IEEE JOURNAL
`
`OF SOLID-STATE
`
`CIRCUITS,
`
`VOL. SC-10, NO. 1, FEBRUARY
`
`1975
`
`47
`
`Distortion in Variable-Capacitance Diodes
`
`ROBERT G. MEYER, SEN1OR MEMBER, IEEE, AND MARK L. STEPHENS
`
`using
`is anatyzed
`diodes
`in variable-capacitance
`Abstract–Distortion
`for
`intermodula-
`expressions
`the Volterra
`series approach.
`Closed-form
`tion
`distortion
`produced
`by
`variable-ezpacitance
`diodes
`in series- and
`parallel-tuned
`circuits
`are derived
`and verified
`by experiment
`at
`fre-
`quencies
`up to 200 MHz.
`The choice of
`the diode capacitance
`law ex-
`ponent
`for minimum
`distortion
`is investigated.
`Distortion
`in multiple-
`diode
`connections
`is arratyzed
`and the advantages
`of
`the back-to-back
`connection
`is shown.
`Calculation
`show the elimination
`of
`third-order
`for n = 0.5 in this comection.
`distortion
`
`I.
`
`INTRODUCTION
`
`v ARIABLE-CAPACITANCE
`
`diodes are being used in-
`of communication
`circuits
`the tuning
`for
`creasingly
`tuning offers the advantages
`This method of
`[1] - [6].
`of compactness and the ability
`to place the tuning control
`re-
`mote from the actual circuit.
`Also, since the circuit
`tuning is
`controlled
`by a dc voltage, electronic
`control
`of
`the tuning
`function
`is possible.
`the above advantages of variable-
`However,
`together with
`capacitance diodes is the disadvantage caused by the introduc-
`tion of distortion
`by the voltage-dependent
`capacit ante.
`In
`order
`to be able to use these diodes effectively,
`it
`is important
`to have explicit
`expressions
`for
`the distortion
`produced in
`such circuits.
`In addition,
`such expressions allow the diode
`designer
`to examine the possibility
`of producing diodes with
`minimum distortion
`contribution.
`diodes has been
`Although
`distortion
`in variable-capacitance
`examined in the past
`[7],
`[8]
`these analyses have been spe-
`cial cases and not generally applicable.
`In this paper, general
`expressions are derived for the distortion
`produced by variable-
`capacitance diodes and these are verified by circuit measure-
`ments at
`frequencies up to 200 MHz. Some general miscon-
`ceptions regarding the optimum value of
`the diode capacitance
`exponent
`are corrected.
`Finally,
`the relative
`advantages
`of
`
`various multiple-diode
`ically
`and experimentally.
`
`connections
`
`are explored
`
`both theoret-
`
`II. SINGLE-DIODE
`
`CIRCUITS
`
`used to
`invariably
`diodes are almost
`Variable-capacitance
`tune resonant circuits and that
`is the situation assumed in this
`paper. Circuits other
`than those considered here can, however,
`be analyzed using the same methods.
`
`“cn=!FF-
`
`(b)
`
`Fig. 1.
`
`(a) Measurement arrangement
`(b) Equivalent
`
`the parallel-resonant
`for
`circuit.
`
`circuit.
`
`A. Parallel-Tuned Circuit
`
`is shown in Fig.
`used for measurements
`The actual circuit
`l(a) and the circuit model used for analysis is shown in Fig.
`l(b).
`In Fig.
`l(a), C’. is a small series capacitor used to inject
`the signals and Cx is stray capacitance.
`cb is a large bypass
`In Fig. 1(b), all capaci-
`capacitor and ~Q is the bias voltage.
`tance has been lumped into a single nonlinear capacitance and
`a Norton equivalent current source input
`is used.
`For small distortion,
`the output
`voltage UOin Fig. 1(b) can
`be expressed as a Volterra series [9]
`in terms of
`the input
`is as
`follows:
`
`UO=Al(@)oi~
`
`+Az(j@l, @Joi~
`
`‘A3(h,
`
`ja2,
`
`jm3)oi~
`
`+“
`
`.
`
`.
`
`(1)
`
`the magnitude and phase
`where the operator sign indicates that
`of each term in i:
`is to be changed by the magnitude and phase
`. . . , jon ). A ~(ja)
`is the normal
`linear
`transfer
`of An(jq,
`function
`the circuit.
`of
`Once the Volterra
`coefficients
`An(jcq,
`“ “ “ , jan)
`are known, any type of distortion
`such as
`cross modulation,
`intermodulation,
`or harmonic distortion
`can
`be calculated [10].
`The capacitance C(V) of a diode can usually be expressed in
`the form
`
`(2)
`
`incremental diode capacitance
`built-in potential
`constant
`total voltage
`power
`law exponent.
