Chapter 6

BASIC PROPERTIES AND CIRCUIT ASPECTS OF OSCILLATORS AND AMPLIFIERS BASED

ON TWO-TERMINAL DEVICES

INTRODUCTION

This chapter will deal with a number of topics which are relevant to the interaction between a microwave circuit and an active two-terminal element. Starting from a general description of this interaction, we will look at oscillator properties such as stability and noise (FM and AM), as well as injection locking. Some typical circuits and impedance data will be presented. We will also derive the gain-bandwidth product for two-terminal reflection amplifiers, and discuss their noise properties.

A BASIC OSCILLATOR MODEL

The Oscillation Condition

We have seen that the equivalent circuit for Gunn, IMPATT and Tun­neling devices essentially consists of a negative resistance and a capacitance, which are either in parallel (Gunn and the p-n-junction tunnel diode, Figure 6.1a) or in series (IMPATT and RTD Figure 6.l.b). It is also possible to use an alternative equivalent circuit with a negative conductance (Figures 6.1c, and 6.1d). One or the other of the four circuits in Fig. 6.1.a through 6.l.d may be more convenient to analyze a particular device. In some cases it may be convenient to transform the circuit, which was derived on the basis of device physics, to another version, in order to simplify the circuit analysis (compare Figure 4.7 and the accompanying discussion). The typical microwave circuit in which the active element is embedded can be represented by an inductive reactance and a load resistance (or a susceptance and a load conductance). First, ignoring reactances, we can easily see (compare Figure 6.1e) that (1) if the total loop resistance is negative, any initially assumed current in the loop will grow rapidly, while (2) if the loop resistance is positive, any initial cur­rent will be damped out. Only if (3) the magnitudes of the (negative) device resistance and the load resistance are equal, will a steady state condition be possible. Condition (1) above is the starting condition for any oscillator, while if (2) applies, it is impossible for the oscillator to start. The initial voltage is usually a noise voltage (or a transient from turning on the power supply). Oscillators reach a stable, steady state, condition because the magnitude of

S. Yngvesson, Microwave Semiconductor Devices© Kluwer Academic Publishers 1991

144 Microwave Semiconductor Devices

the negative resistance decreases as the RF voltage increases, as we saw in some detail for the IMPATT in section 3.6. A typical curve of the device re­sistance versus RF voltage amplitude is shown in Figure 6.2. The decrease of negative resistance with increasing oscillator amplitude is called "saturation" of the negative resistance.

The power generated by the oscillator and dissipated in the load, under steady state conditions, can easily be calculated from Figure 6.1e., and is found to be

1 2 PD = 2RLIRF

where the steady state condition

(6.1)

(6.2)

has been used. We want a large value (at saturation) of IRD I for maximum PD. If we use the more complete equivalent circuit of Figure 6.3 (Kurokawa, 1969), then it becomes clear that in order to maintain a steady state oscillation, we must have zero RF voltage in the loop, and this leads to :

(6.3)

If we take the real part of this equation, we will obtain (6.2), while the imaginary part yields:

(6.4)

The last equation says that the device capacitance must be resonated with the circuit inductance.

Stability of the oscillator when a small perturbation is imposed

We can continue to use the model defined in Figure 6.3, to discuss the stability of the oscillator (Kurohwa, 1969). Of the variables defined in Figure 6.3, RD and XD depend on both frequency wand the loop current 10*, while typical circuit RL'S and XL'S vary only with the frequency. If we have a stable steady state condition for this circuit, then we require that the circuit returns to this condition if we expose it to a small change in either current amplitude or frequency. To discuss this physically, let us first introduce a small increase in amplitude for an oscillator which is in steady state. The increased amplitude will saturate the negative resistance further, and the net loop resistance will become positive instead of zero. The temporary increase of the amplitude will therefore be damped out, and the rate at which the steady state amplitude will be approached will be faster the larger the increase was. If instead we assume a temporary decrease in the amplitude, the negative

* 10 is the steady-state value of IRF. It would, of course, be just as valid to discuss the oscillator in terms of a circuit such as Figure 6.1.c, adding a load admittance YL = GL + jBL . In this case, PD = lIGDJV~F' and PD is maximized for a large IGDI.

Chapter 6 145

-Ro

~ ~\--o -Ro Co

(a) (b)

-Go

~ ~\--o -Go Co

(c) (d)

-RoDe (e)

Figure 6.1. Different ver6ion6 of the equivalent circuit for an active two­terminal device (aJ Parallel circuit with negative re6i6tance (b J Serie. circuit with negative reli6tance (cJ Parallel circuit with negative conductance (dJ Seriel circuit with negative conductance (eJ negative reliltance with a load reliltance (equivalent circuit at relonanceJ.

Steady State Point

,,/ -------. RL

IRF Figure 6.2. Magnitude of the negative reli6tance al a function of the RF

current. The Iteady Itate point i, the point for which 1- RD 1 = RL.

146

AcrrVE ELEMENT

Microwave Semiconductor Devices

L-y-J Load

Figure 6.3. Equivalent circuit used for calculation of oscillator characteris­tics.

resistance will increase, and the amplitude will again be restored to steady state. The situation clearly is one of a stable equilibrium if the amplitude dependence of the negative resistance saturates as shown in Figure 6.2. The imaginary part of the oscillation condition should also be taken into account. For example, if XD depends on the amplitude as it usually does to some extent, then a small change in amplitude will also change the oscillation frequency momentarily. The restoring forces which bring the frequency back to the steady state frequency are generally not quite as effective as those for the amplitude, and random changes in frequency, "FM-noise", will occur which we will discuss below.

In order to make a quantitative estimate of the response of an oscillator to a small perturbation, we shall use the fact that the changes are assumed to be very small. We define small changes in 10 and in w, i.e. we are assuming a Taylor expansion of the variables, retaining only the first order terms:

8RD 8RD 10 -+ 10 + Hj RD -+ RD + 7fTH + 8w 6w + ...

w -+ w + 6wj RL -+ RL + {)8~L ow'" with similar equations for XL,D (6.5)

Making use of the fact that the steady state quantities already are "in balance", we find by substituting (6.5) in (6.3), and neglecting products of small quantities,

-I {)RD H 1 8(RL - RD) 0 = 0 o 81 + 0 8w w (6.6a)

I 8XD H 1 8(XL + X D ) 0 = 0 o 81 + 0 8w w (6.6b)

If the circuit is stable, then a solution to this system of linear equations must exist for which 81 and 8w are zero, the so-called trivial solution. If non­zero solutions exist, then the determinant for the system of equations would

Chapter 6

ACITVE ELEMENT

1

Figure 6.4. Equivalent circuit for an injection-locked o6cillator.

147

be zero, while for the above case the determinant formed from the coefficients must not be zero. This is useful to keep in mind in designing oscillators - it would in general be an accident, though, if the determinant were to cancel.

In order to study the effects of noise and injected signals on the oscillator, we next introduce a small RF voltage in the loop, as shown in Figure 6.4. If we add this voltage to the loop equation, (6.3), then our equations describing incremental changes, (6.6a and 6.6b) will have driving terms on the right-hand side of the equations:

Solving these equations for 61 and 6w, we find by Kramer's rule:

61 =vlcost/J 10 sin tP

8(R£-RD) I 8(X:~XD) /a

8",

where a = determinant of coefficients in (6.7 a,b).

