=Paper=
{{Paper
|id=Vol-3309/paper24
|storemode=property
|title=Model Study of the Q-factor of a Varicap Diode by Its Equivalent Circuits
|pdfUrl=https://ceur-ws.org/Vol-3309/paper24.pdf
|volume=Vol-3309
|authors=Andrii Rudyk,Andriy Semenov,Olena Semenova,Serhii Tsyrulnyk,Volodymyr Tromsyuk
|dblpUrl=https://dblp.org/rec/conf/ittap/RudykSSTT22
}}
==Model Study of the Q-factor of a Varicap Diode by Its Equivalent Circuits==
https://ceur-ws.org/Vol-3309/paper24.pdf
Model Study of the Q-factor of a Varicap Diode by Its Equivalent
Circuits
Andrii V. Rudyka, Andriy O. Semenovb, Olena O. Semenovab, Serhii M. Tsyrulnykc and
Volodymyr D. Tromsyukd
a
National University of Water and Environmental Engineering, Soborna street, 11, 33000, Rivne, Ukraine
b
Vinnytsia National Technical University, Khmelnytske highway, 95, Vinnytsia, 21021, Ukraine
c
Vinnytsia National Agrarian University, str. Sonyachna, 3, City, Vinnytsia, 21008, Ukraine
d
Vinnytsia Technical College, Khmelnytske highway, 91/2, Vinnytsia, 21021, Ukraine
Abstract
In this study, the authors showed that the Q-factor of a varicap diode depends on frequency
and is a function of parameters of its small-signal equivalent circuit. Influence of the equivalent
circuit parameters on the varicap diode Q-factor in ranges of low and high frequencies was
analyzed. Equations for determining the maximum value of the varicap diode Q-factor and a
frequency that corresponds to the extreme value of the Q-factor were obtained. Theoretical
results were experimentally verified for three different varicap diodes of BB1xx types
manufactured by NXP Semiconductors. Deviations of theoretical results from experimental
testing in the (20–500) MHz frequency range do not exceed 8.6%. For varicap diodes of BB149
and BB174 types, the frequency ranges were established, in which their Q-factor was more
than 35; and for a varicap diode of BB181 type the frequency ranges were established, in which
its Q-factor was more than 15. Parameters of elements of series and parallel equivalent circuits
of the varicap diode were determined. Frequency dependences of active and reactive
components of the series and parallel equivalent circuits of the varicap diode were analyzed.
For three types of varicap diodes, frequency ranges were determined in which difference
between an equivalent capacitance of the equivalent circuit and a barrier capacitance was less
than 0.5%, and difference between an equivalent resistance of the equivalent circuit and a loss
resistance in the emitter and base regions was less than 3%. It was proved that in the frequency
range where the varicap diode Q-factor was greater than 35 (BB149 and BB174) or 15
(BB181), the equivalent capacitances of series and parallel equivalent circuits can be regarded
as independent of frequency and equal to the capacitance of the varicap diode junction with an
error of less than 0.5%.
Keywords 1
Varicap diode, Q-factor, model study, mathematical model, equivalent circuit
1. Introduction
Varicap diodes are used as electrically controlled capacitances in oscillatory circuits of
telecommunication and information-measuring systems for various purposes. The operating principle
of a varicap diode is based on dependence of the barrier capacitance of an electrical junction on the
applied reverse voltage [1, 2]. This dependence is the capacitance-voltage characteristic of the varicap
diode.
ITTAP’2022: 2nd International Workshop on Information Technologies: Theoretical and Applied Problems, November 22–24, 2022,
Ternopil, Ukraine
EMAIL: a.v.rudyk@nuwm.edu.ua (A. Rudyk); semenov.a.o@vntu.edu.ua (A. Semenov); semenova.o.o@vntu.edu.ua (O. Semenova);
sovmsvom@gmail.com (S. Tsyrulnyk); 2013tvd@gmail.com (V. Tromsyuk)
ORCID: 0000-0002-5981-3124 (A. Rudyk); 0000-0001-9580-6602 (A. Semenov); 0000-0001-5312-9148 (O. Semenova); 0000-0002-5703-
9761 (S. Tsyrulnyk); 0000-0001-5022-8159 (V. Tromsyuk)
©️ 2021 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
For efficient use of the varicap diode in high-frequency circuits, a frequency range must be
determined, in which its Q-factor will be maximum or exceed a given value, by equivalent parameters
of its small-signal circuit. Also, to simplify analysis in an operating frequency range, the varicap diode
can be represented by one of the simplified equivalent circuits proposed in this paper.