`
`where
`
`C= dQ/dV
`
`@ Kv n
`
`received May 23, 1974; revised September 6, 1974. This
`Manuscript
`work was supported by the U.S. Army Research Office, Durham, unda
`Grant DAHC04-74-G-0151.
`of Electrical Engineering and
`R. G. Meyer
`is with the Department
`Computer Sciences and the Electronics Research Laboratory, University
`of California, Berkeley, Calif. 94720.
`of Electrical Engineering
`M. L. Stephens was with the Department
`and Computer Sciences and the Electronics Research Laboratory, Uni-
`versity of California, Berkeley, Calif. 94720.
`He is now with the
`Signetics Corporation, Sunnyvafe, Calif.
`
`Page 1 of 8
`
`GOOGLE EXHIBIT 1010
`
`

`

`48
`
`IEEE JOURNAL
`
`OF SOLID-STATE
`
`CIRCUITS,
`
`FEBRUARY
`
`If ~Q is the bias voltage and u is a small
`
`fluctuation,
`
`then put
`
`uo~ = IA I (jcd~)l 1~ Cos (u~t + LA (ju~)).
`
`1975
`
`(15)
`
`~=~Q+u.
`
`Equation (2) can then be expressed as
`
`c(u) = (o + ~Q)
`K
`
`‘(l’
`
`fir
`
`i.e.,
`
`C(V)= CO+ CIV+C’2V2
`
`+...
`
`where
`
`K
`co= (~+ VQ)”
`
`n
`
`G=-CQO+VQ
`
`+ 1)
`C2= co rr(n
`2(O+ ~Q)2
`
`“
`
`(3)
`
`(4)
`
`(5)
`
`(6)
`
`(7)
`
`(8)
`
`arises from the cubic term
`intermodulation
`The third-order
`in (1) and the term at frequency (2wa - Ub)
`is
`
`;
`
`I;I~ 1A3(jclJa , jua , -jcd~)l CQS[(ua + u= - co~)t
`
`+ LA3(jcoa,
`
`jwa,
`
`‘j@b)]
`
`.
`
`is defined as the ratio of
`(IM3)
`intermodulation
`Third-order
`the magnitude of
`the third-order
`distortion
`component
`to the
`fundamental
`signal, i.e.,
`
`~ ~~~b l~3(j~a,j@aj
`
`‘j@b)l
`
`1A 1(j~a)l~a
`
`“
`
`IM3
`
`=
`
`However,
`
`lAl(j~a)l
`
`I. = lAl(jQb)l
`
`lb = ~0
`
`(16)
`
`(17)
`
`where VO is the peak value of each fundamental
`i.e.,
`
`output voltage,
`
`can be lumped in
`stray capacitance shunting the circuit
`Any
`with CO.
`It
`is shown in Appendix
`I that
`the Volterra
`coeffi-
`cients for the parallel-tuned
`circuit are given by
`
`(9)
`
`(lo)
`
`A3(jq,
`
`jw2,
`
`jtia) =
`
`(jq
`
`+jw +juJ
`
`(C, Am+ $A (jq)A,
`
`(@&l
`
`I (h)
`
`1
`—+
`Rp
`
`1
`
`( jul
`
`+ ju2
`
`+ ju3)L
`
`+ (jcol + jw2 *ju3)C’0
`
`(11)
`
`(12)
`
`+A l(j@A’(jq
`
`,j~’)].
`
`on inter-
`The experimental work in this paper was performed
`modulation
`distortion generated by two input sinusoidal signals
`at different
`frequencies.
`Let
`
`is =1= COS Cdat
`
`‘+Ib
`
`COS (dbt
`
`(13)
`
`where 1= and ~b are adjusted to produce equal output voltage
`magnitudes.
`Then substituting
`(13)
`in (1) gives for
`the
`fundamentals
`
`U. ~ = 1A1( j~a)l
`
`I. cos (u.
`
`f + LA W))
`
`(14)
`
`(18)
`
`is de-
`(IM2)
`intermodulation
`fashion, second-order
`In a similar
`fined as the ratio of
`the magnitude of
`the distortion
`compo-
`nent at (ul
`+ ti2)
`to the fundamental
`signal and
`
`jtib)l
`lA2(jclJa,
`1M2 = ‘0 1A, (j@
`1A I(j”b)l”
`
`(19)
`
`IM2 and IM3 in the
`of
`The above equations allow calculation
`special case in practice is
`general case. However, an important
`when Ua ==~b and both are close to the tuned circuit center
`frequency.
