Realization of new mutually coupled circuit using CC-CBTAs

Abstract

A novel circuit for the realization of the mutually coupled circuit using three current-controlled current backward transconductance
amplifiers (CC-CBTAs) as active components is proposed. The active mutually coupled circuit structures are also called the
synthetic transformers. The circuit is derived by using three floating simulated inductors that are connected as the T-type
transformer model. The circuit has the following attractive advantages: (i) The values of a primary self-inductance, a secondary
self-inductance and a mutual inductance can be independently tuned by the transconductance gain of the CC-CBTAs; (ii) The
circuit uses three grounded capacitors that are suitable from the point of integrated circuit implementation; (iii) It uses
only three active components; (iv) It has a good sensitivity performance with respect to the tracking errors; (v) Both positive
and negative couplings are achieved and the coupling coefficient is not limited by 1 in magnitude; (vi) Symmetrical coupling
is achieved without necessitating any matching condition; (vii) The proposed circuit has a floating structure.

Full text

Realization of new mutually coupled circuit using CC-CBTAs
M. Koksal, U.E. Ayten and M. Sagbas
Abstract: A novel circuit for the realization of the mutually coupled circuit using three
current-controlled current backward transconductance amplifiers (CC-CBTAs) as active
components is proposed. The active mutually coupled circuit structures are also called the
synthetic transformers. The circuit is derived by using three floating simulated inductors that
are connected as the T-type transformer model. The circuit has the following attractive
advantages: (i) The values of a primary self-inductance, a secondary self inductance and a
mutual inductance can be independently tuned by the transconductance gain of the CC-
CBTAs. (ii) The circuit uses three grounded capacitors that are suitable from the point of
integrated circuit implementation. (iii) It uses only three active components (iv) It has a good
sensitivity performance with respect to the tracking errors. (v) Both positive and negative
couplings are achieved and the coupling coefficient is not limited by 1 in magnitude. (vi)
Symmetrical coupling is achieved without necessitating any matching condition. (vii) Finally,
the proposed circuit has a floating structure.
Keywords: Mutually coupled circuits, Synthetic transformer, Current backward
transconductance amplifier, Active circuits, Band-pass filter, Electronic tunability.
1. Introduction
Mutually coupled circuits have a wide range of applications in instrumentation,
communications, control systems, signal processing, and measurement. They can also be used
for analog filters, particularly for replacing the magnetic transformer in stagger-tuned filters.
Therefore, a variety of operational amplifier (Op-amp) based RC-active networks have been
reported for the design of mutually coupled circuits [1-3]. However, new analog integrated
circuit applications have emerged and the performance requirements for analogue circuits
1

have changed. Conventional op-amp circuits have limited bandwidth at high closed-loop
gains due to the constant gain-bandwidth product. Furthermore, the limited slew-rate of the
operational amplifier affects the large-signal for high frequency operation. When wide
bandwidth, low power consumption, and low voltage operation are needed simultaneously,
the op-amps become very complex [4]. On the other hand, current conveyors or current
conveyor-based active elements play an important role in the analog integrated circuit design
because of providing advantage such as large bandwidth, high linearity, wide dynamic range,
simple circuitry, and low power consumption [5–6].
Therefore, several methods for the realization of mutually coupled circuits using active
elements have been published in the literature [1-3, 7-13]. A mutually coupled circuit
comprises a primary self-inductance Lp, a secondary self-inductance Ls, and a mutual
inductance M. However, there are few circuits which can be independently tune to adjust the
values of all these inductances simultaneously and independently.
Several studies are proposed in the literature for simulating a mutually coupled circuit using
operational amplifiers [1-3], bipolar junction transistors [7, 8], second-generation current
conveyors (CCII) [9-11], and the operational transconductance amplifier (OTA) [8,12].
However, current-conveyor and current-conveyor based active component realizations are
more attractive due to the already mentioned properties; namely, higher signal bandwidths,
greater linearity, wider dynamic range, simple circuitry, and low power consumption [4].
Further, many of the proposed synthetic transformers circuits require so many active
components and passive components and require several passive component matching
constraints [3, 9]. The circuit of [10] requires six second-generation current conveyors
(CCIIs), six resistors, and two capacitors. The circuit in [13] uses six dual-output second-
generation current controlled conveyors (CCCIIs). Another circuit is given in [11] employs
two plus-type CCCIIs, two minus-type CCCIIs, five resistors, and two capacitors. All of these
2