Varicap diodes are being utilized in phase locked loops [3, 4]. Moreover, they are being utilized and
in such devices as voltage-controlled oscillators [5, 6], frequency modulators [7, 8], comparators [9,
10], variable phase shifters [11, 12], bandpass filters [13, 14], power amplifiers [15, 16].
A simple but effective method to reduce reference spurs in analog subsampling phase locked loops
by applying a varicap diode cancellation technique is presented in [3]. A fully synthesizable injection-
locking based phase locked loop with an interpolative phase-coupled oscillator, a digital-to-analog
converter with current output, and a digital varicap diode with fine resolution are discussed in [4]. Here,
all circuits making up the phase locked loop were designed and implemented by standard elements
without any modification.
A variation-aware design methodology for a high-performance MOS-varactor voltage-controlled
ring oscillator (MV-VCRO) in near-threshold-voltage (NTV) mode was proposed in [5]. Delay-models
for conventional, bulk-driven, and dynamic-threshold oscillators allowing for nonlinearity in near-
threshold-voltage mode are considered using MOS-varactor capacitance models and effective drive
current. Effects of supply-induced frequency variations on single-ended tuning LC voltage-controlled
oscillator that degrade the jitter performance of the clock are examined in [6]. Moreover, a
compensation technique is proposed, which applies complementary varactors for increasing the supply
sensitivity of single-ended tuning oscillator. The varactor effective capacitance changes with the supply
voltage, which impacts the supply sensitivity.
A W-band vector modulator, which based on vector summing with a Gilbert cell, is discussed in [7].
Here two error-reducing schemes are applied to a summing circuit. One is to apply current sources,
which consist of short channel transistors, to take advantage of the so-called channel length modulation
effect. The other scheme implements impedance correction varactors at input. It controls gain and phase
with a digital-to-analog converter, which makes the output impedance of the summing circuit constant.
An approach to a frequency modulation by employing a split-ring resonator loaded with a varactor
diode is considered in [8]. There, the modulation occurs because of a continuous variety of the varactor
diode capacitance by changing its bias voltage by a signal which is to be modulated. The frequency
range of modulated signals can be controlled by adjusting varactor parameters.
A differential time-domain comparator, which is formed by two voltage controlled delay lines – one
per input terminal, as well as a binary phase detector for comparison solving were discussed in [9]. A
set of digitally controlled inversion-mode varactors is utilized for adjusting propagation delay through
respective lines. Such varactors provide tuning capabilities to the time-domain comparator; which
features can be used for offset calibration. An offset calibration approach is presented in [10]. This
approach uses dynamic characteristics of a comparator in order to obtain a wide linear tuning range by
locating varactors at two different internal nodes – drains of the input pairs (for high linearity) and
output nodes (for wider compensation range). The comparators are implemented in a 3-bit 1GS/s flash
ADC, which can be integrated into an 8-bit hybrid ADC architecture.
A low loss reflection-type phase shifter is introduced in [11]. This shifter applies a varactor diode as
a load. A performed analysis showed that a real part of the load affects phase variation and insertion
loss. A variable phase shifter is presented in [12]. This shifter has a 360° range and may be employed
in X-band phased arrays. The regarded system consists of a branchline coupler, variable reflection
loads on varactor diodes, and a switched line topology. The reflective loads are quite compact and
require only one control voltage for the varactor diodes, so the control circuit is very simplified.
A tunable suspended integrated stripline bandpass filter based on a varactor is developed in [13].
Surface mount varactors are implemented on the filter that provides frequency tuning by using reverse
biasing of the varactors at different voltages. A microstrip balun bandpass filter having a tunable center
frequency is introduced in [14]. Such structure has open loop resonators on varactor diodes, which are
employed for tuning a resonant frequency.