`It
`is further
`assumed that
`the Q of
`the circuit
`is
`high and the stray capacitance is small. Then, using (6)-(9)
`and (1 1), (18) can be expressed as
`
`I-M?+J (20)
`1M3‘:oORp$v’
`
`and U. =
`frequency
`center
`is the tuned circuit
`where U.
`~~ ~6.)b.
`Although most of
`the frequency dependence of
`the
`is apparently
`eliminated
`by the simplification,
`the
`distortion
`tuned circuit
`center
`frequency U.
`is still a parameter, due to
`the frequency-dependent
`diode capacitance’ nonlinearity.
`In
`addition,
`the frequency
`characteristics
`of
`the tuned circuit are
`contained in the term in parentheses.
`If P is the average output power
`in RP per signal, then
`
`P=~~.
`2 Rp
`
`Using (21)
`
`in (20) and rearranging gives
`
`~M3 = Q~C2P
`2W0 c;
`
`~ _
`
`2n
`1.5(n+l)
`
`(
`
`)
`
`where
`
`QP = UOCORP.
`
`(21)
`
`(22)
`
`(23)
`
`(20) and (22) can be used to calculate IM3 caused
`Equations
`by variable-capacitance
`diode tuning of parallel-tuned
`circuits.
`
`Page 2 of 8
`
`

`

`MEYER
`
`AND STEPHENS:
`
`DISTORTION
`
`IN DIODES
`
`49
`
`-20
`
`1
`
`—
`
`0
`
`calculated
`measured
`
`3–
`
`2–
`
`l–
`
`0
`
`.
`
`;
`.
`.
`cd —
`
`.—
`
`!=-,
`
`7~.
`
`Fig. 2. Plot of
`
`the terms dependent on n in (20) versus diode capaci-
`tance law exponent n.
`
`0
`
`0
`
`\
`
`I
`
`MV 1652
`n=o.47
`
`o MV 1401
`\
`n=l
`.61
`
`‘0:“\,,:,,,
`
`Lo
`
`BIOS voltage
`
`,
`
`V.
`
`10.0
`( volt)
`
`plots for typical diodes described
`Fig. 3. Measured capacitance-voltage
`in this paper.
`
`The term in parentheses in these equations indicates the pos-
`sibility
`of eliminating
`IM3 for a particular
`value of n. This
`occurs for n = 3 and not
`for n = 2 as has previously been sug-
`gested [3],
`[6].
`However,
`the practicality
`of attempting
`to
`realize this situation is doubtful.
`This is shown in Fig. 2, where
`the terms dependent on n in (20) are plotted as a function
`of
`n. The factor n (n + 1) comes from C2 in (20).
`The null at
`n = 3 is quite sharp and errors in realizing the exponent
`can
`lead to high distortion
`levels. This is discussed further below.
`In (20) and (22),
`the first
`term in parentheses is due to the
`C’2V2 term in (5) and the second term is due to second-order
`interaction.
`In this case the second-order
`interaction
`always
`acts to reduce overall distortion
`by cancellation.
`In order to check the above theoretical expressions, measure-
`ments were made on a number of variable-capacitance
`diodes.
`The capacitance versus voltage plots for some of
`these diodes
`are shown in Fig. 3 on log scales. Two of
`the diodes are con-
`ventional
`diffused junctions with
`n s 0.4 while the third is
`hyper-abrupt with n a 1.6. Circuit Q values were kept
`in the
`range 10 to 20 and the input
`frequencies were kept centered
`
`I
`0.1
`
`I
`I
`
`n
`
`1
`10
`
`-50
`
`h\\
`
`0
`
`\
`
`\.
`
`\+ok246610
`
`Bins
`
`voltage,
`
`VQ (volf)
`
`Fig. 4. Calculated and measured distortion
`for the MV 1652 diode in a
`parallel-resonant
`chcuit.
`Signal
`frequencies ranged from 5.2 to 7.5
`MHz and each signal produced 223 mV rms across the tuned circuit.
`
`— c alculoted
`0 measured
`
`\
`
`\
`
`\
`
`\
`
`\
`
`,
`
`\
`
`\
`
`\
`
`b-o,
`
`\
`
`\
`
`0
`\
`
`\
`
`-1011
`
`4
`
`>
`
`:
`:
`---
`
`m
`=
`E
`~ -30.
`=
`
`\
`
`-40-
`
`-50 -
`
`-60-8
`
`Bias
`
`voltage
`
`, V.
`
`(WJt
`
`)
`
`Fig. 5. Calculated and measured distortion
`diode
`the hyperabrupt
`for
`MV 1401 in a parallel-resonant
`circuit. Signal frequencies ranged from
`3.5 to 10.1 MHz and each signal produced 223 mV rms across the
`tuned circuit.
`
`in the band. Thus the frequency of measurement changed as
`the bias voltage (and thus the diode capacitance)
`changed.