(6.70)

(6.7b)

(6.80)

(6.Sb)

To proceed, we make use of Foster's reactance theorem (see for example (Pozar, 1990, sec. 5.2):

[for a parallel resonant circuit 1

8B 2GQ 8w = ---;;;-

8X 2RQ 8w = ---;;;-

(6.90)

(6.9b)

148 Microwave Semiconductor Device,

[for a series resonant circuit 1 If we apply this to our oscillator circuit,we obtain:

Q = W 8(XL + XD) /2RL 8w

Note: The Q-value here is the passive circuit Q, without negative RD!

(6.10)

In order to simplify our equations without sacrificing generality, we assume that the device resistance is essentially independent of the frequency, and that the device reactance is independent of the current amplitude, i.e.

8RD ~ O. 8XD ~ O. (6.11) 8w -, 81 - ,

The solution to the determinantal equation for 6w then becomes

'II 1-!!lJf- 10 cos rP 1 o sin rP _ 'II' sin rP • 10

6w ~ 1-~1 1 8(RL -RD ) 1- 128(XdXD) (6.12) 81 0 0 8", 0 flw o 10 8(XL + XD)/8w

which with the help of Foster's reactance theorem becomes

6w 1 'II sin rP -=-X---j w 2Q 10RL

(6.13)

This is a very important and useful expression. It has been derived using certain approximations, but is easy to interpret because of its simple structure. We shall use it to discuss (1) injection-locking of oscillators and (2) phase(FM)­noise in oscillators. A vector diagram of the voltages in the oscillator circuit is shown in Figure 6.5. Note that the factor !!lJf- cancelled in this derivation. This indicates that the feedback effect which stabilizes the amplitude is ineffective against frequency fluctuations.

INJECTION LOCKING OF OSCILLATORS

The injection locking situation corresponds to the circuit in Figure 6.4, with a small sinusoidal voltage produced by the source'll expjrP, the locking ,Duree. This source must be fairly close in frequency to the frequency produced by the oscillator when it is free-running. Equation (6.13) says that there is a solution to the loop equations for a range of 6w's such that sin <p is not required to be larger than 1. If we increase the amplitude ('II) of the locking source, then the frequency range over which injection locking is possible will increase proportionately. The bandwidth over which locking is possible is called the locking bandwidth. Also, the output power of the locked oscillator divided by the output power of the locking oscillator is termed the locking gain. We can calculate these powers from (6.13) at the limit of the locking range by noting that this limit is defined by <p = 1r /2, i.e. the two sources are in quadrature

Chapter 6 149

or

Figure 6.5. Pha60r diagram which 6how6 the addition 0/ the three different voltage, around the loop 0/ an injection-locked oscillator circuit.

at that point. Thus, the two powers can be calculated separately and are proportional to the respective voltages squared. We find by squaring (6.13):

( 5W) 2 1 Pin ~ max = 4Q2 X P "",I

(6.14)

Here, Pin is the power delivered by the locking oscillator, while P ""', is the output power from the locked oscillator.

This is a useful approximate equation for the locking bandwidth (full BW = 25w). Note the dependence on the Q-value - if we want a wide locking bandwidth, then we must use an oscillator with a low-Q resonant circuit.

Returning to the vector diagram, Figure 6.5, we note again that the locking voltage causes the largest change in frequency when it is in quadrature with the output voltage. Imagine that initially the two voltages are of different frequencies - the locking voltage will attempt to drive the oscillator voltage around in such a way that its frequency will change. The new steady state will occur when the two oscillators are at the same frequency, but out of phase by t/I. It is also clear that if the locking voltage is small compared with the oscillator voltage, it will not be able to change the frequency as much.

Injection locking of solid state oscillators is used in practise quite a lot. It enables a higher power oscillator to be controlled by a lower power one (often less noisy, see the next section). In this manner, the higher power oscillator can for example be made to perform frequency "hopping" or be frequency modulated, provided we stay within the locking bandwidth.

150 Microwave Semiconductor Device5

MODEL FOR FM- AND AM-NOISE IN OSCILLATORS

Equation (6.13) can also be used to predict the amount of random phase­(or frequency-) modulation which will occur in an oscillator. For this situation, we assume that the "locking" source v exp j¢ is a noise voltage. The phase with respect to the oscillator will thus take any random value, and the average of sin2 ¢ will be 1/2. The oscillator spectrum will be spread out due to this modulation, as the noise voltage tries to "lock" the oscillator. If we re-write the equation as follows

I R _ ~ . v sin 4> o L - 2Q (6;)

(6.13a)

then we see that the output voltage with a particular frequency shift 8w drops as the frequency deviation increases. If we set the instantaneous frequency change 8w = Wd and the average W = wo , we find by again squaring (6.13) :

2 ---

( Wd) = _1_ X v 2 sin2 ¢ WO 4Q2 I~Rl

(6.15)

In the simplest case, the noise voltage is the Nyquist-Johnson thermal noise, produced by any resistance RL , and given by (see Chapter 8):

(6.16)

kB = Boltzmann's constant; T = absolute temperature; B = bandwidth;

We shall find that the noise sources in solid-state oscillators produce greater power than that given by pure Nyquist-Johnson noise. In anticipation of this fact, we shall already at this stage introduce the excess noise measure eM' by which we will multiply (6.16) in the expressions given below. For the thermal noise case, we only need to set M = 1.

The average oscillator power at a particular Wd is now found as:

(6.17)

Normally, noise power is referred to the total oscillator power, the "carrier" power, and we divide by this to obtain:

[PNOISE(at "'d)] 1 ("'0)2 kBTB X M

POARRIER SSB,FM = 2Q2 "'d POARRIER (6.18)

Sometimes FM-noise is characterized in terms of "RMS" frequency de­viation instead of the above noise to carrier ratio. The equivalence can be found

Chapter 6 151

by regarding the oscillator output at the frequency 10 + J". = "'·t"" as the following frequency-modulated signal, with a "carrier" voltage amplitude Vo:

.(.) ~ Va ~ {w .• + 'I.' ";;~~M" i} (6.19)

where the coefficient fl./p is the peak frequency deviation (it is easy to check that

_ (d(fl.I/I») fl./p fl.wp = ~ max = 27rJ". X T = 27r.l:J./p)

In general, fl./p may depend on the deviation frequency, J"., in a manner which must be consistent with our previous expression for FM-noise, (6.18). We explore this correspondence by expanding (6.19), using standard trigonometry, and assuming ~ « 1:

{ -fl./,

2/1. P sin [211'(/0 + Itl)t + 1/1] <= vet) ~ Vo cos wot fl./,

C~R - 2/: sin [27r(!0 -Itl)t - 1/1] <=

Upper S.B.} (6.20)

LowerS.B.

For a given J"., we thus have a sideband on either side of the carrier (see

Figure 6.6). The amplitude of each side-band is *- (voltage ratio). The corresponding power ratio is the square of this, and we have the following equivalence between the two measures of FM noise:

VSSB(/.I) = fl. / Pj PSSB(!tl) = (fl./p)2 Vo 2/.1 POARRIER 2/1.

(6.21)

It is more common to use the RMS frequency deviation rather than the peak value, and this gives us a further factor of ..;2, i.e.

PSSB = ! (fl.IRMs)2 POARRIER 2 /.I

(6.22a)

PDSB = (fl.IRMs)2 PCARRIER II.

(6.22b)

i.e. (using (6.18»

fl.IRMs = I. JkBTB)( M Q PCARRIER

(6.23)

For the phase-noise, we have

(6.24)

152 Microwave Semiconductor Devices

P(f)

LSB fo USB

Figure 6.6. The FM noise spectrum of an oscillator. Two sidebands, each at a distance of f tl from the carrier frequency fo, with bandwidth B, are indicated.