A wideband output combiner which can be used for outphasing power amplifier is considered in
[15]. Two non-commensurate transmission lines are superseded with two reconfigurable T-type
networks, with shunt varactors as a load. A multiband switchable low-noise amplifier is presented in
[16]. The operation bands can be tuned by using a varactor-based tunable network as an interstage
matching circuit, moreover it promotes covering the 5G frequency bands N257/N260. The developed
tunable structure operates without sacrificing noise performance and requires no increase in DC
consumption and in a chip size.
The purpose of this work is a model study of the frequency dependence of the varicap diode Q-factor
on elements of its small-signal equivalent circuit and experimental investigation of ВВ1хх-type varicap
diodes to confirm the adequacy of obtained theoretical results.
2. Parameters and equivalent circuits of the varicap diode
The main parameters of the varicap diodes related to the Q-factor are [17]:
X
a nominal Q-factor of the varicap diode QV V , it is the ratio of the varicap diode
RV
reactance XV to the total loss resistance RV at a nominal bias voltage U nom at a given
frequency;
d QV
the temperature coefficient of Q-factor TK QV , it is a relative change in the varicap
QV dT
diode Q-factor when the ambient temperature changes by 1 K in a given temperature range;
the frequency range of the varicap diode f min fmax , which is determined by limiting
frequencies at which QV f min QV f max 1 ;
1 1
limiting frequencies of the varicap diode f min and f max , where
2 CT RT 2 CT rS
rS rE rB is the resistance of losses in emitter and base regions of the varicap diode; RT is the
varicap diode reverse resistance when the reverse voltage U rev is applied; CT is the junction
capacitance (barrier capacitance).
Since the varicap diode is a high quality element, its Q-factor is evaluated by equivalent circuits or
by different measurement methods. When the resonance method is used, the varicap diode is
implemented in a measuring resonant circuit. The varicap diode Q-factor is estimated from known
parameters of the resonant circuit and the measured resonant frequency and Q-factor of the circuit with
the varicap diode. Therefore, in this case, it is important to accurately determine parameters of the
measuring resonant circuit. When the amplitude-phase measurement method [18] is applied to evaluate
the varicap diode Q-factor, the phase shift between voltages on the varicap diode and an exemplary
element and the ratio of amplitudes of these voltages are measured.
The paper [1] presents a general nonlinear model of a p-n junction (varicap diode) for a large signal
mode, that is, when currents and voltages change within arbitrary limits. The main elements of such
model are a loss resistance r S , a reverse junction resistance RT , which allows for a thermal generation
current in the junction and a leakage current, and a barrier capacitance CT . In general, the reverse
junction resistance RT is a function of a reverse voltage U rev . For practical equivalent circuits, RT is
a fixed resistor corresponding to a linear approximation of the reverse branch of the static characteristic
in a given range of change in U rev . Its resistance has a value of tens of kΩ, therefore, with a forward
bias, it practically does not affect the current.
The model presented in [19] differs from the previous one by the presence of a capacitor СB to allow
for parasitic capacitances of the leads and the varicap diode case, a resistor r d to take into account the
differential resistance of the junction and an inductance Lp of the varicap diode leads.
The most common equivalent small-signal varicap diode circuit (Figure 1,a) [17] consists of lead
inductances Lp 3 nH , package capacitance СB 1.5 pF [19], loss resistance r S , reverse p-n junction
resistance RT , differential p-n junction resistance r d , and barrier capacitance CT .
Such a varicap diode model is simplified, since it does not allow for the effect of the loss resistance
modulation in the base region, the recombination current, the breakdown, the frequency dependence of
the barrier capacitance, and other phenomena. To consider such effects, additional components must be
introduced into the varicap diode equivalent circuit. However, complication of the equivalent circuit
when additional components are included in it leads to a complication of its analysis and does not
always increase the reliability of obtained results.
3. Model study of the Q-factor of the varicap diode by its equivalent circuits
The total resistance of the varicap diode by the equivalent small-signal circuit (Figure 1,a) is:
RT rd 1 (1)
1 RT rd iCT
rS
iCB RT rd
1 RT rd
rS
RT rd iCT RT rd iCT RT rd
ZV i i 2 L p i 2 L p
RT rd 1 iCB RT rd
1 iCB rS
1 RT rd iCT RT rd iCT RT rd
rS
i C B RT rd
1
RT rd iCT
RT2 rS rd rd2 RT rS 2 RT rS rd 2CT2 RT2rS rd2
R r C C R r r C R r R r r r C R r
2 2 2 2
T d T B T S d B T S T d S d T T d
i
3 CT2CB RT2 rS2 rd2 2 L p CB RT rS RT rd rS rd CT RT rd
2
R r C C R r r 2 2
T d T B T S d
C 2 R r r R r r R r r r r C R r 2 L R r C C R r r
2 2 2 2 2 2 2 2 2
B T S d T S d T S d S d T T d p T d T B T S d
2 CB RT rS RT rd rS rd CT RT rd
2
ReV i ImV RV i X V .