proposed structures realize only grounded synthetic transformer simulator. There have been
also several studies on the floating simulated inductance circuits [14-26] in the literature so
far. A synthetic floating transformer is realized by using three floating simulated inductors
that are connected as the T-type transformer model. However, such synthetic transformers
employ a large number of active and passive components.
In this paper, a synthetic transformer of this type is proposed by using three inductor
simulator circuits as in [27]. The proposed synthetic transformer is compared with the
synthetic transformer reported in the literature by the use of Table 1. Though the lower
number of active components in an active realization is not the only criterion to justify the
advantage of the circuit since the size, functionality and electronic properties of each active
component are also important, this table which also includes the floating property of the
synthetic inductance and the number of passive components used to give a rough idea about
the comparison of different simulations appeared in the literature.
The proposed mutually coupled circuit uses three CC-CBTAs, and three grounded capacitors.
The primary self-inductance, the secondary self-inductance, and the mutual inductance can be
independently controlled and can be tuned electronically. The validity of the proposed circuits
is demonstrated by PSPICE simulations.
TABLE 1: COMPARISON OF THE SYNTHETIC TRANSFORMER
Ref. # of Active Element FT Implementation # of C # of R
[10] 6 CCII No 2 6
[11] 2 CCCII-, 2 CCCII+ No 2 5
[13] 5 DO-CCCII No 2 -
[14] 6 DO-CCCII Yes 3 -
[15] 6 DO-CCII Yes 3 3
[16] 6 DVCC Yes 3 6
[17] 6 FTFN Yes 3 12
[18] 3 DO-CCCII, 3 Op-Amp Yes - -
[19] 6 DO-CCII Yes 3 6
[20] 3 OTA, 3 DO-CCII Yes 3 3
[21] 3 MCFOA Yes 3 6
Proposed 3 CC-CBTA Yes 3 -
3

2. Current Controlled Current Backward Transconductance Amplifier
The CC-CBTA is obtained by modifying the CBTA, an active component that is introduced
in [27]. The circuit symbol of the CC-CBTA is shown in Fig. 1. Where p and n are input
terminals, and w, z are output terminals. This active element is equivalent to the circuit in Fig.
1(b), which involves dependent current and voltage sources. The terminal equations of the
CC-CBTA can be defined as
Iz=gm(s)(Vp-Vn), Vw=RwIw+µw(s)Vz , Ip=αp(s)Iw, In= -αn(s)Iw(1)
Figure 1. a) Block diagram of the CC-CBTA. b) Equivalent circuit of the CC-CBTA.
where, Rw is the parasitic resistance of the w terminal.
)(s
p
α
,
)(s
n
α
and
)(s
w
µ
are the
current and voltage gains, respectively.
)(sg
m
is the transconductance gain. They can be
expressed as;
)(
)1(
)(
p
pp
p
s
s
ω
εω
α
+−
=
,
)(
)1(
)(
n
nn
n
s
s
ω
εω
α
+−
=
,
)(
)1(
)(
gm
gmgm
om
s
gsg
ω
εω
+−
=
, and
)(
)1(
)(
µ
µµ
ω
εω
µ
+−
=
s
s
w
, with
1||
<<
p
ε
,
1||
<<
n
ε
,
1||
<<
gm
ε
, and
1||
<<
µ
ε
. Where, go is the
DC transconductance gain. Also,
p
ε
and
n
ε
denote the current tracking errors,
µ
ε
denotes
the voltage tracking error,
gm
ε
denote transconductance error and ωp, ωn, ωgm, ωµ denote
corner frequencies. Note that, in the ideal case, the voltage and current gains are
1)(
=
s
w
µ
and
1)()(
==
ss
np
αα
, respectively.
4

The CMOS implementation of the CC-CBTA that consists of current controlled current
conveyors [28] and transconductance section [29] is given in Fig.2. It is obtained by
connecting appropriate outputs of these sections. The dimensions of the MOS transistors used
in the CC-CBTA implementation are given in Table 2.
Figure 2. CMOS implementation of the CC-CBTA.
In Fig 2, the transconductance section is realized by using the transistor M20–M33 that is
formed by MOS coupled pair and current mirrors. Where, vin is the differential input voltage
(vin = vp − vn), io is the output current of the transconductance section and IB is the bias current.
We will assume that all MOS devices operate in the saturation region. Let us assume that M20
and M21 are perfectly matched and the current mirrors have unity current gain. io can be given
by
inBinmo
vKIvgi )2(
==
(2)
Where, the transconductance parameter
LWCK
ox
2/
µ
=
,
µ
is the mobility of the carrier, Cox
is the gate-oxide capacitance per unit area, W is the effective channel width, L is the effective
channel length.
In the circuit of Fig. 2, the current controlled current conveyor part is realized by the
transistors M1 through M19. The control of the parasitic resistance Rw by the current Io in this
section is achieved according to the following equation.
KI
R
o
w
16
1
=
(3)
5