In either case, the bandwidth over which we measure the noise must be specified. In scaling between different bandwidths (all sources seem to quote noise data for oscillators with different bandwidths!), the noise power to car­rier power ratios scale proportional to the bandwidth, as we can see from the Nyquist thermal noise formula. The RMS frequency deviation, AfRMs, how­ever, scales with the square root of the bandwidth, compare (6.22). All of the above assumes that the basic noise mechanism gives "white", frequency­independent noise. We also note from the above results, that the noise to carrier power ratio falls as one over the deviation frequency squared (20 dB per decade), while the RMS frequency deviation is independent of the devia­tion frequency. Typical plots of the two quantities thus would look as shown in Figure 6.7.

AM-noise in Oscillators

The AM noise of an oscillator can be found by using (6.8) with 8w = o. This results in the following expression

[ PNOISE ] = 2(kTB X M)/~CARRIER

PCARRIER AM,SSB 4Q2 (!'!.<I.) + S2 e.zt Ul o

The factor S is the saturation factor given by:

S __ 8RD/81 X 10 •

- RL '

(6.25)

(6.26)

Chapter 6 153

I Slope = 20 dB/decade

'120 dB -J------I I

I I I I

'140 dB -t------+-------I I I

I I I I I I I I I

-1----+---+---+---- fd(Hz)

100 103 104 loS

I

I I I

MRMs I

I I

10,2 Hz f I I I

I I

fd(Hz)

100 103 104 loS

Figure 6.'1, Two equivalent repre,entotiom of the FM noile .pectrum of an o,eillator. The driving noile ,ouree i. a"umed to have the eharaeteriltic. of white noile.

154 Microwave Semiconductor Devices

A large value for S is obtained if the negative resistance changes rapidly with the oscillator amplitude, and this means that the feed-back mechanism which brings the amplitude back toward the steady state value is more effective, as we discussed above, based on a simple physical model. Because 52 normally

is quite large compared with 4Q2 (~f near the carrier, we find that the

AM noise is typically much less per unit bandwidth than the FM noise, at moderately large Id. We can estimate a typical value of 5 for an IMPATT oscillator from Figure 3.27a, and use the RF voltage dependence instead of the RF current dependence - these should be equivalent. At the point of the curve with maximum output power, S is about 2.5. The saturation factor will be the dominant term in the denominator of (6.25) up to frequencies of about 50 MHz in this case, if we assume that the Q-value is 250. Thus we predict a l1at AM noise spectrum up to about 50 MHz, and a frequency-dependence as 1/!l above that.

l/C or "flicker" noise

The noise spectrum of oscillators can have the most deleterious effects for low values of the deviation frequency, Id. Examples of systems which require low close-to-carrier noise are doppler radars. At these low frequencies (kHz or 10-100 kHz) the basic noise mechanism which modulates the oscillator is often not the "white" thermal noise or some other "white" noise mechanism, but 1/ I or "l1icker" noise. The latter type of noise occurs in many semiconductor devices, as well as in many other random processes. The 1/1 character arises because there is a continuous distribution of "time-constants" of the different events which cause the noise. The most common cause of 1/ I noise in semi­conductors is surface effects, such as surface traps which can randomly capture or release electrons. The quality of the surface, and the processes used to clean the surface, have a major effect on the amount of 1/ I noise observed in the de­vice. Noise processes of the 1/1 type will be discussed in some further detail in Chapter 8. If 1/ I noise dominates at low frequencies (baseband), nonlinearities in the device can convert this baseband noise to microwave frequencies near the carrier frequency. The up-converted noise source can now be substituted for the thermal noise which we used in (6.18) and (6.25). The up-converted noise (as a function of h) follows the same 1/ I-dependence. The consequence is that an additional li l factor is added to the frequency dependence. This additional frequency factor will result in the following dependencies for the noise expressions quoted above

FM - Noise: {

( PNOISB ) ex (I )-3 } POARRIBR SSB d

t1IRMS ex (fd)-1/2

AM N · (PNOISE) (I )-1 - Olse: ex d PCARRIER SSB

(6.27)

(6.28)

(6.29)

Chapter 6 155

Effects due to 1/1 noise are most commonly observed in Gunn oscillators, which have a much better noise performance than IMPATT's, as we shall see below.

FM- Noise in Injection-locked Oscillators

Kurokawa (1969) has derived an expression for the RMS frequency devi­ation of an injection-locked oscillator (DSB case):

( AI )z_lJxkBTXBXM/Po. ~ RMS - 2 '

(Q.~t~) + ~cosztP (6.30)

Here, Po is the output power, Pi is the injected power and tP is the phase­angle as defined earlier in this chapter. The factor Po/Pi is usually called the "locking gain".

This expression tells us that the FM-noise of an injection-locked oscillator can be made as small as that of the locking oscillator, for low values of Ill, as illustrated in Figure 6.8. At higher la, the noise reverts to that which is obtained in the free-running oscillator.

The AM noise is not at all affected by the injection-locking, however.

ACTUAL NOISE OBSERVED IN TWO-TERMINAL SOLID STATE DEVICES

Noise in Gunn devices

In both Gunn and IMP ATT devices, the measured noise is higher than what is predicted from the above theoretical model if a value of M = 1 is assumed for the noise measure.

Several sources (quoted in the figure caption) have been consulted to ex­tract the curve of FM noise (expressed as the noise-to-carrier power ratio), versus la, in a 1 Hz bandwidth, for typical 5-10 GHz Gunn oscillator devices, shown in Figure 6.9. The corresponding ~/RMs is typically from less than 1 Hz to a few Hz, in a 1Hz bandwidth, at ill = 1 kHz. The frequency-dependence of the noise-to-carrier ratio is closer to liS than liz, indicating that the basic noise mechanism at low frequencies is of the 1/ I-type. A white noise mech­anism takes over at III = 100 kHz to 1 MHz (frequency-dependence liZ). The dependence of the FM noise on cavity Q-value is about as expected from (6.18), if the same device is used in different cavities. FM noise data for Gunn oscillators at millimeter waves (40-60 GHz) show a somewhat higher level, see Figure 6.10, for both InP and GaA6 devices. The AM-noise in a 1 Hz band­width is much lower, typically -140 dB below the carrier, even for a millimeter wave device, see Figure 6.11. As expected from (6.25), the AM-noise should

156 Microwave Semiconductor Device,

100 r------7,;-;;;--;;;;;;;;;;;;::;rs=;;::::::::::::J

10

1.0

Pose = IOOMW

Fose ,. FSVNC "9.72 GH,

Ox :20

/

/

/-\000 /

",'/ ~40D8 0.1 F;c::~"-----

0.01

--------- CALCULATED DATA

0.001 L. ____ ---L _____ --'-____ ---'

II( 10K lOOK MODULATION FR£OUENCV IN HERTZ

Figure 6.8. FM noise, plotted a. 1::..IRMs (units Hz/Hz bandwidth) lor an injection-locked o.cillator. The different curve, corre,pond to different value, for the locking gain, a. marked. Adapted from ASHLEY, J.R. and PALKA, F.M. (1970). "Mea.ured FM Reduction by Injection Pha.e Locking," Proc. IEEE, 58, 155, @1970 IEEE.

be independent of /d and the Q-value, at low deviation frequencies. The AM noise depends weakly on frequency away from the carrier. H this is taken as evidence ofa 1// noise process, then the up-conversion of the 1// noise to the carrier frequency apparently follows a different dependence on ttl than that for the FM-noise. The very low AM noise makes Gunn devices ideal as local oscillators for mixers. The noise temperature of a mixer receiver is influenced by the AM noise of the local oscillator, but not by the FM-noise (see Chapter 9). It is the noise power of the oscillator at a frequency away from the car­rier, equal to the IF frequency of the mixer, which can be converted to the IF frequency, and thus appear at the receiver output. For ttl equal to a typical IF-frequency of 1 GHI, the AM noise level of a mm-wave Gunn oscillator was measured to be -170 dB below the carrier (Kuno, 1981).