(a) (b)
(c) (d)
Figure 1: Varicap diode equivalent circuits: (a) Complete circuit for small signal; (b) Simplified; (c) Low-
frequency simplified; (d) High-frequency simplified
The varicap diode Q-factor by its total small-signal equivalent circuit (Figure 1,a) is:
QV
ImV
X V
3 CT2 CB RT2 rS2 rd2 2 L p CB RT rS RT rd rS rd CT RT rd
2
(2)
ReV RV RT2 rS rd rd2 RT rS 2 RT rS rd
CB 2 RT rS rd RT rS rd RT2 rS2 rd2 rS2 rd2 CT RT2 rd2 2L p RT rd 2CT CB RT rS rd
2
.
C R r r
2 2
T
2
T S d
2
The authors examined frequency dependencies for the Q-factor of varicap diodes of ВВ149, ВВ174
and ВВ181 types produced by NXP Semiconductors. Table 1 presents SPICE Model parameters for
these varicap diodes [20-22].
Table 1
SPICE Model Parameters of varicap diodes
CT Urev rs RT rd RP CB LP
Type
(pF) (V) (Ω) (kΩ) (kΩ) (kΩ) (pF) (nH)
ВВ149 1.9-19.5 0.5-28 0.75 85 7.33 6.75 0.22 0.85
ВВ174 1.95-21.26 0.5-28 0.6 78 6.75 6.21 0.15 0.65
ВВ181 0.7-17.0 0.5-28 3.0 125 7.5 7.08 0.35 1.25
Allowing for formula (2), the authors constructed graphs of QV f CT , f (Figure 2) at the change
in junction capacitance in the (2-20) pF range for varicap diodes of BB149 and BB174 types and in the
(1-17) pF range for the varicap diode of BB181 type, with frequency changing in the range (20-500)
MHz.
The Q-factor lies in the (15-25) range for different types of varicap diodes at frequencies of the
(20~50) MHz order and at a minimum reverse voltage of (0.5-1) V. The value of the varicap diode Q-
factor decreases with increasing reverse voltage (with decreasing barrier capacitance). As the frequency
increases, the varicap diode Q-factor increases and reaches its extremum. With a further increase in the
frequency for the varicap diode equivalent circuit (Figure 1,a) we obtain a negative Q-factor at
frequencies above 700 MHz. This can be explained by the fact that at high frequencies the varicap diode
reactance of becomes inductive, since the inductive resistance of the varicap diode leads becomes
greater than the junction capacitance (Figure 3).
Formula (2) is difficult for practical evaluation of the varicap diode Q-factor. Therefore, at
frequencies up to several hundreds of MHz, one can neglect parameters Lp and СB of the equivalent
RT rd
small-signal circuit, denote RP , and limit to a simplified equivalent circuit (Figure 1,b). In
RT rd
this case, the typical value of reverse resistance of the varicap diode junction is RT 50 k [23].
The total resistance of the varicap diode according to the simplified equivalent circuit (Figure 1,b)
is:
1 (3)
RP
iCT R r С R
2 2 2
r i C R 2
ZV i rS
P S T P S T P
RP
1 1 С
2 2 2
T R P
iCT
RP rS 2СT2 RP2 rS CT RP2
i ReV 1 i ImV 1 RV 1 i X V 1 .
1 2СT2 RP2 1 2СT2 RP2
(а)
(b)
(c)
Figure 2: Graphs of dependencies QV f CT , f for varicap diodes of BB149(a), BB174(b) and
BB181(c) types
Figure 3: Dependence QV f CT , f for the varicap diode of BB181 type in the (20-1000) MHz
frequency range
The varicap diode Q-factor according to the simplified equivalent circuit (Figure 1,b) is:
Im X V 1 CT RP2 (4)
QV 1 V 1 .
