Exhibit 1044

COMMUNICATION CIRCUITS: ANALYSIS AND DESIGN

KENNETH K. CLARKE DONALD T. HESS

Oarke-Hess Communications Research Corporation Formerly: Polytechnic Institute of Brooklyn

ADDISON-WESLEY PUBLISHING COMPANY Reading, Massachusetts Menlo Park, California London Don Mills, Ontario

RPX-Farmwald Ex. 1044, p 1

This book is in the ADDISON-WESLEY SERIES IN ELECTRICAL ENGINEERING

Consulting Editors DAVID K. CHENG LEONARD A. GOULD FRED K. MANASSE

Copyright@ 1971 by Addison-Wesley Publishing Company, Inc. Philippines copyright 1971 by Addison-Wesley Publishing Company, Inc. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. Published simultaneously in Canada. Library of Congress Catalog Card No. 78-125610.


8.2 AMPLITUDE MODULATION TECHNIQUES 353

Even direct sideband filtering of suppressed carrier AM to produce SSB has its practical limitations. It is a difficult matter to design a sideband filter which cuts off sufficiently rapidly to attenuate one sideband while not distorting the other sideband either in magnitude or in phase. For SSB voice modulation, mechanical, crystal, or ceramic bandpass filters are usually employed to remove the undesired sideband (See Fig. 7.6-1 for an example). Even though these filters have considerable ripple in magnitude as well as significant nonlinearity in phase in the passband, their effect on the intelligibility of average speech is negligible. For other forms of modula-tion the sideband filter usually has to be hand-tailored to the modulation to minimize distortion. In all cases, however, the sharp cutoff required of the bandpass sideband filter is possible only if the filter center frequency is not too high (the required Q of the tuned circuits in the filter to maintain a fixed BW is directly proportional to w0 ). Consequently, in almost all SSB transmitters, the information is suppressed-carrier-modulated at a reasonably low carrier frequency (50 kHz to 500 kHz for voice modu-lation) where sideband filtering is accomplished, and then the resultant SSB signal is heterodyned to the desired carrier frequency w0

In addition to the complications that SSB creates at the transmitter, its demodu-lation is possible only by synchronous detection, which requires a reference oscillator at the radian frequency w0 Since it is impossible to derive this frequency from the SSB signal itself, a small pilot carrier is usually transmitted along with the SSB signal to provide the reference at the receiver. The demodulation of SSB will be pur-sued in more detail in Chapter 10.

In the subsequent sections of this chapter we shall consider the theoretical methods by which amplitude modulation (or multiplication of two signals) may be accomplished. We shall then examine some practical circuits which implement the theoretical methods.

8.2 AMPLITUDE MODULATION TECHNIQUES In this section we investigate the theoretical methods by which we can multiply or modulate cos w0t by g(t) to obtain the AM signal

v{t) = g(t) cos Wot = A[l + mf(t)] cos w0 t. (8.2-1)

In general, there are four basic methods by which amplitude modulation can be accomplished : a) analog multiplication, b) chopper modulation, c) nonlinear device modulation, and d) direct tuned-circuit modulation. As we shall see, all these methods can be employed to generate normal AM, whereas only direct analog multiplication and chopper modulation can be employed to gen-erate suppressed carrier AM. In addition, with the exception of the direct tuned-circuit modulator, modulation is accomplished at low power levels and amplified


354 AMPLITUDE MODULATION 8.2

(class B-see Section 4.2 and 9.2) to the desired output level. The direct tuned-circuit modulator directly modulates the amplitude of a high-power carrier which has been amplified by more efficient class C amplifiers to the desired level.

Analog Modulation Analog modulation (or multiplication) is accomplished in any device whose output [v0 (t)] is directly proportional to two inputs [v1(t) and v2(t)], that is,

v0(t) = Kv 1(t)v2(t). (8.2-2) Clearly, if v1(t) = cos


desired output. Thus if v1 = A[l + mf(t)] and v2 = V1 cos w 0 t, where f(t) is the modulation information (cf. Eq. 8.1-2), then

A' ~

v0(t) = 4K5 ViA[1 + mf(t)] cos w0t,

provided that A(l - m) - Vi ~ 0 or equivalently

Vi m < 1--

- A'

(8.2-7)

(8.2-8)

Since Vi > 0, we observe that, with half-square-law devices in the circuit of Fig. 8.2-1, the modulation index is constrained to be less than unity and thus the circuit clearly cannot be used to generate suppressed carrier AM.

