534 IEEEJOURNAL OF SOLID-STATECIRCUITS,VOL. SC-21,NO.

4,AUGUST 1986

Intermodulation in High-Frequency Bipolar

Transistor Integrated-Circuit Mixers
ROBERT G. MEYER, FELLOW, IEEE

,4/mtracf —Intermodulation in bipolar-transistor donble-balanced mixers

Jb 1102
at high frequencies is analyzed theoretically and by computer simulation.
A
The dependence of the distortion on a relatively few nornmhzed parame-
ters is ilhss@ated. Computed residts are compared with measurements on a
+ 0, 02 Q~
monolithic quad mixer,
“o A L

I. INTRODUCTION

HE DOUBLE-BALANCED mixer configuration

T shown in Fig. 1 is widely used for frequency conver-
sion systems fabricated using bipolar-transistor
integrated-circtiit technology [1]–[5]. This circuit is attrac-
tive because a large applied oscillator voltage V. causes the
Q
~

.
210

Fig, 1. Double-balanced quad mixer using bipolar transistors.

transistor quad QI – Q4 to behave as a set of almost perfect
switches. The signal input us is converted to a current i$
by the differential pair Q5, Q6 which can have arbitrarily II. LARGE-SIGNAL ANALYSIS
large linearizing resistors R ~ (limited by noise figure de-
gradation and gain loss). The current i, is then switched The circuit of Fig. 1 maybe analyzed ,at high frequencies
back and forth by the quad switch, ideally producing by considering one pair of transistors in the quad and
frequency conversion with no distortion (such as intermod- assuming this is driven by the oscillator voltage VOat the
ulation) of the signal input. base and fed by an ideal current source at the einitter. This
In practice it is not possible to achieve instantaneous is shown in the large-signal equivalent circuit of Fig. 2 and
switching of the quad, so that there is a time period in each is justified by the symmetry of the quad and the fact that
cycle when all four devices QI – QA are on and the current any residual distortion due to Q5, Q6 can be considered
is may be subject to distortion from the nonlinear device independently. Note that in the equivalent circuit of Fig. 2
transfer characteristics. At low frequencies, even this possi- the effects of r. and r= are neglected. Simulation has
bility causes few problems as the exponential base–einitter shown this to be a good approximation at high frequen-
nonlinearities of QI – Qq tend to cancel [6], leaving only cies. Also the effect of Cy in the quad is small and it also is
very small distortion caused by extrinsic base and emitter neglected. Although base resistance r~ is a function of Ic,
resistances during the switching interval. At high frequen- the variation is typically not large and equal and constant

cies, however, charge storage in the quad devices and the values of r~ are assumed for the two devices. The large-sig-
influence of their base resistances causes significant distor- nal input capacitance Cmrepresents charge storage in the
tion in ~the ideal mixing process. The most important device due to emitter-base depletion capacitance CJ, and
manifestation of this is the creation of third-order inter- base charging capacitance so that
modulation products in the converted output signal.
In this paper the mechanisms of intermodulation crea- (1)
tion in the basic mixer of Fig. 1 are examined, and
methods are derived for estimating distortion from device dIcz dVz
parameters and signal levels. I~z=~l— + cJe
—dt . (2)
dt

Note that equal and constant transit times rl are assumed

Manuscript received Au ust 13, 1985; rewsed January 16, 1986. This in the two devices, as are equal and fixed values of CJe.
work was supported by ti e U.S. Army Research Office under Grant
DAAG299-84-K-O043, The former assumption restricts the validity of the analysis
The author is with the Department of Electrical Engineering and to lC levels below high-current ~= roll-off. The assumption
Computer Sciences and the Electronics Research Laboratory, University
of California, Berkeley, CA 94720. of constant CJ, is justified by simulations showing that
IEEE Log Number 8608916. almost the same distortion is generated if the actual Cj, is

0018 -9200/86/0800-0534$01 .00 01986 IEEE

MEYER:
INTERMODULATION
INBIPOLAR
TRANSISTOR
MIXERS 535

local oscillator frequency and u~l and U,z are frequencies

of two input signals in i,, then the if. fundamentals in the
output have frequencies (a. – Q,l) and ( tio – 0,2). Usually
ti~l = 0,2 = @o. The IM3 products in the output have fre-
quencies (tio – 2U,1 + U,2) and (00 – 2a~z + Q,I).
Computer simulation of the circuit of Fig. 1 was per-
Fig. 2. Large-signaf high-frequency equivalent circuit of Fig. 1 formed using the program SPICE. Ideal current inputs
(IQ i i,) were applied at the emitters of the quad and the
replaced by a constant value equal to the bias value. This output was taken as (101– 10a). Simulation showed that
is plausible because of the relatively slow variation of C,, the same values of lM3 were present in 101 and 102
with bias voltage. individually, as expected. The large-signal analysis routine
The analysis proceeds as follows. The transistor transfer in SPICE was used to generate a time-domain waveform of
characteristics are 15000 points which was then Fourier analyzed. Careful
checks using known waveforms as inputs verified the abil-
ICI = I~ev’lv’ (3) ity of the program to compute accurately IM3 values of
– 90 dB. This requires appropriate manipulation of the
IC2 = I~ev’lv’ (4) program tolerance limits and care to allow transients to
where equal values of Is are assumed and VT= kT/q. decay to adequately low levels.
From Fig. 2 Equations (8) and (9) together describe the large-signal
high-frequency behavior of the mixer. A general closed-
VO= l~lrb + VI – V2– I~2rb (5) form solution of these equations is not available but an
IQ+i, =IB1+Ic1+IB2+Ic2. (6) approximate and practically useful solution can be found
using computer simulation. Equations (8) and (9) show
Substituting (l)-(4) in (5) we find that the normalized output currents lcl/lQ and lCZ/lQ
depend on a relatively few normalized parameters:

lQ lQ \ ‘T ‘Q I
where

A = tiorlrbIQ/V~ (11)
Assuming a sinusoidal local oscillator waveform VO=
in (7) we find
Vo~ cos Uot and normalizing time to t‘= LJo,( B = LJoCJeVT/IQ (12)
C = uorbCJ,. (13)

Computer simulation and examination of the circuit to-

ZQ d IC2 geth& with (8) and (9) shows that lM3 is only very weakly
– ‘OTlrb<

d
~
H~

ICI IQ
dependent on factors uo~l and C and further that the
functional dependence on the remaining variables of IM3
in the output current can be approximately expressed as
+ C,erbuo~ : —
in — — + in — ;:2 . (8)
() IQ IC2
Substituting (l)–(4) in (6) we find

i, d ICI ICI d I(.2 where

1 +~=LI)Orl~ —IQ +—+LJOT1—
IQ —–
dt ‘ H 1(2 i. = 1~~ cos a~lt + I~Mcos a~2t. (15)
H
I VT d ICI IC2
+ : + uoc,e~ ~ in — — Note that IM3 in the mixer varies as the square of the
Q Q IQ IQ “ ‘9) signal input amplitude l~M, just as in amplifiers. Thus
when distortion is calculated for one particular amplitude
of input signal, distortion values for any other amplitude
III. COMPUTERSIMULATION can be derived from this if all else is constant. By conven-
tion, equal amplitudes are assumed for all components in
Third-order intermodulation 1it4~in the mixer is defined is.
as the ratio of the amplitude of the third-order intermod- Equations (8), (9), and (14) show two major sources of
ulation product in the output to the intermediate frequency intermodulation in the mixer. First, if C,, = O then (9)
(if.) fundamental signal in the output. Thus if a. @ the becomes a linear equation and from (8) the parameter
 536 lkm JUUKNAL VI’ SULILJ-SIA lli GIKUJSTS,VOL.SC-11,NO.4, AUGUSTlY?5b

WoC,eVT /10
A ,’-2 ,0-1
-40 ‘0”3 1 I *

rb =0

Is.M/lo =02
I;M/Io=o.2 -50
VOM = 500 mV

-70 ~~
IM3 ---- --,* IM3 -60
---G- ______ ----- (dB)
(OB)
I -70

—Wo~ rbIQ/VT =06 -80

:/
-90 ‘---- Wo~ ~bl./ VT= O3
-90

t t

Fig. 3. Computed values of IM3 versus normalized local oscillator Fig. 5. Computed valued of IM3 versus q&VT/ZQ with rb = O, Vo~
voltage amplitude with C,, = Oand l~~/IQ ==0.2. = 500 mV, and l~~/IQ = 0.2.

depends on the factor C,erbuOis not an important source

of distortion in the mixer. For purposes of distortion
0.I
-6000’ prediction, CC, and CP of the driver transistors Q5 and Qh