ReV 1 RV 1 RP rS 2СT2 RP2 rS
According to formula (4), the resistance r S determines the varicap diode Q-factor at high
frequencies. To increase the Q-factor when an n -region is introduced into the varicap diode, resistance
of the base must be reduced and so does resistance of the contact by choosing its material and increasing
concentration of impurities in the n -region of the base [24].
To evaluate the reliability of the simplified equivalent circuit of the varicap diode (Figure 1,b), the
authors determined dependences of relative deviation of the Q-factor estimate
Q
sQV 1 V 1 100% for varicap diodes of BB149, BB174, and BB181 types (Figure 4) on
QV
junction capacitance and frequency considering the equivalent circuits in Figures 1,a,b (formulas (2)
and (4)).
The value of the Q-factor relative deviation increases when the frequency increases and the reverse
voltage decreases. The maximum deviation of the Q-factor was in varicap diodes with maximum barrier
capacitance (among studied ones, this is a varicap diode of SMV2023-011LF type).
In this study the authors estimated the maximum frequency of an operating frequency range of the
varicap diode equivalent circuits (Figures 1,a,b) in case when the reactance of the varicap diode was at
least 10 times greater than the inductive resistance of its outputs. For the complete equivalent circuit
(Figure 1,a), this condition is as follows:
2
2 5 LP CB2 CT2 rS2 RP2 3 CT2CB RP2 rS2 2 LP CB2 RP rS CT RP2 CT 2C B
(5)
T B P S
2
1 2
C C R r
CB RP rS CT RP2 2 LP
2
10 L .
CB RP rS CT RP
2 p
2
The authors estimated graphically the maximum frequency of the operating frequency range of the
varicap diodes equivalent circuits (Figures 1,a,b) under this condition as shown in Figure 5.
(a)
(b)
(c)
Figure 4: Dependences of relative deviation of the Q-factor sQV f CT , f for varicap diodes of
BB149(a), BB174(b) and BB181(d) types
(a)
(b)
(c)
Figure 5: Dependences of varicap diode reactance XV f CT , f for estimating the maximum
frequency of the operating frequency range of equivalent circuits for varicap diodes of BB149(a),
BB174(b) and BB181(c) types
The operating frequency ranges for varicap diodes of the selected types are from 390 MHz at
maximum capacitance of 19.5 pF to 1100 MHz at minimum capacitance of 1.95 pF for the BB149 type
varicap diode, from 430 MHz at maximum capacitance of 21.26 pF to 1200 MHz at minimum
capacitance of 1.95 pF for the BB174 type varicap diode, from 330 MHz at maximum capacitance of
17 pF to 1300 MHz at minimum capacitance of 0.7 pF for the BB181 type varicap diode (Figure 5).
The authors concluded that the decrease in the maximum operating frequency of the varicap diode is
determined by an increase in its barrier capacitance and depends on the reverse voltage Urev .
At lower frequencies at СT rS 1 , the varicap diode equivalent circuit is a parallel connection of
RP and CT (Figure 1,c), its Q-factor is determined by the formula QV 1.LF CT RP and it linearly
depends on frequency. At high frequencies at CT RP 1 , the equivalent circuit is a series connection
1
of rS and CT (Figure 1,d), the varicap diode Q-factor is determined by the formula QV 1.HF
СT rS
and decreases when frequency increases. To increase the varicap diode Q-factor at high frequencies,
the resistance rS must be reduced, i.e. the thickness of n-region in the base is to be reduced. Taking
into account all above-mentioned, we may conclude that the frequency dependence of the varicap diode
Q-factor has a maximum with the following coordinates:
2
d QV 1 CT RP RP rS CT RP rS
2 2 2
;
(6)
d
2
RP rS 2CT2 RP2 rS
RP (7)
1
d QV 1 rS
0 extr ;
d CT RP
RP (8)
QV 1.max QV 1 extr .