Even though the circuit does not produce a 100 % modulated AM wave (which is desirable for efficient transmission), the modulation index at the output can be increased by subtracting some of the excess carrier from the undermodulated signal. This technique is referred to as carrier cancellation. For example, if we subtract D cos w0 t from v(t) = A[l + mf(t)] cos w0t, we obtain v0(t) in the form

A' m'

~ [ r--;t;;" J v0(t) = (A - D) 1 + A _ DJ (t) COS w0t , (8.2-9) from which it is apparent that we can increase the resultant modulation index m' to unity by choosing D such that (A - D)/A = m or D/A = 1 - m. Chopper Modulation Chopper modulation is accomplished by chopping g(t) at the carrier frequency rate and placing the resultant signal through a bandpass filter centered at the carrier frequency. The basic skeleton circuit of the chopper modulator is shown in Fig. 8.2-2, in which the switch, which is controlled by cos w0t, remains open for A cos w0t ~ 0 and closed for A cos w0 t < 0. To demonstrate that this circuit accomplishes amplitude modulation, we first write v0 (t) in the form

V0(t) = g(t)S(t), where S(t) is a switching function having the properties

S(t) ={ ~: COS Wot ~ 0, COS Wot< 0. (8.2-10)

(8.2-11)

Clearly, then, S(t) is a square wave with unity amplitude which may be expanded in a Fourier series of the form

1 2 2 S(t) = - + - cos w0t - - cos 3w0t + 2 n 3n (8.2-12)



R

+ + V,,(I) H(jw) v.(I)

I A coswol I

" (] [ CJ LI I

Fig. 8.2-2 Chopper modulator.

to yield

g(t) 2g(t) 2g(t) v (t) = - + --cos w0 t - --cos 3w0t +

a 2 1t 3n (8.2-13)

From Eq. (8.2-13) we note that v0 (t) is the superposition of AM waves centered at w0 , 3w0 , 5w0 , If the bandpass filter H(jw) attenuates the low-frequency com-ponents of va(t) as well as the AM components of va(t) in the vicinity of 3w0 , 5w0 , , then the output v0 (t) is given by

(8.2-14)

where hdt) is the impulse response of the low-pass equivalent of the bandpass filter and O(w0) is the plane angle of H(jw) at w = ro0 . If the low-pass equivalent filter is



fiat over the band of frequencies occupied by g(t), then v0(t) simplifies to the desired form

(8.2-15)

where HL(jw) is the Fourier transform of hL(t). Note that chopper modulation is possible only if the spectrum of the desired

AM wave does not overlap the spectra of any of the other components of va(t). A typical sketch of IV.,(w)I, where V.,(w) is the Fourier transform of v.,(t), is presented in

lv. I -..--...IGI

w ..

IGI! 3n """

Wo 1

I 3Wo I w-Wo - w.. wo+w.. ~-w.. 3w0 +w ..

Fig. 8.2-3 Spectrum of v0(t).

Fig. 8.2-3 for the case where g(t) is band-limited to Wm. It is apparent from the figure that, unless

(8.2-16) then the spectra overlap and chopper modulation is impossible. In addition, the closer Wm is to w0/2, the more complex the bandpass filter must be to effect the frequency separation.

Note that no restriction has been placed on the modulation index of g(t); there-fore, normal AM with a modulation index of unity as well as suppressed carrier AM may be generated with the chopper modulator.

In order to relax the inequality of Eq. (8.2-16) and in turn make the design of the filter H(jw) simpler, the single-pole voltage-controlled switch of Fig. 8.2-2 may be replaced by the double-pole voltage-controlled reversing switch shown in Fig. 8.2-4. The reversing switch has the effect of making va(t) symmetrical about zero,

S'(t)

b'

j cos1

w01 j

+ v.(I) H(jw)

Fig. 8.2-4 Balanced chopper modulator.

+

v,(I)



thus eliminating the low-frequency component of v0(t). Consequently, with the chopper modulator of Fig. 8.2-4, modulation is possible if

(8.2-17) Equation (8.2-17) ensures that the spectra of the desired AM wave centered at w0 and the AM wave centered at 3w0 do not overlap.

On a rigorous basis we observe that for the balanced chopper modulator of Fig. 8.2-4

where v0 (t) = g(t)S'(t)

S'(t) = { l, -1,

COS Wot 2:: 0, COS Wot< 0.