`Measured values of
`IM2 and IM3 for
`the MV 1652 (n= 0.47)
`are compared with calculated values in Fig. 4 and good agree-
`ment
`is seen. The calculated values were determined from (18)
`and (19), although (20) and (22) are quite accurate under these
`conditions
`and could be used for
`IM3.
`The value of n found
`from Fig. 3 was used to calculate the coefficients
`in (5).
`Measured distortion
`for
`the hyper-abrupt
`diode MV 1401 is
`shown in Fig. 5 and is seen to exhibit anomalous ripples.
`If a
`power
`law of
`the form of (4)
`is assumed for
`the capacitance
`variation,
`this behavior
`cannot be predicted
`and significant
`errors
`are found
`between measurements
`and calculations.
`
`Page 3 of 8
`
`

`

`50
`
`IEEE JOURNAL
`
`OF SOLID-STATE
`
`CIRCUITS,
`
`FEBRUARY
`
`1975
`
`L
`
`v.
`
`R,
`
`“’~r
`
`‘m
`
`Cb
`
`(a)
`
`(b)
`
`Fig. 6.
`
`(a) Measurement
`
`arrangement
`(b) Equivalent
`
`the
`for
`circuit.
`
`series-resonant
`
`circuit.
`
`in (5) were
`the coefficients
`if
`that
`it was found
`However,
`routine from
`at each bias point by a curve-fitting
`determined
`careful measured capacit ante data,
`the observed behavior was
`well predicted by (18) and (19) as seen in Fig. 5. Thus these
`hyper-abrupt
`diodes ekhibit
`small
`ripples in the C-V curves
`which
`are too small
`to be observed visually but
`show up
`strongly
`in distortion measurements.
`The reason for
`this be-
`havior
`can be found
`in the method of
`fabrication
`of
`these
`devices [2].
`The required doping profile is realized as a step-
`wise approximation
`leading to small perturbations
`in the C-V
`curve.
`it would appear difficult
`the above reasons,
`For
`~ = 3 with sufficient
`precision to achieve the null
`discussed previously.
`
`to achieve
`in distortion
`
`B. Series-Tuned Circuit
`
`The circuit used for measurements is shown in Fig. 6(a) and
`the circuit model used for analysis is shown in Fig. 6(b). C~ is
`a large coupling capacitor and cb is a large bypass capacitor.
`Rb is a large bias resistor.
`voltage UOin Fig.
`express the output
`For small distortion,
`6(b) as a Volterra series in terms of the input us as follows:
`
`U. =Bl(ju)
`
`o us +Bz(jq,
`
`jq)o
`
`u:
`
`+B3(jq,
`
`ja2;
`
`jo3)ou~+
`
`o...
`
`(24)
`
`to that
`An analysis similar
`coefficients
`for this case as
`
`in Appendix
`
`I gives the Volterra
`
`D1 (jw) =
`
`1
`
`1 + juCoR~ + ( jco)2LCo
`
`(27a)
`
`(27b)
`
`is defined as in (12).
`and DID2
`(18) and (19) again give
`Following the previous development,
`coefficients
`l?n (jti ~,
`the circuit
`distortion
`if
`the Volterra
`o “ “ , jan)
`are used.
`Again taking the special case of U. =
`Ua = Ub it can be shown that
`the third-order
`int ermodulat
`ion
`for the series-tuned circuit
`is
`
`IM3=+V;
`
`C2
`--Q,
`
`3
`
`1-
`
`(
`
`2n
`
`1.5(n+
`
`1))
`
`where
`
`Q.=
`
`1
`
`U. CoR~
`
`(28)
`
`(29)
`
`and all other quantities are previously defined. Equation (28)
`is similar
`in form to (20)
`for
`the parallel-tuned
`case.
`If P is the
`average out put power
`in R~ per signal, then
`P=;ii
`
`(30)
`
`s
`
`B1 (JcJ)
`
`jaCoR~
`+ (jti)2LC0
`= ~ ~jUCoR~
`
`If (30)
`
`is substituted in (28), we obtain
`
`(25)
`
`B2(jti1,
`
`jc.02) =
`
`L
`
`1 + (jtil
`
`+jo2)CoR~
`
`+ (jai
`
`+jti2)2-LCo
`
`(CID1D2+D,(jul)Dl(jti2)Dl(ju3))
`R,(RJ1 +IkI+ +h)
`1 + (jUI +juz+ju3) CoR~ + (jai
`+ ja2 +jti3)2LC0
`where
`
`*M3 = Q~C2P
`2(4 c;
`
`~ _
`
`2n
`
`1.5(n+
`
`1)
`
`“
`
`(31)
`
`(
`)
`the circuit Q values are
`if
`to (22)
`is identical
`(31)
`Equation
`equal. That
`is, the same formula applies to both the series and
`parallel cases.