2 rS RP rS
For the varicap diodes the condition RP rS is satisfied (Table 1), so formulas (4) and (8) are
simplified:
CT RP (9)
QV 1 ;
1 2СT2 RP rS
1 (10)
extr ;
CT RP rS
RP (11)
QV 1.max QV 1 extr 0.5 .
rS
Allowing for obtained analytical relationships (7) and (8), frequency theoretical dependences of
varicap diode Q-factors were constructed (solid lines in Figure 6), and coordinates of the extremum of
these dependences were calculated for varicap diodes of BB149, BB174, and BB181 types
manufactured by NXP Semiconductors [20–22]:
the ВВ149 type varicap diode ( rS 0.75 , RP 6.75 k , СT 13 pF ): fextr 172.153 MHz ;
QV 1.max 47.434;
the ВВ174 type varicap diode ( rS 0.6 , RP 6.21 k , СT 9 pF ): fextr 289.852 MHz ;
QV 1.max 50.867;
the ВВ181 type varicap diode ( rS 3 , RP 7.08 k , СT 9 pF ): fextr 121.401 MHz ;
QV 1.max 24.29.
The results calculated by formulas (10) and (11) differ from those calculated by formulas (7) and
(8) by less than 0.1%.
60
50
Varactor Q factor
40
30
20
10
0
0 100 200 300 400 500
Frequency, MHz
Figure 6: Theoretical (solid lines) and experimental (dashed lines) dependencies QV f for
the varicap diodes BB149 (blue and brown), BB174 (grey and yellow) and BB181 (red and green)
A test installation is shown in Figure 7.
Figure 7: A test installation
Figure 6 illustrates experimental dependencies QV f for the varicap diodes of BB149
(brown), BB174 (yellow) and BB181 (green) types with dotted lines. As can be seen, these
experimental dependences represent theoretical ones quite accurately with a relative error no
more than 8.6%. For the experimental dependences, the extremum coordinates are:
varicap diode of ВВ149 type: fextr 171.421 MHz ; QV 1.max 46.883;
varicap diode of ВВ174 type: fextr 260.734 MHz ; QV 1.max 52.446;
varicap diode of ВВ181 type: fextr 118.337 MHz ; QV 1.max 25.323.
When the temperature increases, the varicap diode Q-factor decreases due to increase in the
resistance rS . As the reverse bias increases, the capacitance CT and resistance r S decrease, and the Q-
factor increases accordingly. The decrease of rS when the reverse bias increases can be explained by
expansion of the junction and decrease in the base thickness in n-region of the varicap diode structure
[25].
The authors estimated the minimum voltage amplitude of the varicap diode in large-signal mode by
linear section lengths of the Capacitance vs Reverse Voltage characteristic [20–22], which was about
(2–3) V for all types of considered varicap diodes. The measurements were carried out in small-signal
mode at a 0.5 V voltage across the varicap diode, (1-15) mA currents and (1~10) mW power.
4. Parameters of series and parallel equivalent circuits of the varicap diode
A varicap diode can be represented by series (Figure 8,a) or parallel (Figure 8,b) equivalent circuits
[17, 26] with their parameters being determined by following relationships:
(a) (b)
Figure 8: Equivalent circuits of the varicap diode: (a) Series; (b) Parallel.
- the series equivalent circuit:
RP rS 2СT2 RP2 rS RP 1 2СT2 RP rS (12)
RV . series ,
1 2СT2 RP2 1 2СT2 RP2
1 2СT2 RP2 (13)
СV . series ,
2 CT RP2
CT RP2 (14)
X V . series ,
1 2СT2 RP2
RP 1 2СT2 RP rS iCT RP2 (15)
ZV i RV . serial iX V . serial ;
1 2СT2 RP2
- the parallel equivalent circuit:
R r С R r R 1 С r , (16)
2 2 2 2 2 2 2 2
RV . parallel
P S T P S P T S
2СT2 RP2 rS RP rS 1 2СT2 RP rS
СT RP2 СT (17)
СV . parallel ,
2 CT2 RP2 rS2 RP rS 1 2 CT2 rS2
2
2 CT2 RP2 rS2 RP rS (18)
2
1 2 CT2 rS2
X V . parallel ,
СT RP2 СT
i RV . parallel X V . parallel RP 1 2CT2 rS2 (19)
Z B i .
RV . parallel i X V . parallel 1 2CT2 RP rS iCT RP
Figure 9 demonstrates frequency dependencies of series and parallel equivalent circuit parameters
for varicap diodes BB149 ( СT 13 pF ), BB174 ( СT 9 pF ) and BB181 ( СT 8 pF ) types
considering relationships (12), (13), (16), (17).