[7”
effectively add directly to C,, of Q1–QA. Equations (8) and
-65 (9) indicate that the factor aO<,V~/lQ is now the only
factor affecting the nonlinear behavior of the circuit and
IM3 -70 this has been verified by computer simulation. Computed
(dB)
values of IllJ in the mixer versus LOOCJeV~/IQ with rb = O,
-75 l~~/ZQ = 0.2, and VOM= 500 mV are shown in Fig. 5.
Simulations of typical quad mixers with complete device
-80 c,, =0 models (including all parasitic and allowing CJ= to vary
l~f.4/I,3=02
V’M=500 mV with V~E) yield values of IlfJ that are usually close to the
-85
result obtained by simply adding the distortion products
due to the two mechanisms, as represented by (14). Thus
Fig. 4. Computed values of !fM3 versus LOo~eVT/IQ with ~e = O, Vo~
= 500 mV, and l~~/IQ = 0.2. in any particular application the designer can estimate in
advance whether a given circuit can meet a desired lkf~
specification. Appropriate and necessary bias levels can be
@O~lr#~/ VT together with local oscillator drive VoM/ VT estimated and, if necessary, alternative fabrication
and signal input i, /lQ determines the nonlinear behavior. processes giving more favorable combinations of device
Simulated values of IM3 as a function of VoM/V~ for two parameters can be proposed.
different values of uorlrbIQ/V~ with ~,= O and l~~/IQ
= 0.2 are shown in Fig. 3. Note that for low values of IV. MEASUREMENTS
VCM/V~, W increases as Vo~/V~ decreases because the
quad transistors spend more time per cycle in the state Measurements were made on a monolithic quad mixer
where all four are on and thus generate higher distortion. using R ~ = 200 0 external to the chip. This was sufficient
The physical origins of the increase in IA43for large values to make distortion due to Q5, Q6 negligible at the current
of Vo~/ VT are not known but this effect is seen in both levels used. The output was taken single ended from one
computed and measured data. Simulations with ~1 and/or common collector driving a 50-Ki load. Device parameters
rb equal to zero gave lkf~ near – 100 dB with numerical were rb = 35 Q, Cj,O= 2.4 pF, CICO = 0.86 pF, and 71= 320
limitations becoming apparent. The value of the parameter ps. (For frequencies ~.= 100 MHz, j~l = 99 MHz, and
tio~l in (9) had almost no effeet on the simulated distor- f.z = 98.9 MHz, measured and computed values of Ikfs are
tion. Simulated values of 1143 versus uO~lrJ~/ VT with shown in Figs. 6 and 7.) The computation was performed
Cj, = O, l~~/IQ = 0.2, and Vo~ = 500 mV are plotted in with the full device model and examination showed that
Fig. 4. both distortion mechanisms considered previously were
The second major source of nonlinearity in the mixer is contributing significantly. The overall agreement between
the depletion capacitance Cje. Simulations show that the computed and measured values is reasonably close and
distortion produced by C,, is essentially independent of indicates that the major distortion sources and mecha-
both the value of rl and of rb. This is similar to the nisms in the circuit have been adequately modeled.
situation in a common-base amplifier at high frequencies. As an example of the use of Figs. 4 and 5, consider lkf~
Note that in the mixer, when the quad devices are conduct- in the test circuit for Vo~ = 500 rnV, 1~~ = ().8 ti, and
ing they act as common-base stages so that this result is IQ= 4.2 mA. The total predicted distortion from Fig. 7 is
plausible. This result indicates that the term in (8) which 1A43= – 58 dB. The value of aO~lrbIQ/V~ = 1.14 and Fig.
MEYER:1NTERMODULATION
IN BIPOLARTRANSISTOR
MIXERS 537

predicted from the simulation using the complete circuit

model.

1
— Computed curve
. Measured pomis
V. CONCLUSIONS
-50

IM3
Intermodulation in bipolar-transistor integrated-circuit
(d El)
●
mixers has been analyzed theoretically and shown to de-
-60 ●
pend on a relatively few normalized parameters. This has
11 VOM= 280 mV been verified by computer simulation. The analysis and
●
lSM:08 mA computer simulation were able to successfully predict the
-70
fo : IOOMHZ
f~, : 99 MHZ magnitude and parameter dependence of intermodulation
t
fs2. 989 MHz in an experimental monolithic quad mixer.
Fig. 6. Computed and measured values of ZM3 versus bias current for a
monolithic quad mixer.
ACKNOWLEDGMENT
v~(volts)
~i The author wishes to acknowledge reviewer’s comments
o 01 02 03 04 05 06 0.7 08
-40 1 , 1 I I I 1 lr---%Q& which significantly improved the paper.
— Computed curve

IM3
(d8)
-50
I ●
. Meosured po!nts

●
REFERENCES

[1] B. Gilbert, “A precise four-quadrant multiplier with subnanosecond

responsej’ IEEE J. Solid-State Circuits, vol. SC-3, no. 4, pp. 365-373,
Dec. 1968.
-60 ●
[2] A. Bilotti, “Applications of a monolithic analog multiplier,” IEEE J.
b
Solid-State Circuits, vol. SC-3,no. 4, pp. 373-380,Dec.1968.

I
●
IsM=08mA
[3] R. G. Meyer,“Integratedcircuitmixersflin IEEE NEREM Rec.,
IQ =42mA VO1. 12, 1970, pp. 62–63.
-70 [4] C. Yamada et al., “A 470 Mf-Iz 5V CATV tuner,” in IEEE ISSCC
Dig., Feb. 1985, pp. 28-29.
t
[5] E. H. Nordholt, H. C. Nauta, and C. A. M. Boo:, “A high-
Fig. 7. Computed and measured values of IM3 versus local oscillator dynamic-range front end for an up-conversion car-racho receiver:
voltage for a monolithic quad mixer. IEEE J. Solid-State Circuits, vol. SC-20, no. 3, pp. 688-696, June
1985,
[6] W. M. C. Sansen and R. G. Meyer, ” Distortion in bipolar-transistor
variable-gain amplifiers,” IEEE J. Solid-State Circuits, vol. SC-8, no.
4 predicts IM3 due to rb is – 65 dB when Z~~/IQ = 0.19. 4, pp. 275-282, Aug. 1973.
The effective value of C,, at the operating point is 4.1 pF,
giving uOC,,V~/IQ = 0.016, and from Fig. 5 IM~ due to Cjt
is – 62 dB. When these values of IM3 are directly added Robert G. Meyer (S’64-M68-SM’74- F’81), for photograph and biogra-
the result is – 57.4 dB, which is very close to the value phy please see this issue, p. 533.