`of Fig.
`the series resonant circuit
`for
`Measured distortion
`6(a) with diode IN 5461 (n = 0.42)
`is shown in Fig. 7 together
`with
`distortion
`calculated
`from (18) and (19).
`Reasonable
`agreement
`is seen. The deviations are believed due to the dif-
`ficulties
`in realizing accurately
`the required low value of Rs at
`high frequencies.
`The voltage levels are much smaller than the
`
`Page 4 of 8
`
`

`

`MEYER
`
`AND STEPHENS:
`
`DISTORTION
`
`IN DIODES
`
`51
`
`—
`
`0
`
`calculated
`
`measured
`
`—
`
`0
`o
`
`calculated
`measured
`measured
`
`~
`
`BIOS voltage,
`
`V.
`
`( volt)
`
`10
`
`~.
`
`Bias
`
`voltage,
`
`V~
`
`(volt)
`
`Fig. 7. Calculated and measured distortion
`in
`diode
`the IN 5461
`for
`ranged
`from 31 to 48
`a series-resonant
`circuit.
`Signal
`frequencies
`MHz and each signal produced
`22.3 mV rms across the series resistor
`RP
`
`the MV 2101 diode in
`for
`Fig. 9. Calculated and measured distortion
`the circuit of Fig. 8. Signal frequencies ranged from 100 to 137 MHz
`and each signal produced 22.3 mV rms across the resistor R$.
`
`*CS L
`
`Vo
`
`R$
`
`T
`
`“z
`
`d
`
`v,
`
`&c’
`
`c
`
`, LTIT’T+ %Vc
`
`(0)
`
`R,
`
`‘(b)
`
`Fig. 8.
`
`(a) Resonant
`
`the measurement
`for
`circuit
`Equivalent
`circuit.
`
`of distortion.
`
`(b)
`
`resonant case because the series resistance R~ is much
`than Rp and results in more output
`power
`for a given
`
`parallel
`smaller
`voltage.
`comparison with multiple-diode
`the purposes of
`Finally,
`circuits, distortion measurements were made in the resonant
`circuit of Fig. 8(a) where C~ is a small coupling capacitor.
`This
`circuit
`is convenient
`for high-frequency measurements and the
`equivalent
`circuit
`is shown in Fig. 8(b).
`The results of mea-
`surements on the MV 2101 diode in this circuit are shown in
`Fig. 9 and compared with calculated vahres where good agree-
`ment
`is seen. The expressions for distortion
`in this c~se are the
`same as for the series resonant circuit of Fig. 6 for given output
`signal
`levels.
`
`III. MULTIPLE-DIODE CIRCUITS
`
`schemes have been suggested to reduce distortion
`Various
`caused by variable -capacit ante diode tuning [7],
`[8].
`Two of
`these are the antiparallel
`connection
`shown in Fig. 10 and the
`back-to-back
`(or
`composite
`common
`cathode)
`connection
`shown in Fig. 11. Capacitors Co are large bypass capacitors.
`Using the analysis methods developed in this paper,
`these
`
`Fig. 10. Antiparallel
`
`diode connection.
`
`;, 4m J
`
`+
`
`V2
`
`Rb
`
`7.
`
`Icb
`
`v,
`
`Fig. 11. Back-to-back
`
`or composite
`
`common
`
`cathode
`
`diode connection.
`
`can now be analyzed and compared with the single-
`circuits
`diode circuit
`for distortion
`performance.
`
`A. Antiparallel Connection
`
`the capacitance of each diode by a power series as
`Represent
`of
`the incremental
`applied voltage u in a similar
`a function
`fashion to (5):
`
`C1(U)=KO +K1V+K*U2
`
`+..”
`
`C2(V)=L0-
`
`L1U+L2V2+ .
`
`. .
`
`(32)
`
`(33)
`
`the minus
`where
`connection.
`The total capacitance is
`
`sign
`
`occurs
`
`because
`
`of
`
`the reversed
`
`CT=(KO+
`
`LO)+ (KI-LI)U+(K2+L2)U2
`
`+.
`
`. . .
`
`(34)
`
`For two matched diodes
`
`CT=2KO+
`
`2K2V2 +..”.
`
`(35)
`
`Page 5 of 8
`
`

`

`52
`
`IEEE JOURNAL
`
`OF SOLID-STATE
`
`CIRCUITS,
`
`FEBRUARY
`
`1975
`
`—
`
`calculated
`
`-20–
`
`cI
`\
`
`-30-
`
`-40-
`
`x.