Chapter 6 157

-so -40

- -50 ID

" -60 -u Q, -70 ....... z Q,

-80 iii II) -90 .... D z -100 I z IL -110

-120

-1S0 10· 10" 1011 10·

FD (HZ)

Figure 6.9. T!ipical SSB FM noi.e power per unit bandwidth, normalized with re.pect to the carrier power, lor Gunn o.cillator. in the 5 - 10 GHz range. Selleral .ouree. halle been u.ed, a repre.entatille one i. JOSENHANS, J. (1966). "Noi.e Spectra 01 Read Diode and Gunn O.cillator.," Proc. IEEE, 5-1, 1-178, @1966 IEEE.

The excess noise measure, M I can be derived from FM noise data similar to those in Figure 6.10, for a 35 GHII oscillator, see Figure 6.12. Even in the region above 1 MHz, for which 1/ I-noise is negligible, the excess noise measure is about 25 dB. The driving noise process in the device is thus one which is much more powerful than thermal noise. Gunn elements with a low n x L­product can also be used as low-level amplifiers, as discussed later on in this chapter. These amplifiers show a noise figure of about 10-15 dB, typically, indicating an internal noise source with a noise power per unit bandwidth about ten times that of thermal noise. It therefore appears that the excess noise process in Gunn devices depends on the level of excitation as well. The

158 Microwave Semiconductor Devices

o ----- -1 --I -1---

1k 10k u,"" 1M 10M , 100M

O"sel from carrier-Hz

Figure 6.10. FM noise (in dBcl Hz bandwidth, SSB) of InP and GaA" Gunn oscillator. in the 40-60 GHz band. Data for the oscillators repre.ented by the following symbol" are: e-InP, 60 m W, Q. = 125,- o-GaA", 29 m W, Q. = 690,­·-GaA", 12 mW, Q. = 1390. Reprinted from EDDISON, I.G. and DAVIES, I. (1982). "InP May be the Power at Millimeter Wavelengths," Microwaves tJ RF, Vol. 21, October 1982, p. 77, with permission.

reason for the high noise measure is at least qualitatively known, as discussed below.

One possible source of the excess noise might be the fact that the electrons heat up to an elevated "electron temperature", T., as we discussed in Chapter 2. The value of T. is of the order of 1,000 to 2,000 degrees, however, which would only explain an excess noise measure of a few dB. We apparently need to look a little more closely at the fluctuations in the electron current under conditions of negative differential mobility and high electric fields. Several

Chapter 6 159

-140

-145 -al 'C -u -150 a. ...... z a.. -155 uj en H 0 -160 z I

:E ~

-165

-170 10· 10" 1011 10·

FD (HZ)

Figure 6.11. Typical SSB AM noise power per unit bandwidth, normalized with respect to the carrier power, lor Gunn oscillator •. Several .ouree. have been used, a representative one is JOSENHANS, J. (1966). "Noise Spectra 01 Read Diode and Gunn Oscillator,," Proc. IEEE, 54, 1478, @1966 IEEE.

different types of fluctuations, i.e. "noise mechanisms", are described in some detail in Chapter 8. In particular, we will use the results from that chapter, regarding the "diffusion-impedance-field" model, introduced by Shockley et al. (1966). In this model, the noise voltage produced at the terminals of the device by a small section of the device is found. The contributions from such individual sections are then weighted by the "impedance-field", and added, to give the total noise voltage. The impedance-field converts current fluctuations to voltage fluctuations, and is defined as:

VZTX = E(z)/I (6.31)

160 Microwave Semiconductor Devices

50

201L~~~~1~O--~~~1LO~2~~~~1LO~3~~~~1LO~'~

Offset frequency from carrier (103 Hz)

Figure 6.12. Comparilon of the noile measure (M) for 35 GHz Gunn oldl­lator •. The upper curve i. for InP, and the lower one for GaA •. Reprinted from EDDISON, 1.G., (1984). "Indium Pho.phide and Gallium Arsenide Tranlferred-Electron Device.," in Infrared and Millimeter Wave" K.J. But". ton, Ed., Academic Preu, Orlando, FL., Vol. 11, Ch. 1, p. 1, with permil­.ion.

Here, both E(z) and I are ac small-signal quantities. In Chapter 8, we derive (8.53). which is a general expression for the noise in a bulk semiconductor, valid under "hot-electron" conditions. We also show there that for the special case of thermal equilibrium, this expression reduces to the normal Nyquist thermal noise formula. The more general noise mechanism is usually termed "diffusion noise". In analogy with this equation, we find for the voltage fluctuations at the terminals of the device (Thim, 1971):

(6.32)

The diffusion constant under high electric field conditions thus plays a major role in determining the noise output from the device. Figure 6.13 shows D"

Chapter 6 161

versus electric field for GaAB. * In analogy with our discussion of the mobility for GaAB in Chapter 2, we can distinguish contributions to Dn from the two valleys, as well as from fluctuations due to electrons transferring between the valleys:

(6.33)

In the second term, 111 and 112 are drift velocities in the two valleys, and 1112 and 1121 the inter-valley scattering frequencies. The peak in the diffusion constant occurs close to the critical field, where the velocity also has a peak. Close to this electric field, inter-valley transfers of the electrons are common, while for either higher or lower fields the electrons mostly stay in one or the other of the valleys - this explains why the diffusion constant has a maximum at the critical field.

Thim (1971) first assumed a uniform electric field through the device, and calculated the noise figure of a stable Gunn amplifier (see Chapter 8 for the definition of amplifier noise figure, F) based on the above method. The resulting expression is quite complicated, but simplifies to the following one if the n x L-product is very small « 1010 cm- 2 ):

F = 1 + eD .. x _1_ (6.34) kBTo 1- I'NI

The magnitude of the NDM is used in this expression. If D .. = 300 cm2/B is assumed, as well as I'N = -2800 cm2 /V 8, we find a noise figure of 7.1 dB. Typical early measured noise figures for GaAs Gunn amplifiers were 15 dB, with 10.5 dB the lowest ever obtained by the mid-1970s. Thim (1971) shows, however, that if the electric-field distribution is made non-uniform, the noise figure always increases. The typical electric-field distribution is in fact more like the one shown in Figure 2.9., and we can obtain reasonable agreement between experiments and theory, using numerical calculations, which take into account the correct field-distribution (Sitch and Robson, 1976). The diffusion constant for InP is smaller, and lower noise figures are therefore predicted for this material. The noise figure has a minimum for n x L close to 1010 cm- 2 • It is understandable that the noise figure increases for larger n x L, since space­charge fluctuations grow more quickly, the larger n x L is, as we saw in Chapter 2 - this should make random fluctuations grow faster, and give rise to larger noise.