(8.2-18)

(8.2-19)

Since S'(t) is a square wave with a peak-to-peak amplitude of 2 and zero average value, S'(t) may be expanded in a Fourier series of the form

S'(t) = ~cos w0 t - 34 cos 3w0 t + . (8.2-20) 1t 1t Hence

4g(t) 4g(t) va(t) = --cos w0 t - - 3- cos 3w0 t + , 1t 1t (8.2- 21)

which, as expected, has no low-frequency component. Equation (8.2-21) also indicates that the reversing switch permits us to obtain twice as much output from the modulator of Fig. 8.2-4 as from the modulator of Fig. 8.2-2.

In many practical cases the switch S(t) in the chopper modulator of Fig. 8.2-2 has a resistance r in series with it which prevents complete attenuation of g(t) when S(t) is closed. Consequently, v0 (t) is switched between g(t) and rg(t)/(r + R) in lieu of g(t) and 0. This may be expressed mathematically as

( ) = ( )S( ) g(t)[l - S(t)]r V0 t gt t + R . r+

It is apparent that the fundamental component of v0(t) is given by 2g(t)( 1 - _r_) cos w0t =

2g(t) ~cos Wot, 1t r+R 1t R+r

and in tum v0(t) is given by

v0(t) = [2~t) * hL(t)J R: r cos [wot + O(wo)].

(8.2-22)

(8.2-23)

(8.2-24)

Thus we see that the only effect of the series resistance r is to attenuate the output by the factor R/(R + r). If r = R, the output level is reduced by a factor of 2.



Nonlinear Device Modulation Nonlinear device modulation is accomplished by summing the modulation and the carrier, applying them to a nonlinear device, and then passing the device output through a bandpass filter centered at ro0 to extract the desired AM signal. A block diagram of a nonlinear device modulator is shown in Fig. 8.2-5. As we shall see, the nonlinear device modulator has more restrictions for its proper operation than any

IH(jw)I

v1(t)=g(t)=A[I +m/(t)) Wo w-

: : ~ I ~ Ji-----1~ V2(t)= V1 COSWQI

Fig. 8.2-S Block diagram of nonlinear modulator.

of the previously considered modulators. First of all the nonlinear device must have no greater than a second-order (square-law) nonlinearity. Second, the maximum modulation frequency Wm must be less than ro0/3; and third, if the nonlinear device contains a "half-square-law" term, 100 % modulation or suppressed carrier modu-lation is not possible.

To determine the reason for these restrictions as well as an expression for the output signal v0(t), let us first express the output of the nonlinear device va(t) in the form of a MacLauren series,

where

(8.2-25)

With V; = v1 + v2 , Va reduces to

(8.2-26)

A little thought indicates that, if V2(t) = Vi cos root and V1(t) = g(t), the following components of Va (as well as a number of other components) have frequency spectra



8.4 PRACTICAL CHOPPER MODULATORS

The key component in a chopper modulator is the voltage-controlled switch which opens and closes at the carrier rate. Therefore, in this section we shall look at two voltage-controlled single-pole single-throw (SPST) switches-one employing a diode bridge and the other employing a single FET-and then we shall consider the problem of employing these single-pole switches to construct a voltage-controlled reversing switch. The SPST switch is used in the single-ended chopper modulator shown in Fig. 8.2-2, whereas the reversing switch is employed in the balanced chopper modulator shown in Fig. 8.2-4.

Note that, although the voltage-controlled switch is being discussed in con-junction with chopper modulators, it functions equally well as a synchronous de-modulator or a mixer in the same configuration as Fig. 8.2-2. For a synchronous detector, however, the output filter must be replaced by a low-pass filter, whereas for a mixer the output bandpass filter must be tuned to the intermediate frequency.

Diode-Bridge Modulator Almost all single-ended chopper modulators in use today employ the diode bridge as the voltage-controlled switch. Figure 8.4-1 illustrates a typical diode-bridge modulator in which a positive value of v1(t) causes all the bridge diodes to conduct thereby bringing v0(t) close to ground potential, and in which a negative value of v1 reverse-biases all the bridge diodes, thereby permitting va(t) to follow g(t).