`
`=.-
`
`3= %s -
`
`:
`
`\
`
`0
`
`\
`
`.
`
`-~~o
`
`Bias
`
`voltage,
`
`V~
`
`( volt)
`
`are
`distortion
`circuit
`tuned
`for
`analyses
`previous
`The
`now applicable with
`these new coefficients.
`From (10),
`However,
`the
`lA2(jul,
`j~z)l
`= O and IM2 is ideally
`zero.
`is now lost
`canceling effect of
`the second-order
`interaction
`(see (20) and (22)) and thus IM3 will be somewhat higher
`than
`for a single diode.
`is compared with calculated distortion
`Measured distortion
`for
`this case in Fig. 12 using MV2101
`diodes and good agree-
`ment
`is seen for
`IM3.
`Second-order distortion
`IM2 was not
`measurable.
`These measurements were made in the circuit of
`Fig. 8. A somewhat
`lower Q value for Fig. 12 prevents a direct
`comparison with the single-diode data of Fig. 9, but
`the distor-
`tion data in the two figures are seen to be comparable.
`
`B. Composite Common Cathode
`
`The analysis of the circuit of Fig. 11 can again be approached
`by deriving an equivalent power series for
`the capacitance of
`the multiple-diode
`connection.
`voltage is shared
`the incremental
`In the circuit
`of Fig. 11,
`between the diodes.
`Let u be the total
`incremental
`voltage.
`Then
`
`C1(V1)=KO +K1U1+K2U;
`
`+”””
`
`C2(U2)= L0-L1V2+L2V;
`
`+”””
`
`where
`
`(36)
`
`(37)
`
`diode
`the antiparallel
`for
`distortion
`and measured
`Fig. 12. Calculated
`Signal
`8 using MV 2101
`diodes.
`connection
`in the circuit
`of Fig.
`frequencies
`ranged
`from 76 to 110 MHz
`and each signal produced
`22.3 mV rms across the resistor R~.
`
`the resultant capacitance of
`that
`II
`is shown in Appendix
`It
`composite common cathode connection is
`
`the
`
`c(v) =p~-
`o
`
`*V+%2P;
`0
`
`P;
`
`-POP2)S? +..
`
`.
`
`(39)
`
`where
`
`Po=~+J-
`KO Lo
`
`pi=_—
`
`KI
`
`2K;
`
`LI
`—
`+ 2L;
`
`p2 = K:/2
`
`- KoKz/3
`
`~ L:/2
`
`K:
`
`- LoLz/3
`L:
`
`“
`
`For matched diodes
`
`C=
`
`E!+KE
`28
`
`1
`
`[1
`
`1“5K~
`KOK2
`
`02 +...
`
`(40)
`
`(41)
`
`(42)
`
`(43)
`
`term is halved and the
`the constant
`shows that
`(43)
`Equation
`linear
`term disappears
`as expected.
`Thus
`second-order
`distor-
`tion
`is ideally
`zero.
`The term in V2 (which
`causes third-order
`~
`distortion)
`shows
`two
`significant
`effects.
`First,
`the factor
`the
`arises from the fact
`that each diode has approximately
`half
`ac voltage
`across it and thus third-order
`distortion
`is reduced
`by
`(*)3.
`Secondly,
`the term in parentheses
`shows a second-
`order
`interaction
`term and the possibility
`of
`reducing
`this co-
`efficient
`to zero.
`This will occur
`if
`
`I. SK:
`—=1.
`KOK2
`
`(44)
`
`is n = ~.
`(7), and (8) the condition imposed by (44)
`Using (6),
`it
`is relatively easy to produce variable capacitance
`In practice,
`diodes with exponents
`close to this value and hence to achieve
`very low levels of second- and third-order
`distortion
`using the
`composite common cathode connection.
`Measured distortion
`for
`the composite common cathode con-
`nection in the circuit of Fig. 8 using MV 2101 diodes is shown
`in Fig. 13. Good agreement
`is seen between calculated and
`measured distortion
`for
`IM3. The errors in IM2 prediction
`are
`due to the sensitivity
`of
`the circuit
`to unbalance caused by
`stray
`capacitance
`associated with
`bias resistor
`llb.
`Again,
`direct comparison
`cannot be made with the single-diode data
`of Fig. 9 due to differing Q values. However,
`the general trend
`of much reduced distortion
`for
`the composite common cath-
`ode connection is apparent.
`
`IV. CONCLUSION
`
`Closed-form expressions have been derived for distortion
`caused by variable capacitance diodes in series- and parallel-
`tuned circuits.