Recent work, especially on InP amplifiers, has confirmed the lower noise figure for InP, and an 8dB noise figure has been measured at 40 GHz, as well as 10 dB at 60 GHz. The slow rise of the noise measure with frequency is unusual and advantageous. Recent developments of three-terminal devices, such as MESFETs and HFETs (Chapters 10 and 11), are rapidly making Gunn

* Earlier curves of the electric field-dependence of D .. are fairly different. Glisson et al. (1980) argue that their new measurements are more correct.

162 Microwave Semiconductor Devices

10

Vl 8 <' ...

E u

N 6 0

.2 4 >

.~

:::l :::: a 2

Electric field, c (kY/cm)

Figure 6.13. The measured diffusion coefficient for GaAs as a function of electric field. Reprinted from GLISSON, T.H., SADLER, R.A., HAUSER, J.R., and LITTLEJOHN, M.A. (1980). "Circuit Effects in Time-of-Flight Diffusivity Measurements," Solid State Electron., 23, 631, @1980Pergamon Pre" pic.

amplifiers obsolete, however. It is worth noting that diffusion noise constitutes the main noise source in these devices as well, and we will discuss these topics at length in later chapters.

Large-signal Gunn amplifiers use material with larger n x L products, which results in a larger negative resistance. The noise figure is consequently also higher, approaching the noise measure we mentioned earlier for Gunn oscillators (about 25 dB in the white noise region, see Figure 6.12). In the oscillator case, GaAs and InP devices have essentially equal noise properties. No generally accepted theory exists for the oscillator case, which represents a much more difficult, nonlinear, problem, with growing high-field domains, rather than a stable electric field distribution.

Chapter 6 163

Noise in IMPATT Devices

Typical FM noise for silicon-based IMPATT oscillators has been plotted in Figure 6.14. (Chang and Kuno, 1990). It is not very different from that ofa typical Gunn diode, although fairly large variations (±10 dB) may be found in individual devices of either type. The frequency-dependence of the FM noise for Si IMPATTs indicates a white noise process. The AM noise ofIMPATTs is especially sensitive to the bias circuit, which needs to have a high impedance for low noise. Some measured IMPATT data probablY show high FM noise levels due to insufficient attention to such effects. Selected IMPATTs may have as low AM noise as Gunns at Id's in the kHz range, see Figure 6.15. (Chang and Kuno, 1990). At frequencies relevant for local oscillator applications, however, IMPATTs typically have at least 10 dB higher AM noise, as also shown in this figure. Thus, Gunns are preferable as local oscillators for mixers, as discussed above.

Silicon and GoAl IMPATT oscillators have noticeably different noise lev­els, as shown in Figure 6.16., which gives data from (Okamoto, 1975) for 50 GHz devices. The GoA., device shows evidence of a 1/ I-type noise process. The excess noise measure, M, was derived from these data with the help of equations equivalent to (6.18) and (6.25), and is noted in the figures. M is about 10 dB lower (27.1 dB) for the GoA., device, a noise measure which is comparable to that for Gunn oscillators. It is also found that the noise measure increases rapidly close to maximum output power conditions for the oscillator. The noise figure for amplifiers using the same devices was found to agree well with the noise measure derived for the oscillators.

Noise processes in avalanche devices are reviewed by Gupta (1971). Ini­tial theories, such as the one due to Hines (1966), assumed small-signal noise voltages. The diode was assumed to consist of separate avalanche and drift regions, with the noise originating in the avalanche region, only. The key fea­ture of the avalanche process is the fact that the avalanche starts out with a very small number of carriers, and then grows by a large factor. Thus, small fluctuations in the time at which the avalanche is initiated can give rise to a large fluctuating component in the current. One can also see why silicon devices are noisier than GoAB devices: For GoAB, an = a p , and the effective distance for ionization is about equal to WA , the avalanche region width. For silicon, an » a p , and the lower ionization rate for the holes means that the effective ionization width must be large than WA. In practice, WA is given, and the silicon device will require a larger number of ionization events for the same current, making it even more susceptible to "jitter" in the timing of the avalanche, i.e. the silicon device is more noisy.

Hines (1966) found the following expression for the noise measure of an amplifier:

(6.35)

164 Microwave Semiconductor Devices

-40

-60 -lEI 'tI -CJ -80 Il. "-Z 11.

uj -100 en f-f 0 Z I -120

2: II..

-140

101 10<4 iOIl iO· i07

FD (HZ)

Figure 6.14. Typical DSB FM noise power per unit bandwidth, normalized with respect to the carrier power, Jor silicon-based IMPATT device,. Adapted from CHANG, K., and KUNO, H.J. (1990). "IMPATT and Related Tran,it­Time Device,," in Handbook of Microwave and Optical Components, K. Chang, Ed., John Wiley & Sons, New York, Vol. 2, Ch. 7, 305, and other sources.

Here, VA is the voltage across the I\valanche region, and TA is the transit time across this region. The factor' is the coefficient in the expression a = ao x E(. As an example, for silicon, = 6 and with c.I = 2c.1A and VA = 3V, we find M :::: 40 dB. Equation (6.35) was derived with the assumption that a" = a p and is valid for IMPATT (small-signal) amplifiers, not oscillators. One feature of (6.35) is that it has a maximum for a frequency equal to the avalanche frequency, c.lA. The minimum noise occurs at about twice c.lA (Sze and Ryder, 1971). Since the maximum negative conductance occurs between

Chapter 6

o ~ '" ~~ ~~ uo

-140

O~ -150 '1m WN "':>: 0 .... Zz ~-« -160 m '" o

165

94 GHz

-170~1--~----KH~Z~~~~--~--~~~~~ir----~~-------G~~-Z--~~~-J· FREQUENCY AWAY FROM CARRIER (FM)

Figure 6.15. DSB AM noi,e power per unit bandwidth, normalized with re­'pect to the carrier power, for ,everal millimeter wave device,. Solid line, in­dicate ,ilicon-ba.ted IMPATT" da,hed line, Gunn device8, and da.thed-dotted line.t, kly.ttron,. Reprintedfrom CHANG, K., and KUNO, H.J. (1990). "IM­PATT and Related Transit-Time Device,," in Handbook of Microwave and Optical Components, K. Chang, Ed., John Wiley tJ Son., New York, Vol. f, Ch. 7, 305, with permiuion.

these values, some compromise between maximum gain (or oscillator power), and noise is necessary.

While the above theories are valid for the small-signal case, Statz et al. (1976) developed II. large-signal theory of noise in Read-type IMPATTs. The large-signal treatment of the oscillator current wave-form follows the general method presented in Chapter 3 of this book, i.e. the voltage is assumed sinu­soidal, and the current is expanded in terms of modified Bessel functions. A crucial additional feature is the introduction of the reverse saturation current: this is the value of the current before the avalanche starts (compare Figures 3.10 and 3.15, in which the current is assumed to go to zero at this point; the smallest current can not be zero, but is equal to the saturation current!). If the saturation current is low, then shot noise fluctuations in I'Gt will be large, and these result in large fluctuations in the output current after it has been

166 Microwave Semiconductor Devices

• b

-". r""··-·-""·"~""~ -10

-20 iii -125 iii S S -30

" -130 . u 0. ~ -~O .... 'iii iii -50 ., -135 ., e e

, -"'[ z -60 0.

-70 ..; uI ., '" -BO ~ -145 H H-27.laB 0 z z -90 >: x c -ltJO . ...