It is apparent that the amplitude Vi of v1(t) must be sufficiently large when v1 is negative to keep all the diodes reverse-biased. A little thought indicates that for g(t) ;;::: 0, and with v1(t) = - Vi, D2 and D3 are on the verge of conduction when g(t) = Vi + 2V0 ; and that for g(t) < 0, D1 and D4 are on the verge of conduction for jg(t)I = V1 + 2V0 Hence to ensure that all the bridge diodes remain reverse-biased for v1 = - V1 , we require that

Vi > g(t) - 2V0 (8.4-1)

for all t. It is also apparent that Vi must be sufficiently large when v1 = +Vi to keep all

the diodes forward-biased so that the bridge presents a low impedance to ground; that is, v0(t) should be a small voltage with v1 = Vi. To determine the required magni-tude of V1 in this case, we define the current leaving the v1 source as I 2 and the current leaving the g(t) source as i 1{t). Since, in general, the four bridge diodes are integrated on a single chip with identical geometries, the bridge is balanced and i1(t) and / 2 split equally between the two bridge arms; thus

. 12 -i 1(t). ID1(t) = 2 = ID4(t),

(8.4-2)


8.4

+

g(t) 'V

x

~

V1(t)

Vo,

2n Wo

PRACTICAL CHOPPER MODULATORS

+

D3

v. g,.v. t RL L c

D4

~ Wo

Fig. 8.4-1 Chopper modulator employing diode bridge.

377

+

v.(t)

If we assume that each diode is characterized by the volt-ampere relationship (8.4-3)

then

_ _ _ kT(l 12 + i 1(t) l / 2 - i1(t)) Va - Vv2 Vv1 - q n 2/s n 2ls (8.4-4)

= kT ln (1 + i1(t)//2 ) q 1 - i1(t)//2

By expanding v0(t) in a MacLauren series in i 1//2 , we obtain

va(t) = r4i1( 1 + ;;~ + ;J1 + ). (8.4-5) where r4 = (kT/q)(2/I2 ) is the small-signal diode resistance with I 2/2 as a bias current. It is apparent that, if we wish to keep nonlinear components ofi1 [which is propor-tional to the modulation g(t)] out of the output to avoid envelope distortion in v0(t), then iV31~ 1. With this restriction v0 (t) = r4i 1(t) and the forward-biased diode



bridge may be modeled as a single resistor of value rd shunting v0 ; hence the diode bridge takes the form of an ideal voltage-controlled switch in series with a resistance

With this model

while

I _ Vi - 2V0

2-R2

thus to keep ii/31~ < 0.01,

and 2R 2 r - .

d - q(Vi - 2V0 )/kT'

(8.4--0)

for all t. The inequalities of Eqs. (8.4-1) and (8.4-6) may be satisfied simultaneously by choosing Vi > g(t) - 2V0 and choosing R 2 of the order of -foRi. For example, if lg(t)lmax = 10 V, then the bridge remains open with Vi = - Vi for Vi > 8.5 V (V0 = i V). If we select Vi = 9 V, then if

R - Ri +rd R1 2

- 7.75 ~ 7.75' the bridge appears as a resistor rd with v1 = Vi. .

A complete diode-bridge modulator which incorporates the floating source vi (t) as well as the output filter is shown in Fig. 8.4-2. In this circuit the transformer

L c

~ . + lc= alc

v0 (t)

l - v,;

Fig. 8.4--2 Practical balanced modulator.


8.4 PRACTICAL CHOPPER MODULATORS 379

is a closely coupled transformer operating in its midband range, and therefore functions as an ideal transformer. In addition, (1 + f3)RE is large in comparison with R 1 so that the transistor does not load the bridge. Consequently, if the output-tuned circuit is broad enough to pass the modulation and yet narrow enough to remove the low-frequency and higher-harmonic components of v0(t), then from Eq. (8.2-19)

2g(t) R 1 Vo(t) = -R R cos Wot + Vee (8.4-7) 1t E I +rd

Equation (8.4-7) assumes, of course, that Eqs. (8.4-1) and (8.4--6) have been satisfied and that the transistor remains in its active region.

The control voltage v1(t) may be supplied by a sufficiently large sine wave of radian frequency w0 instead of a square wave. If, as shown in Fig. 8.4-3, Vi is large in comparison with V,. and Vp (Eqs. 8.4-1 and 8.4-6), then the sine wave functions in essentially the same fashion as the square wave in controlling the states of the bridge.

Switch open

V. jg(t) j..,.,- 2 V0......__--+--+--1r----+-------+----

Fig. 8.4-3 Sinusoidal control voltage.