`The optimum value for
`the diode capacitance
`law exponent
`is shown to be 3 for
`the minimization
`of
`third-
`order distortion,
`but
`there are practical difficulties
`in the reali-
`zation of
`this distortion
`null. One of
`these difficulties
`is the
`presence of small
`irregularities
`in the C-V plots of hyperabrupt
`diodes which cause anomalous ripples in the distortion
`charac-
`teristics of
`these diodes.
`the composite
`of
`the superiority
`In multiple-diode
`circuits,
`over the antiparallel
`connection
`common
`cathode connection
`is shown to be due to two factors.
`These are the sharing of
`voltage across the two series diodes in the composite common
`cathode connection
`and also the existence of a second-order
`
`Page 6 of 8
`
`

`

`MEYER AND STEPHENS: D1STORTION IN DIODES
`
`53
`
`-40 1
`
`Substituting
`
`(1) in (47) and equating first-order
`
`terms gives
`
`i.e.,
`
`Al(jU)=
`
`~
`
`11
`
`.
`
`— + jcdCO
`R<+ jcoL
`
`Equating second-order
`
`terms gives
`
`O = (jcol +jcdz) CoA2(jcol
`
`,jc+)
`
`c,
`+ ( jcdl + jcoz) ~Al(jol)Al(jco2)+
`
`(48)
`
`(9)
`
`A2(jcol,
`
`ja2)
`
`(jtil
`
`+j~2)L
`
`-70~o
`
`Bias
`
`voliage,
`
`VQ
`
`I volt)
`
`i.e.,
`
`Fig. 13. Calculated and measured distortion for the composite common
`cathode diode connection
`in the circuit of Fig. 8 using MV 2101
`diodes.
`Signal
`frequencies ranged from 135 to 178 MHz and each
`signal produced 22.3 mV rms across the resistor R$.
`
`Equation (11)
`order terms.
`
`is derived in a similar
`
`fashion by equating third-
`
`interaction
`for n = 0.5.
`
`term which causes a null
`
`in third-order
`
`distortion
`
`APPENDIX II
`
`APPENDIX I
`
`PARALLEL RESONANT CIRCUIT ANALYSIS
`
`Consider the parallel resonant circuit shown in Fig. 1(b). The
`incremental
`capacitance is given by
`
`c(v)= d#=co+c1u+c2v2+”
`
`. .,
`
`(5)
`
`i.e.,
`
`COMPOSITE COMMON CATHODE CONNECTION
`
`the diodes of Fig. 11. Represent
`Consider
`capacitance of
`the diodes by
`
`the incremental
`
`c1 (VJ =
`
`dQ ~
`—=
`dvl
`
`KO+KIVI+K2V:
`
`+.””
`
`dQ2
`C* (V2) = — = LO- L1V2+L2V;
`dvz
`
`+”””.
`
`charges on the capacitances are
`
`(49)
`
`(50)
`
`C(UO)=CO +C1UO+C2V; +”””
`
`where Co, Cl, and C2 are defined
`capacitor current
`ic is given by
`
`dQ _ dQ duo
`——
`
`._
`‘c-%-duo
`
`dt’
`
`The incremental
`
`(45)
`
`by (6),
`
`(7), and (8);
`
`i.e.,
`
`Q,
`
`=
`
`“’ Cl(vl)dvl,
`
`J
`
`o
`
`i.e.,
`
`i.e.,
`
`Cl dv:
`duo
`ic=co —+ ——+—
`dt
`2
`dt
`
`Cz dv:
`— +“””.
`3
`dt
`
`Ql=Ko
`
`Vl+K~V~+K~v:
`
`+...
`
`(46)
`
`Q2=Lov2-
`
`Ll~L23
`yv2+7v2
`
`+”””.
`
`Represent VOas a Volterra series in terms of
`
`is as follows:
`
`Since this is a series connection
`
`V. =Al(j@)
`
`oi~+Az(j@l,
`
`j@Joi~
`
`+A3(jwl,
`
`ja2,
`
`jti3)oi~
`
`+----
`
`Kirchhoff’s
`
`current
`
`law for Fig.
`
`l(b) gives
`
`i~=iC+—
`
`1
`L J
`
`vodt + a.
`RP
`
`(51)
`
`(52)
`
`(
`
`(53)
`
`Q,= Q2=Q.
`
`‘1)
`
`Using (53) and performing
`gives
`
`a series reversionon(51)
`
`and (52)
`
`(47)
`
`‘Q
`‘1 ‘<-2K0
`
`~Q2+z
`
`1E-%)Q’-”””
`
`“4)
`
`Page 7 of 8
`
`

`

`Now the total voltage is
`
`V= V1+U2.
`
`Using (56), adding (54) and (55), and performing
`reversion gives
`
`(56)
`
`another series
`
`Q=&v
`
`pi
`
`~
`
`+#2P;
`
`-POP2)U3-
`
`. . .