-100 ',--------1~5 J,.......J...l.--'---LJ.J., .... ! '1",,11 -110 ",111 I "II"" ,

I~' 10' 10' 10' 10' 10· 10' 10' 10· 10·

FU UIZ) FDOtZ)

Figure 6.16. Comparison of noise spectra for Si and GaA6 IMPATT device6 at 50 GHz. (a) AM noise (b) FM noi6e. Both figure6 give DSB noise power per unit bandwidth, normalized with respect to carrier power. Adapted from OKAMOTO, H. (1975). "Noise Characteristic6 of GaA" and Si IMPATT Diode" for 50-GHz Range Operation," IEEE Tran". Electron Device", ED-22, 558, @1975 IEEE.

Chapter 6

MFM lPOU• [dB] [watt]

55 5

50 4

45 3

40 2

35- I

8 9 10

Frequency [GHz J

167

Diode A

MFM Experiment

Ide' 0.45 [omp]

Figure 6.17. Theoretical and ezperimental noi6e mea6ure al a function 01 frequency lor an IMPATT diode. Reprinted from STATZ, H., PUCEL, R.A., SIMPSON, J.E., and HAUS, H.A. (1976). "Noi6e in Gallium Ar­unide Avalanche Read Diodel," IEEE Tran6. Electron Device6, ED-23, 1086, @1976 IEEE.

"amplified" by the avalanche process (and vice versa for a larger I,at). The saturation current was varied by changing the temperature, and good correla­tion was obtained between I,al and the noise measure, M. Figure 6.17 shows the good agreement obtained between measured and calculated noise measures as a function of frequency for one case.

ELECTRONIC TUNING OF SOLID STATE OSCILLATORS

Solid-state oscillators are typically tuned electronically by adding a var­actor diode to the resonant circuit, and then varying the capacitance of the varactor with a DC bias voltage. The schematic circuit is as shown in Figure 6.18.

168 Microwave Semiconductor Devices

Figure 6.1S. Equivalent circuit of a Gunn or IMPATT oscillator, which is electronically tuned by mean" of a varactor.

We can derive some general properties of such circuits by using Slater's perturbation theorem for resonant circuits (Harrington, 1961):

a/ aw.-awm

/0 Wm+W. (6.36)

Here, W. is the stored electric energy and Wm is the stored magnetic en­ergy in the circuit. The quantities with a are small changes in these energies. We also need to introduce the stored (electric in this case) energy of the varac­tor, Ww• This stored energy can be changed by a factor of ±"Ww, by changing the capacitance. The stored energy in the oscillator circuit is Woo Therefore, at resonance, the total stored energy is

W. + Wm = 2W. = 2(Wo + Ww)

If we apply (6.36), we find that the tuning range of the oscillator is

a/ _ ± "W. /0 - 2(Wo + W.)

From the definition of the Q-value

(6.37)

(6.38)

Q = 211" x STORED ENERGY/ENERGY LOST PER CYCLE (6.39)

and setting Pw = power lost in the varactor, and Po = power lost in the load plus the oscillator circuit, we get

(6.40)

Chapter 6 169

If the oscillator circuit losses are small, then Po is approximately equal to the output power. We further introduce p .. as the available power from the negative resistance, i.e.

p .. = Po + P~ (6.41)

If we introduce the normalized variable 21 through the relation

_ (P .. - Po)Q~ Z = PoQo

(6.42)

we can write the equation for the frequency tuning as

(6.43)

In order to maximize the tuning range, we need to maximize 21, which requires a maximum value for the varactor Q-value, Qv' Typical values for

_ Cm .. " - Cmin ""' 0 25 'Y - Cm"" + Cmin - .

(6.44)

and typical Q's are of the order of 10-30 for both the circuit and the varactor. If further about half of the available power P" appears as output power, Po, and the other half is dissipated in the varactor, we find that z is about 1. The expected tuning range then is:

AlII -, ~ ±0.25 x - x - ~ ±6%

o 2 2 (6.45)

At higher frequencies, Q-values are smaller and the tuning range is smaller as well.

A type of tunable oscillator which has a substantially wider tuning range uses a magnetically resonant YIG-sphere as the tuning element. The YIG sphere has a very high equivalent Q-value, and can also be tuned by a larger factor than the limited gamma for a varactor allows. YIG-tuned oscillators thus have tuning capabilities in the octave range.

One should note that tunable oscillators generally have lower Q-values than fixed tuned ones (small oscillator Qo is required to make z large if Q~ is limited), and thus they also exhibit higher FM-noise. Another factor to consider is the post-tuning drift which often occurs due to for example to the thermal time-constants of the device, and which lasts for a few hundred nanoseconds.

170 Microwave Semiconductor Devicell

~ -8 ::>

~ -12

-16

-204:-----:--T·-·~ ---t-- -~-- '-(6 - -1\ 12 13 FREQUENCY - GHz

Figure 6.19. Conductance and susceptance of a Gunn element, biased to 12 V. The device length i! 12 p.m. Reprinted from STERZER, F. (1971). "Tranllferred Electron (Gunn) Amplifier! and Ollcillatorll for Microwave Ap­plicationll," Proc. IEEE, 59, 1155, @1971 IEEE.

EXAMPLES OF ACTUAL CIRCUITS AND IMPEDANCE DIAGRAMS FOR GUNN AND IMPATT OSCILLATORS

ExaIIlples of Device IIIlpedance Plots

We illustrate the general treatment so far in this chapter by showing ex­amples of some actual circuits used for Gunn and IMPATT oscillators, and some actual measured data for the device impedance.

The negative conductance and the susceptance for a Gunn-device is quoted from (Sterzer, 1971) in Figure 6.19. Note the wide bandwidth over which a negative conductance exists. In this case the maximum magnitude of the negative conductance corresponds to about 50 ohms, i.e. this is not likely to be a particularly high-power device, which would require a larger area and lower impedance level. A different way of displaying such data is to use the complex impedance plane, as in Figure 6.20, which shows -GD and BD for an IMPATT device at a number of different frequencies and RF voltage levels. If we follow an iso-frequency curve, such as the one for 8.4 GHz, we can see that the negative conductance saturates, and that there is also a small change in the susceptance. Yet a different way of graphing device impedance data is given

Chapter 6 171

160

I~

140

130

18% 120

110

100 uJ <..)

90 z ... t-

eo 11. uJ <..)

10 '" :::>

'" 60

~O

40

30

f 0 VOLTS AC 20

10

0 ·25 ·20 ·15 ·10 ·5 0

CONOUCTANCE"

Figure 6.20. Complez admittance for a Read diode at a number of different bia6 voltage. and frequenciell, as marked. Reprinted from SCHARFETTER, D.L., and GUMMEL, H.K. (1969). "Large Signal AnalY6i6 of a Silicon Read Diode Oscillator," IEEE Tranll. Electron Devicell, ED-16, 64, @1969 IEEE.

in Figure 6.21, from (Masse' et al., 1985). In this case, the DC bias current and the diode area are used as parameters. The small-signal susceptance is about proportional to the area, while for large-signal conditions, the situation becomes more complicated. For example, it is important to take the series resistance into account. As shown in one of the problems, an approximate correction to the output power is obtained from the following expression:

(6.46)

The maximum power is not obtained for the largest negative conductance as a result, and there is an optimum area which yields the largest output power. The actual operating point is found by drawing the "circuit line" in the -G D / BD plot, as indicated in Figure 6.22. Also, for small areas, the current density approaches high values for which the operating frequency becomes close

172

-500

G (mS)

Microwave Semiconductor Devices

til E III

Figure 6.21. Complez admittance of a 40 GHz double-drift, GaAs, IMPATT diode with the bias current and the frequency as parameters, a .. marked. Con­stant output power contours are also shown. Reprinted from MASSE', D., ADLERSTEIN, M.G., and HOLWAY, Jr., L.H. (1985). "Millimeter Wave GaAs IMPATT Diodes," in Infrared and Millimeter Waves, K.J. Button, Ed., Academic Preu, Orlando, FL, Vol. 14, Ch. 5, £91, with permisBion.

to the avalanche resonant frequencYi thus I - G D I decreases beyond a certain value for the current.