(a)
(b)
(c)
(d)
Figure 9: Frequency dependencies of equivalent circuit parameters for BB149 (purple), BB174 (green)
and BB181 (blue) varicap diodes: (a) resistance of the series equivalent circuit (Ohm); (b) capacity of
the series equivalent circuit (pF); (c) resistance of the parallel equivalent circuit (Ohm); (d) capacity of
the parallel equivalent circuit (pF)
From the dependencies shown in Figure 9 we may made the following conclusions:
the dependencies RV . series f and RV . parallel f verge to the value RP rS RP at low
frequencies and to rS at high frequencies (Figures 9,a,c);
the dependence CV . series f verges to infinity at low frequencies and to the value C T at high
frequencies (Figure 9,b);
the dependence СV . parallel f is almost equal to the value C T at low frequencies and verges to
zero at high frequencies (Figure 9,d).
The frequency f1 , at which the capacity CV . series differs from the capacity C T by less than 0.5%, is:
1 12СT2 RP2 7.071 2.252 (20)
1.005 СT f1 .
12 CT R p2 CT RP CT RP
For the BB149 type varicap diode at СT 13 pF the frequency f1 equals 25.66 MHz, for the ВВ174
type varicap diode at СT 9 pF it is equal to 40.29 MHz and for the ВВ181 type varicap diode at
СT 8 pF it is 39.76 MHz.
The frequency f 2 , at which the active resistance RV . series differs from the resistance rS by less than
3%, is:
RP rS 22СT2 RP2 rS 2.887 0.919 (21)
1.03 rS f2 .
1 С R 2
2
2
T
2
P CT RP rS CT RP rS
For the ВВ149 type varicap diode at СT 13 pF the frequency f 2 equals 993.5 MHz, for the
ВВ174 type varicap diode at СT 9 pF it is equal to 1672.8 MHz and for the ВВ181 type varicap
diode at СT 8 pF it is 788.2 MHz.
The frequency f 3 , at which the capacity СV . parallel differs from the capacity СT by less than 0.5%,
is
СT RP2 1 0.0113 (22)
0.99 СT f3 .
C R r RP rS 2 199 CT rS
2
2 2 2 2 CT rS
3 T P S
For the ВВ149 type varicap diode at СT 13 pF the frequency f 3 equals 1159 MHz, for the ВВ174
type varicap diode at СT 9 pF it is equal to 2093 MHz and for the ВВ181 type varicap diode at
СT 8 pF it is 470.8 MHz.
The frequency f 4 , at which the active resistance RV . parallel differs from the resistance rS by less than
3%, is:
R r С R r 1.03 r (23)
2 2 2 2 2
1 0.919
f4
P S 4 T P S
.
24СT2 RP2 rS RP rS 2 0.03 CT rS
S
CT rS
For the varicap diode of ВВ149 type at СT 13 pF the frequency f 4 equals 94.256 GHz, for the
varicap diode of ВВ174 type at СT 9 pF it is equal to 170.185 GHz and for the varicap diode of
ВВ181 type at СT 8 pF it is 38.292 GHz.
If the varicap diode frequency range is limited by the range QV QV 0 , we obtain:
QV 0 (24)
f min ,
2 CT RP
1 (25)
f max .
2 QV 0CT rS
For the ВВ149 type varicap diode for QV 35 this frequency range fmin fmax corresponds to
63.51 466.62 MHz, for the ВВ174 type varicap diode for QV 35 it corresponds to
99.72 842.52 MHz and for the ВВ181 type varicap diode for QV 15 it corresponds to
42.17 442.32 MHz.
The Q-factors of 35 and 15 were chosen considering the frequency characteristics (Figure 6) about
the passband level, that is a 0.707 level from the maximum value of the varicap diode Q-factor.
