics by another set oftransfer functions. S12 and S2i can be thought to becomposed of dynamic symmetrical part, which is cascadedwith a dynamic unsymmetrical part. This representationallows one to directly draw some conclusions concerningthe modulator performance. The most important of themare:

(a) The extra noise components (mentioned in Section2) originate in distortions which are caused by the dynamicnetworks.

(b) Distortions due to the symmetrical network can becompensated by adding an equalising (linear) network. Thisis not the case for distortions, which are caused by theunsymmetrical network.

(c) The maximum bit rate is limited by the bandwidth ofthe symmetrical network, because the bandwidth of theunsymmetrical network has turned out to be much larger.

(of) The dynamic networks, and hence the transientimpedances of the p-/-«-diode, can be neglected if thefrequency components due to these networks are beyondthe bandwidth of the communication system.

Our next task is to investigate the influence of the diodeparameters on the dynamic networks. The question iswhether or not an equalising effect can be achieved by aproper choice of these parameters. To this end a p-/-«-diodewith the following typical data will be regarded:

A = 10~5cm\ v = 103cm2/Fs, Dt = 25cm2/s

Dc = l-5cm2/s, rt - 30 ns, rc = 0-3 ns,

Nc = 1018cm"3, «; = 2x 1010cm"3

The r.f. has been set to 34 GHz. Variable parameters are theMayer width w, the forward and reverse currents IF0 andIR0 and the time constant (3 of the time-dependent currentwaveforms. The results of these investigations are:

(i) The bandwidths of the transfer functions are thelarger the narrower w. There exists an optimum /-layerwidth. For large widths the inductive turn-on component ofthe transient impedence diminishes but the capacitive turn-off component increases. Just the opposite is valid for smallwidths.

(ii) The symmetry of the transfer functions can largelybe influenced by the ratio of the switching currents IFQ/IRO-

The deviations of iS12(co) from S2l(o}) can be minimised byletting IFo/IRo ^ 1 / 2 .

(iii) Increasing the current level of constant co andconstant IFO/IRO broadens the transfer function whichrepresents the turn-on transient, while the other one isnearly unchanged.

(iv) A finite slope of the waveform of the turn-on current(0=^0) narrows the bandwidth of the transfer functionconcerned. Hence, the driver circuit should be capable ofsupporting an ideal squarewave current during turn-on. (Asuperposed spike at the beginning of the turn-on pulsewould even be more favourable.) A fmite slope at the endof the turn-off current pulse has, on the other hand, a negli-gibly small influence on the transfer function. This seems tobe important because such a slope cannot be avoided inpractice.

In the foregoing, we have used an alternative formulationof the principal problem: The performance of the modu-lator is optimum if S12(cS) = S2l(cS). This statement ismore general than the corresponding one we have used inSection 2: The transient input reflection coefficients shouldapproach a.m. traces. For iS12 =£ S21 even in the case of a.m.traces, if the switching speed during the turn-on transient isdifferent from the switching speed during the turn-off tran-sient.

These guidlines have been taken into account whendesigning an optimum p-i-w-diode. The transfer functionsfor a phase modulator at 34 GHz are shown in Fig. 8; theparameters have been chosen according to w = 0-5/im,IF = 5 mA, IFo/IRo = 1/2, (3 = 0. One sees that the differ-ence between S12(cS) and S21(tS) can indeed be small.

ISI

0-5

335 345t.GHz

20 s

0#

21

-20°

-30°

200'

180*

160*

335 345f.GHz

Fig. 8 Magnitude and phase of the dynamic transfer Junctionsagainst frequency (1 corresponds to the forward, 2 to the reversesteady state; the diode parameters are given in the text)

MICRO WA VES, OPTICS AND ACOUSTICS, SEPTEMBER 1979, Vol. 3, No. 5 203

In order to confirm the theoretical results, measurementshave been performed on a phase modulator at 33-6 GHz.The modulator network is similar to that described inReference 9. The p-i-n-diode is mounted at the end of acoaxial line, which is coupled to a rectangular waveguide.The diode has been fabricated by a double-diffusion tech-nique;11 its dimensions are A = 10~5 cm2, w = 1-1-5/im.The transient reflection coefficients of the modulator areshown in Fig. 9; the measurement set up has already been

forwardstate

Fig. 9 Measured transient reflection coefficients of a phasemodulator at 33-6 GHz

the time interval between two crosses is 02 ns

described in Reference 9. The turn-on trace indeed showsan inductive component as forecast by the above diodemodel. Furthermore, the switching speed decreases whenthe forward steady state is approached. The turn-on timehas been measured to 14 ns. The turn-off transient is dueto an IR0 = 2IF0, its switching speed increases at the end ofthe turn-off period (toff = 1 ns). Another detail, which canbe observed, is a finite capacitive component at the end ofthe turn-off transient. One can conclude that a close quali-tative agreement between theory and measurements hasbeen established.