Examples of Oscillator Circuits

In general, we need a circuit which has a low impedance, and also a con­venient way for introducing the bias. One example of such a circuit is shown in Figure 6.23. This general type of circuit is known from waveguide/ coax transitions to give a low impedance over a reasonably large bandwidth. Note the choke at the top which is used to stop the RF from leaking out the bias line. This circuit was analyzed for oscillator applications by Chang and Ebert (1980). A different method of obtaining a low impedance is to use a "top-hat" radial line resonator, see Figure 6.24, which uses a post type of arrangement for impedance transformation from the high waveguide impedance. Two mi-

Chapter 6

Constant frequency

lines

VAC = 0 V

-G

173

5V 10 V B

__ :J::==---r Operating point

it- Circuit I line

Figure 6.22. nlustration of a method for finding the crossing of the circuit and device admittance curves for an IMPATT device. Note that the su"ceptance of the circuit varie8 rapidly with frequency if the circuit has a high Q-value. In plotting the circuit admittance, the sign is reversed for both the real and the imaginary parts. The operating point at the crossing of the two curve" then satisfies (6.3).

crostrip circuits are shown in Figure 6.25. The device typically has to be positioned in a hole in the substrate in this case, so that adequate heat-sinking can be provided from the ground-plane. A circuit which can be used both for single device oscillators and for power-combining is shown in Figure 6.26. The center-conductor of the coax couples via the magnetic field to the cavity at its resonant frequency. Other frequencies on the other hand are terminated in the matched load, and spurious oscillations are thus avoided. This circuit will be discussed further in the next chapter.

NEGATIVE RESISTANCE DEVICES USED AS AMPLIFIERS

If the real part of the circuit admittance (Re{Y.}) is always greater than the magnitude of the negative conductance of the device, then oscillation can not occur, and instead we have the possibility of an unconditionally stable amplifier. Typically, the amplifier is connected via a circulator, as shown in Figure 6.27. The gain is found by calculating the reflection coefficient

VOUI Zo - Zl P = -- = --- == Voltage gain "g".

Vi" Zo + Zl (6.47)

174 Microwave Semiconductor Devices

Figure 6.23. One type 01 waveguide circuit u.ed lor two-terminal device •. Adapted from CHANG, K., and KUNO, H.J. (1990). "IMPATT and Re­lated Transit-Time Devices," in Handbook of Microwave and Optical Components, K. Chang, Ed., John Wiley & Son" New York, Vol. J, Ch.7, 305, with permiBlion.

Figure 6.24. A waveguide circuit lor u.e with two-terminal device., which employ, a radial line "top-hat" re,onator. Adapted from CHANG, K., and KUNO, H.J. (1990). "IMPATT and Related Tmn,it-Time Device,," in Handbook of Microwave and Optical Components, K. Chang, Ed., John Wiley & Son" New York, Vol. 2, Ch. 7, 305, with permis,ion.

In order to be able to make specific calculations we assume that the equiv­alent circuit inside the black box is as in Figure 6.28. We take ZD to be = -R for simplicity, since any device reactance can be included in the series resonant circuit. This circuit has the voltage gain

Zo + R - j(wLs __ 1_) 9 _ wOs

- Zo - R + j(wLs __ 1_) wOs

(6.48)

Chapter 6

DIODE

CAP

DIODE

81AS

_OUTPUT

(a)

_ OUTPUT

L-7r----'

BIAS

175

Figure 6.25. Two ezample.t of microstrip circuit6, u6eful for two-terminal de­vice 06cillator6. Adapted from CHANG, K., and KUNO, H.J. (1990). "IM­PATT and Related Transit- Time Device"," in Handbook of Microwave and Optical Components, K. Chang, Ed., Vol. 2, Ch. 7, 305, with per­m16""on.

Impalt

Diode

_____ 0 1

Figure 6.26. A coazial circuit module for use with IMPATT diode .tingle­element or power-combining o.tcillatorl. Adaptedfrom ALDERSTEIN, M.G., and FINES, J. (1989). "A Multi-IMPATT Injection-Locked Oldllator at 35 GHz, " IEEE TranI. Microw. Theory Tech., MTT-37, 571, @1989 IEEE.

176 Microwave Semiconductor Devices

/ Transmission line imp = Zo(Y 0)

Pin --.. ·- --.... - Pout> Pin

~ p=2

Y1 or Zl

Y d' Y c ["Black Box"]

Ip I > 1!

Figure 6.27. A circulator-coupled negative-resistance type one-port amplifier. The negative resistance is inside the "black boz".

Figure 6.28. A ,ingle-tuned series resonant representation of what is inside the "black boz" in Figure 6.27.

Chapter 6

We make the usual approximations for a high-Q circuit:

j(wLs __ 1_) = j wLs(1 _ w~) = ... ~ 2jLsll.wi wCs w2

and then find the power gain

1 12 = (Zo + R)2 + 4L~ll.w2 = (Zo + R) 2 X

9 (Zo - R)2 + 4L~ll.w2 Zo - R

( Zo + R)2 == g~ Zo-R

is the maximum gain (at ll.w = 0)

177

(6.49)

(6.50)

In practise, 90 » 1, and Zo ~ R, i.e. Zo + R ~ 2R. At the 3 dB points:

We find the bandwidth, B == D."'':dB, by approximating (6.49):

12 2 1 R R

19 3dB ~ 9" x 1 + L~ll.w29~/ R2 => ll.W3dB ~ 90 L s i B ~ 1I"90 LS;

and a voltage gain-bandwidth product of

R 9.B~ -L

11" 5

The corresponding expression for a parallel resonant circuit is

1/R g"B~ -C

11" 5

(6.51)

(6.52)

(6.53)

(6.54)

An alternative way is to express the gain-bandwidth product in terms of the external Q-value

2 woLs B g"b~ -Q where Q • .,t = -Z i b= -, i

ez:t tI 0

(6.55)

A reflection amplifier, such as described above, may be used with a Gunn element. Note that in order to exhibit a stable electric field distribution (no domain formation) the Gunn device needs to have a product

n xL < 5 X lOll cm- 2 (for GaAs) (6.56)

Under these conditions, the device presents a negative differential (AC) resis­tance, if biased above the threshold field, Ec>it.

If n x L exceeds this value, domains will form, and stable amplification will not be possible. At present, InP reflection amplifiers produce higher power

178 Microwave Semiconductor Devices

-G

DEVICE CIRCUIT

Figure 6.29. Equivalent circuit of a two-terminal device oscillator, including the series resistance. For Problem 1.

than three-terminal devices at frequencies from 30 - 94 GHz, and are finding some use. Their noise properties were discussed earlier in this chapter.

IMP ATTs are ususally not used in the pure reflection mode, since this decreases the output power considerably (Masse' et al., 1985). Instead the "quasi-stable mode" may be employed. If there is no power present at the input, the device will oscillate, "somewhat noisily", at reduced power level.