The capacitance of varicap diodes of ВВ149, ВВ174 and ВВ181 types in any equivalent circuit
diverges from the varicap diode junction equivalent capacitance in these frequency ranges by no more
than 0.5%. Therefore, we can assume that СV . series СV . parallel CT . Eventually, for the ВВ149
type varicap diode in the 25.66 1159 MHz frequency range, for the ВВ174 type varicap diode in
the 40.29 2093 MHz frequency range and for the ВВ181 type varicap diode in the
39.76 470.8 MHz frequency range the next formulas can be written with an error less than 0.5%:
RP 1 2СT2 RP rS 1
(26)
ZV .series i i ,
1 2СT2 RP2 CT
RP 1 2СT2 rS2 (27)
ZV . parallel i ,
1 2СT2 RP rS iCT RP 1 2СT2 rS2
1 2СT2 RP2 (28)
QV .series i QV . parallel i ,
CT RP 1 2СT2 RP rS
that is, the varicap diode in the equivalent circuit can be superseded by a series or parallel connection
of a resistor with resistance being a function of frequency, and a capacitor with capacitance being
independent of frequency and determined by the equivalent varicap diode junction capacitance [17].
For both serial and parallel equivalent circuits of the varicap diode, its quality factor is determined
by formula (28).
If the noise is present in the measuring instrument channels, then a random component of uncertainty
arises; the authors propose to estimate it by the next formula [18]:
(29)
q q2 q1q2 sin
sN 1 ,
q1q2
P1 P
where q1 2
and q2 22 are the signal-to-noise ratio in the measuring instrument channels; si is the
s1 s2
root-mean-square values of noise in the measuring instrument channels; Pi is a power of signal; is a
phase shift proportional to the time delay of a signal in the channel.
The Allan variance method was proposed in [27] to be applied for identifying the noise structure in
the measuring instrument channels.
5. Conclusion
In the paper the authors showed that the varicap diode Q-factor is determined by parameters of the
small-signal equivalent circuit and is a function of frequency. We analyzed the effect of barrier
capacitance, junction resistance, and loss resistance of the varicap diode on its Q-factor in low and high
frequency ranges. We obtained formulas that provide determining the maximum value of the varicap
diode Q-factor and the frequency corresponding to its maximum Q-factor. We experimentally tested
theoretical results for three different varicap diodes of BB149, BB174 and BB181 types manufactured
by NXP Semiconductors. Deviations of the theoretical results from experimental testing in the (20
500) MHz frequency range were less than 8.6%. We established that the frequency range in which the
Q-factor QV 35 for varicap diodes of BB149 and BB174 types was 63.51 466.62 MHz and
99.72 842.52 MHz, and for the BB181 type varicap diode at QV 15 it was 42.17 442.32 MHz.
We obtained formulas that determine active and reactive parameters of the elements in the series
and parallel equivalent circuits of the varicap diode. We analyzed frequency dependences of the varicap
diode Q-quality factor and parameters of the series and parallel equivalent circuits. This promoted to
simplify the formulas for determining parameters of the elements in the series and parallel equivalent
circuits in the operating frequency range, where the varicap diode Q-factor is more than 0.707QV .max .
We proved that in the frequency ranges of 25.66 1159 MHz for the BB149 varicap diode,
40.29 2093 MHz for the BB174 varicap diode, and MHz for the 39.76 470.8 BB181 varicap
diode, equivalent capacitances of the series and parallel equivalent circuits differed from the junction
barrier capacitance by less than 0.5%. Therefore, in the specified frequency ranges, the varicap diode
in the equivalent circuit can be replaced by a series or parallel connection of a resistor, whose resistance
is a function of frequency and is determined by formulas (12) or (16), and a capacitor, whose
capacitance does not depend on frequency and is determined by the equivalent capacitance of the
varicap diode junction.
Theoretical assumptions and simulation results are confirmed by the experimental testing results.
The authors proposed a method for estimating the maximum frequency of the operating frequency range
of a varicap diode using SPICE Model Parameters, provided that the reactive capacitance of the varicap
diode is an order of magnitude greater than the inductive resistance of its leads.
The practical application of the study is that the authors had obtained formulas for determining the
frequency spectrum with the Q-factor higher than a given value.
Further development of the study is evaluating the influence of the varicap diode characteristic
nonlinearity in the of a large-signal mode with more than 2 V amplitude, various types of noise and
temperature instabilities on the frequency dependence of the varicap diode Q-factor.
6. Acknowledgements
This work was supported by the Ministry of Education and Science of Ukraine, grant
No. 0121U109722 “Methods and devices for forming and processing chaotic signals, access control
and positioning in robotic and infocommunication systems”.
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