The various transfer functions of the modulator equi-valent network are shown in Fig. 10. The bandwidth of thetransmission coefficient of the linear circuitry (represented

10

08

06

04

02

0

" /

/

f- /

32-4 33 2 340f,GHz

348 356

Fig. 10 Magnitudes of the transfer functions of the experimentalmodulator against frequency

transmission through the ABCD networkturn-on transientturn-off transient

204

by its ABCD matrix in Fig. 7) is more than four times largerthan that of the dynamic networks. The latter transferfunctions have been calculated by inserting the data of Fig.9 into eqn. 50. A slight asymmetry in the magnitudes ofSn and S21 c a n be observed.

Concluding this Section, the above considerations will beextended to cover 90° switches. The numerical results inReference 1 concerning the overall performance (i.e. signal/noise ratio) of a modulator cascaded with bandpass filtersand a demodulator show that the optimum trace of thetransient reflection coefficients is a Euclidean straight line,which we will call a secant trace. Secant traces can, how-ever, only be realised by introducing a lossy transformation.Hence, it seems more favourable to realise a purely resistivetransient trace, which is called an conduction transition-type trace in Reference 1. Such a trace coincides with aconstant reactance segment in the Smith chart. It can berealised by taking a 180° modulator and transforming itshyperbolic middle-point impedance (i.e. the matchingpoint impedance) into a well-defined point. (This procedurecan be deduced from the ideas laid down in Reference 10.)This task can be fulfilled by a lossless two port. We mayhence conclude that Sl2(co) = S21(w) is also a design goalfor the 90° modulator.

6 ConclusionsAn equivalent circuit for a digital phase modulator has beenderived, which is suited for considering the effect of theswitching transients on the overall system performance. Itconsists of an ideal switch, a dynamic network describingthe transient impedances of the p-i-n-diode, and an idealbandpass filter, which takes the diode embedding intoaccount. Based on the equivalent circuit, guidelines can bederived for an optimum choice of various parameters. Itcan be shown that the agreement between theory andmeasurements is good, although some simplifying assump-tions had to be made.

7 References

1 KUREMATSU, H., DOOI, Y., and SAIKAWA, T.: 'Error ratecaused from transient responses of microwave PSK modulator',Fujitsu Sci. Tech. J., 1972, 8, pp. 83-107

2 HUTCHISON, P.T., BARNES, C.E., BROSTRUP-JENSEN, P.,HARKLESS, E.T., MUISE, R.W., amd NARDI, A.J.:'Repeaterdesign and performance in the WT4 system'. Proceedings of theIEEE international conference of millimetric waveguide systems,London, November 1976, pp. 175-178

3 BOSCH, F., and PETERSON, O.G.: 'Switching performance ofmm-wave pin-diodes for ultra high data rates', Proceedings of theIEEE international microwave symposium, San Diego, June1977, pp. 212-215 .

4 YAMAMOTO, H., and SHITA, H.: '20 GHz band pin-diodehigh speed quadriphase modulator', Rev. Electr. Commun. Labs.,1977, 25, pp. 343-352

5 SCHUNEMANN, K., and MULLER, J.: 'A charge-control modelof the pin-diode', IEEE Trans., 1976, ED-23, pp. 1150-1158

6 SCHLANGENOTTO, H., and GERLACH, W.: 'On the effectivecarrier lifetime in psn-rectifiers of high injection levels', Solid-State Electron., 1969, 12, pp. 267-275

7 LADANY, I.: 'Analysis of inertial inductances in a junctiondiode', IRE Trans.. 1960, ED-7, pp. 303-310

8 SZABO, L., SCHUNEMANN, K., and SPORLEDER, F.: Pin-diode transients in high speed switching applications'. Proceed-ings of the 5th european microwave conference, Hamburg,September 1975, pp. 138-141

9 SCHUNEMANN, K., MULLER, J., DORSCHNER, T.A., andSPORLEDER, F.: 'A reflection-type phase modulator with fastswitching pin-diodes', NTZ, 1975, 28, pp. 319-322

10 DORSCHNER, T.A.: 'Characterization of reflection phasemodulators using hyperbolic geometry'. Proceedings of the 3rdeuropean microwave conference, Brussels 1973, pp. A.9.1-A.9.4

11 MULLER, J.: 'Pin-diodes for the modulation of mm-wavefrequencies', IEEE Trans., 1976, ED-23, pp. 61-63

MICROWAVES, OPTICS AND ACOUSTICS, SEPTEMBER 1979, Vol. 3, No. 5