`
`(57)
`
`where
`
`(40)
`
`(41)
`
`(42)
`
`Equation (57) can be differentiated
`(39).
`
`to give C(u)= cZQ/drJas in
`
`REFERENCES
`
`[1]
`
`[2]
`
`[3]
`
`[4]
`
`[5]
`
`[6]
`
`[7]
`
`K. E. Manchester, J. D. McDougall, O. Tkal, and T. W. Chu,
`“Monolithic
`varactor
`tuned
`rf amplifier
`using ion implantation,”
`in ISSCC Dig. Tech. Papers, pp. 186–1 87, 1973.
`diode
`tuning
`P. M. Norris
`and P. Heidenreich,
`“Hyperabrupt
`Trans. Broadcast
`theory
`and application
`to AM radio,”
`IEEE
`pp. 87-91, July 1967.
`Telev. Receivers,
`vol.BTR-13,
`M. H. Norwood and E. Shatz, “Voltage variable capacitor tuning:
`a review,” Proc
`IEEE,
`vol. 56, pp. 788-798,
`May 1968.
`to am and
`G. Oswald,
`“Application
`of multiple
`varactor
`diodes
`1968.
`fm tuners~’
`in ISSCCDig.
`Tech. Papers, pp. 138-139,
`IEEE Trans.
`L. Sokoloff,
`“IC voltage variable
`capacitors
`(WC)j’
`vol. BTR-15, pp. 33-40, Feb. 1969.
`Broadcast
`Telev. Receivers,
`R. De Cola, “Varactor tuning applied to AM-FM receivers;’ IEEE
`Trans. Broadcast
`Telev. Receivers,
`vol. BTR-13,
`pp. 82-86,
`July
`1967.
`production
`“The
`and W. John,
`H. Henrici
`circuits
`with
`voltagedependent
`resonant
`entech. Z., vol. 16, no. 10, pp. 517-522,
`
`of cross-modulation
`capacitors,”
`Nachricht-
`1963.
`
`in
`
`IEEE JOURNAL
`
`OF SOLID-STATE
`
`CIRCUITS,
`
`FEBRUARY
`
`1975
`
`[8]
`
`[9]
`
`[10]
`
`distortion with capacitor
`“Nonlinear
`H. Keller and O. Dietrich,
`no. 4, pp. 266-269, 1967.
`diodes~’ Radio Mentor,
`S. Narayanan, “Transistor distortion analysisusing Volterra series
`representation,” Bell Syst.
`Tech.
`J.,
`vol.
`46,
`pp.
`991-1024,
`May/June
`1967.
`“Cross modula-
`R. G. Meyer, M. J. Shensa, and R. Esehenbach,
`tion and intermodulation
`in amplitlers
`at high frequencies,”
`IEEE
`J. Solid-State
`Circuits,
`vol. SC-7, pp. 16-23,
`Feb. 1972.
`
`was born
`(S’64-M68-SM’74)
`Robert G. Meyer
`in Melbourne,
`Australia,
`on July 21, 1942. He
`received
`the B. E., M. Eng. Sci. and Ph.D.
`de-
`grees
`in
`electrical
`engineering
`from the Uni-
`versity
`of Melbourne
`in 1963,
`1965, and 1968,
`respectively.
`as an Assistant
`he was employed
`In
`1968,
`Lecturer
`in Electrical
`Engineering
`at
`the Uni-
`versity
`of Melbourne.
`Since September
`1968,
`he has been
`employed
`in the Department
`of
`Electrical Engineering and Computer Sciences,
`University of California, Berkeley, where he is now an Associate Profes-
`sor. His current
`research interests are in high-frequency
`distortion
`in
`amplifiers and analog integrated circuit design. He is a consultant
`to
`Exar Integrated Systems,
`Inc.
`Dr. Meyer is a Member of Sigma Xi.
`
`in Salt Lake City,
`L. Stephens was born
`Mark
`He received
`the
`on November
`29, 1947.
`Utah,
`Institute
`of Tech-
`B. S.E.E. degree from Carnegie
`1969,
`and
`the
`nology,
`Pittsburgh,
`Pa.,
`in
`M. S.E.E.
`degree
`from the University
`of Cali-
`fornia
`at Berkeley
`in 1974.
`Design Engineer
`He has worked
`as a Circuit
`for Hewlett-Packard
`Co., Avondale,
`Pa., and as
`a Research
`Assistant
`for
`the Electronics
`Re-
`search
`Laboratory
`the University
`of Cali-
`at
`fornia,
`Berkeley.
`He is presently
`employed
`by
`Signetics Corporation, Sunnyvale, Calif.
`
`Page 8 of 8
`
`