Problems, Chapter 6

1. For the IMPATT diode oscillator circuit in Figure 6.29, show that the power delivered to the load is (assuming steady state oscillation):

Hint: Use the Q-value.

2. Use the Table below to make a plot of the output power versus VRF for the IMP ATT diode for which these data were measured. Assume two cases: a) Rs = OJ b) Rs = 0.50j Find the optimum power in each case. Note the equation given in Problem 1: The value of B is 50 mS.

3. Derive Eq. (6.25) for AM-noise. Start with (6.8a) and (6.8b) and set 6w = OJ Make reasonable approximations, similar to those made for FM­noise case (eq. (6.18».

Chapter 6 179

4. a) Calculate Il.fRMS from the InP Gunn data shown in Fig. 6.10 (the '.'-points) and plot versus fd. Assume B = 1 kHz (the data in Fig. 6.10 are for B = 1 Hz!). What is the frequency dependence you find? b) Use the data given in the figure to estimate and plot the noise measure (M).

Assume an RF frequency = 60 GHz ("M-band"?) Compare your results with Fig. 6.12.

5. Calculate the initial growth rate for oscillations in an oscillator circuit by using the complex frequency p = iw+a. Use a Taylor expansion ofp valid for small growth rate (i.e. regard a as a small change in p ~ iw). Define gwo == Q where Wo is the oscillation frequency. You can assume that XD is a constant capacitance. The quantity sought is an expression for 9 in terms of circuit parameters.

6. An oscillator has IGDI = Go(l- !3ViF) (GD is negative). Find the maxi­mum output power and the load conductance for which this occurs. Define the saturation factor

Find the value of S at the maximum power· point (G L = load at this point). Find the AM noise power in a 1 Hz BW, for fd = 10 kHz; M = 16 dB, Pea"i .. = 10 mW, Q.zt = 50; Wo = 211" X 10 GHz; use S from above;

7. (a) Given an FM-noise spectrum:

What type of noise mechanism causes the FM-noise? (b) What is the noise mechanism if Il.fRMS is independent of fd?

8. What is the locking range for a 100 mW IMPATT oscillator locked by a 1 mW Gunn oscillator, with Q = 40?

9. The IMPATT oscillator in Problem (8) has M = 35 dB, white noue, and Q.zt = 40. Plot Il./RMs versus fd a) without injection locking b) with injection locking, in the middle of the band (note, cos t/J ~ 1) [use eq. (6.30)). Assume that the IMPATT oscillator is locked to a Gunn oscillator, which has a noise measure, M = 25 dB, and Q •• t = 200.

REFERENCES

CHANG, K., and EBERT, R.L. (1980). "W-Band Power Combiner Design," IEEE Tran~. Microw. Theory Tech., MTT-28, 295.

180 Microwave Semiconductor Devices

--, and KUNO, H.J. (1990). "IMPATT and Related Transit-Time De­vices," in Handbook of Microwave and Optical Components (K. Chang, Ed.), Vol. 2, Ch. 7, 305.

CONSTANT, E. (1976). "Noise in Microwave Injection Transit Time and Transferred Electron Devices," Physica, 83B, 24.

EDDISON, I.G. (1984). "Indium Phosphide and Gallium Arsenide Trans ferred-Electron Devices", in Infrared and Millimeter Waves, K.J. Button, Ed., Vol. 11, Ch. 1, p. 1.

GLISSON, T.H., SADLER, R.A., HAUSER, J.R., and LITTLEJOHN, M.A. (1980). "Circuit Effects in Time-of Flight Diffusivity Measurements," Solid State Electron., 23, 637.

GUMMEL, H.K. and BLUE, J.L. (1967). "A Small-Signal Theory of Avalanche Noise on IMPATT Diodes," IEEE Trans. Electron Devices, ED-14, 569.

GUPTA, M.S. (1971). "Noise in Avalanche Transit-Time Devices," Proc. IEEE, 59, 1674.

___ , (1977). "Electrical Noise: Fundamentals and Sources," IEEE Press, New York.

HARRINGTON, R.F. (1961). "Time-Harmonic Electromagnetic Fields," McGraw­Hill, New York.

HINES, M.E. (1966). "Noise Theory for the Read Type Avalanche Diode," IEEE Trans. Microw. Theory Tech., MTT-16, 738.

JOSENHANS,J. (1966). "Noise Spectra of Read Diode and GUNN Oscilla­tors," Proc. IEEE Lett., 54, 1478.

KUNO, H.J. (1981). "Solid-State Millimeter Wave Power Sources and Com­biners," Microw. J., Vol. 24, June 1981, p. 21.

KUROKAWA, K. (1968). "Noise in Synchronized Oscillators," IEEE Trans. Microw. Theory and Tech., MTT-16, 234.

---, (1969). "Some Basic Characteristics of Broad-Band Negative Resis­tance Oscillator Circuits," Bell System Techn. J., 48, 1937.

MASSE', D., ADLERSTEIN, M.G., and HOLWAY, Jr., L.H. (1985). "Millimeter­Wave GaAs IMPATT Diodes," in Infrared and Millimeter Waves, K.J. But­ton, Ed., Vol. 14, Ch. 5, 291.

OKAMOTO, H. (1975). "Noise Characteristics of GaAs and Si IMPATT Diodes for 50-GHz Range Operation," IEEE Trans. Electron Devices, ED-22,558.

POZAR, D.M. (1990). "Microwave Engineering," Addison-Wesley, Reading, MA.

Chapter 6 181

SHOCKLEY, W., COPELAND, J.A., and JAMES, R.P. (1966). "The Impedance Field Method of Noise Calculation in Active Semiconductor Devices," in Quantum Theory of Atom" Molecule, and the Solid State, (P.O. Lowdin, Ed.) Academic Press, New York.

SITCH, J.E., and ROBSON, P.N. (1976). "The Noise Measure of GaA. and InP Transferred Electron Amplifiers," IEEE Tran,. Electron. Device" ED-23,1086.

STERZER, F. (1971). "Transferred Electron (Gunn) Amplifiers and Oscilla­tors for Microwave Applications," Proc. IEEE, 59, 1155.

STATZ, H., PUCEL, R.A., SIMPSON, J.E., and HAUS, H.A. (1976). IEEE Tran •. Electron. Device" ED-23, 1075.

SZE, S.M. and RYDER, R.M. (1971). "Microwave Avalanche Diodes," Proc. IEEE, 59, 1140.

THIM, H.W. (1971). "Noise Reduction in Bulk Negative-Resistance Ampli­fiers," Electronic, Letter., 7, 106.

FURTHER READING

ADLER, R. (1946). "A Study of Locking Phenomena in Oscillators," Proc. IRE, 34, 351.

CARROL, J.E. (1970). "Hot Electron Generators," American Elsevier, New York (see Chapter 11).

EDSON, W.A. (1953). "Vacuum Tube Oscillators," John Wiley, New York.

---, (1960). "Noise in Oscillators," Proc. IRE, 48, 1454.

KUROKAWA, K. (1969a). "An Introduction to the Theory of Microwave Circuits," Academic Press, New York.

---, (1976). "Microwave Solid State Oscillator Circuits," in Microwave Device., M.J. Howes and D.V. Morgan, editors, Chapter 5, John Wiley, New York.

VENDELIN, G.D. (1982). "Design of Amplifiers and Oscillators by the S­Parameter Method," John Wiley, New York.