One main advantage of a sine-wave drive is that the transformer coupling v1(t) to the diode bridge need not be nearly as broadband. On the other hand, the larger value of Vi with a sine-wave drive requires a much higher breakdown voltage for the bridge diodes.

Whether v1(t) is a sine wave or a square wave, in practical diode bridges short-duration transient "spikes" appear on v0(t) in the vicinity of the bridge transitions from open to closed because of parasitic capacitance and diode charge storage. These spikes are, in general, of little consequence, since they contain sufficiently high-frequency components so that they are not transmitted to the output through the bandpass filter H(jw). FET Modulator A junction or an insulated-gate FET may be employed instead of the diode bridge as the voltage-controlled switch in a chopper modulator. Figure 8.4-4 illustrates a



2n V1(I) w

L

- Vi

O+P)RR1

+ Ri

v.

+ + D

G v.(t) g(t) rv

1 V1(t) s

Fig. 8.4-4 N-channel junction FET chopper modulator. typical N-channel junction FET chopper modulator. In this circuit, with v1(t) = - Vi < Vp (Vp is the pinch-off voltage of the FET), the FET opens, permitting v0 (t) to follow g(t). On the other hand, with v1(t) = 0, then the FET functions as an ohmic conductance gDss of value

- 21DSS(1 Vas) 21Dss gDss--- -- --- Vp Vp vas=o - Vp'

(8.4-8)

provided that lvDsl = lv0I < lOOmV. Consequently the FET may be modeled as an ideal voltage-controlled switch in series with a resistance rDss = 1/gDss For typical junction and insulated-gate FET's, rDss varies from several ohms to several thousand ohms.

With V1(t) = 0, V (t) _ _ g(t)rDSS . a - VDs - ,

Ri + rDss thus to ensure that lvDs(t)I remains less than 100 mV for all t we require R 1 to be sufficiently large so that

R ( lg(t~max 1) (8.4-9) 1 > rDSS lOOmV - . For example, if lg(t~max = 5 V and rDss = 500 n, then R 1 > 24.5 kn To avoid loading by the output transistor when the FET is reverse biased, the resistor R1



should not be chosen too much greater than this value. If, on the other hand, Rt is chosen to be less than 24.5 kll, then va(t) exceeds 100-mV, r 0 ss becomes nonlinear, and v0(t) is no longer a linear function of g(t); consequently nonlinear envelope dis-tortion begins to appear on the output AM wave.

If RE is sufficiently large so that transistor loading can be neglected, then v0(t) is given by (cf. Eq. 8.4-7)

2g(t) Rt v0(t) = -R R cos w0(t) + Vee

7t E 1 + rDSS (8.4-10)

However, if RF. is not sufficiently large, then the loading must be incorporated with the g(t)-R 1 network as shown in Fig. 8.4-5 by forming a Thevenin equivalent net-work. Clearly g(t) is decreased by a factor of 'I because of the loading; however, in addition, a de bias V' is added in series with g(t). If g(t) = 0, as it is for suppressed carrier modulation, then the presence of V' produces an average component in the modulation voltage being chopped and thus a nonzero carrier at the output. To eliminate this undesired carrier component, either RE must be increased relative to Rt or an isolation stage such as a source follower must be inserted between the chopper stage and the output transistor.

R] a

v-------aa

+ '"\, g(t)

-

Ji;E -Vo

'----------oa' a'

n = (l+IJ)R V'= o~E - Vo)R1' R] = R1 !1 (l P>RE (I+ P>RE+ R1 (I+ P>Rc+ R1

Fig. 8.4-5 Effect of transistor loading.

In addition to the diode bridge or the FET chopper, a bipolar transistor being switched between saturation and cutoff may be employed as the voltage-controlled switch. However, when saturated, the transistor may be modeled as a resistor in series with a de voltage source of approximately 100 m V (for silicon). This voltage source has the effect of introducing a carrier component at the modulator output, which is quite undesirable if suppressed carrier AM is being generated. This saturation voltage may be largely balanced out by placing two transistors in series (emitter to emitter) and placing the switching voltage between their bases.



Balanced Chopper Modulator Figure 8.4---6 indicates how two diode bridges can be employed to alternately apply + g(t) and - g(t) across R1 and thus produce the effect of a reversing switch. Note that the bridges are arranged so that one bridge is open when the other is closed. It is apparent that the closed bridge is unaffected by the open bridge and thus Eq. (8.4-6) still determines the value of Vi required to ensure that the bridge remains

2n Wo

,_

Fig. 8.4--6 Reversing switch for balanced chopper modulator.

closed for all t. On the other hand, the closed bridge does affect the open bridge in that it increases the voltage across the open bridge to

g(t)(l + R Ri ); 1 +rd

hence with the arguments employed to obtain Eq. (8.4-1) we require

Vi > g(t)(l + R Ri ) - 2V0 1 +rd (8:4-11)

for all t to ensure that all the diodes in the open bridge remain reverse biased. Figure 8.4-7 illustrates a practical chopper modulator in which both g(t) and

v1(t) are supplied from grounded sources. The transformer T1 is a closely coupled center-tapped audio transformer with a midband frequency range sufficient to pass the frequency components of g(t) (see Section 2.2), while the transformer T2 uses a



v. c L ic-......;;:_

J_ J{c

Fig. 8.4-7 Practical balanced chopper modulator.

closely coupled transformer with a midband capable of passing the main frequency components of v1(t). If v1(t) is a large-amplitude sine wave, then the restrictions on T2 are nominal. [Although unity turns ratios are indicated for the two transformers, other turns ratios merely introduce a scale-factor change in g(t) and v1(t).] If we assume that R1 is not loaded by the transistor and that Eqs. (8.4-6) and (8.4-11) are satisfied, then v0{t) may be expressed as

(8.4-12)

where S'(t) is given by Eq. (8.2-19). And if we assume that the transistor does not saturate, we may write



and in turn

(8.4-13)

where z11L(t) is the low-pass equivalent impulse response of the output parallel RLC circuit. If the output filter passes g(t) undistorted while removing the 3w0 component of v0(t), then v0(t) reduces to the desired form

4cxRL R1 vo(t) = Vee - -- g(t) cos Wot.

ttRE R1 +rd (8.4-14)

8.5 SQUARE-LAW MODULATOR The square-law device, although quite attractive as a mixer, finds very little application as an amplitude modulator. The basic reason for this is that most physical devices have half-square-law characteristics rather than full-square-law characteristics. As we saw in Section 8.2, unless a full-square-law characteristic exists, not only is sup-pressed carrier modulation impossible but also normal AM with 100% modulation is impossible. Consequently, unless a "quick and dirty" low-index modulator will satisfy the requirements of the situation, the other modulators discussed in this chapter are usually employed. Therefore, we shall look only briefly at one square-law modulator constructed with a junction FET operating within its saturation region.

A typical square-law FET modulator is shown in Fig. 8.5-1. If for this circuit we assume that the FET operates within its saturation (square-law) region and that RL is much less than the output impedance of the FET, then we may approximate the drain current as

iD = IDss( 1 - v;:)2. (8.5-1)

where Vp is the pinch-off voltage and I Dss is the drain current with vGs = 0 and vDs = - Vp. For the bias arrangement shown in Fig. 8.5-1, iD reduces to

. ( ) _ I (Vi + V2) 2 _ I {Vi cos w0t + A[l + mf(t)]) 2 ID t - DSS Vp - DSS Vp .

Since the component of iD(t) centered about w0 is

21DssViA V/ [1 + mf(t)] cos w0 t,

(8.5-2)

then with the assumption that the output filter removes the low-frequency and second-harmonic components of iD, v0(t) is given by

21DssViA Vo(t) = VDD - v~ [1 + mf(t)]. z llL(t) cos Wot. (8.5-3)


8.5 SQUARE-LAW MODULATOR 385

L c

+ Y,. l'Gs~ ~ io

v. = vos

l + i0 =loss(l- ~)2 - -

_ 1 ( v1 + v2)2 V2 = A[I m/(t)) - DSS Y,.

IV, ! I V1 = Jl1 COSWQI Wo = vie-

Fig. 8.S-1 FET square-law modulator.

If, in addition, the output filter is fiat over the band of frequencies occupied by the AM signal, v0(t) simplifies to the desired form

21DssViARL V0(t) = VDD - V~ (1 + mf(t)] COS Wot. (8.5-4)

To realize v0(t) in the form ofEq. (8.5-4) we must restrict v0(t) > -I Vpl for all time to ensure that operation remains within the saturation region. This may be accom-plished for any FET parameters by choosing a sufficiently small value of RL .

In addition,

for all time to keep the FET from being cut off and

for all time to keep the gate-to-source diode from turning on. These two restrictions


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