Table 2. Dimension of the CMOS transistors.
PMOS Transistors W(µm)/L(µm)
M3-M4 50/0.5
M5-M11 15/0.5
M22-M27 2.5/0.25
M28-M29 10/0.25
NMOS Transistors W(µm)/L(µm)
M1, M225/0.5
M12-M19 5/0.5
M20, M21 1/0.5
M30, M31 2/0.25
M32, M33 10/0.25
3. The proposed mutually coupled circuit
The transformer can be modeled by a two-port network with the following matrix equation:












=






2
1
21
12
2
1
I
I
LM
ML
s
V
V
s
p
or











+
+
=






2
1
2221
12111
2
1
I
I
MLM
MML
s
V
V
s
(4)
where, Lp and Ls are the primary and secondary self-inductances of the transformer,
respectively. M12 and M21 are the mutual inductances of the transformer.
The proposed simulated transformer circuit is shown in Fig. 3. Routine analysis, shown that if
the high frequency non-ideal gain effects are ignored, the voltages V1 and V2 can be computed
as in Eq. (4) where,
11
11
13
3
1
1
m
w
pn
p
g
CR
L
µαα
α
=
, (5a)
22
22
23
3
2
1
m
w
pn
p
g
CR
L
µαα
α
=
,
(5b)
33
33
32
2
2212
1
m
w
np
n
g
CR
MM
µαα
α
==
,
(5c)
6

33
33
31
1
1121
1
m
w
np
n
g
CR
MM
µαα
α
==
.
(5d)
Here, αpj, αnj and µwj ( j=1, 2, 3 ) are the corresponding current and voltage non-ideal low
frequency gains, respectively, of the jth CC-CBTA.
Fig. 3. Proposed synthetic floating transformer circuit.
Fig. 4. a) The symbol of the mutually coupled circuit, b) its equivalent circuit.
It is well known that a magnetic transformer shown in Fig. 4a has an equivalent T-model
circuit shown in Fig. 4b. In order to ensure a symmetrical coupling as in this circuit, the
simulated transformer in Fig. 3 should have M12=M21=M . If Eqs. (5c) and (5d) are examined,
it is true that this equivalence is easily achieved if
β
α
α
α
α
==
2
2
1
1
p
n
p
n
(6)
which is practically possible due to the same active element used in the design (CC-CBTA1
7

and CC-CBTA2). There are not any other matching conditions for symmetrical coupling. With
this condition the mutual inductance becomes
33
33
3
2112
1
m
w
n
g
CR
MM
µα
β
==
, (7a)
and the primary and secondary inductances become
33
33
311
11
13
3
11
m
w
nm
w
pn
p
p
g
CR
g
CR
L
µα
β
µαα
α
+=
,
(7b)
33
33
322
22
23
3
11
m
w
nm
w
pn
p
s
g
CR
g
CR
L
µα
β
µαα
α
+=
, (7c)
Under the ideal conditions, Eqs. (5) reduces to
1
11
1
m
w
g
CR
L
=
,
2
22
2
m
w
g
CR
L
=
,
3
33
m
w
g
CR
M
=
, (8a)
and
3
33
1
11
m
w
m
w
p
g
CR
g
CR
L
+=
,
3
33
2
22
m
w
m
w
sg
CR
g
CR
L+=
.
(8b)
Another important concept in mutual coupling is the coupling coefficient defined by
sp
LLMk /
=
which is equal to 1 in perfect coupling. The coupling coefficient for the
proposed transformer circuit
))((
/
3
31
2
22
3
31
1
11
333
m
w
m
w
m
w
m
w
mw
g
RC
g
RC
g
RC
g
RC
gRC
k
++
=
(9)
If the circuit parameter are chosen as gm1=gm2=gm3=gm and Rw1=Rw2=Rw3=R,
3 1 3 2 3
/ ( )( )k C C C C C= + +
. Obviously if C1 and C2 are chosen very small with respect to C3
8

then k ≈ 1.
This is a very restrictive condition because it yields Lp=Ls=M and the same situation can be
achieved by removing CC-CBTA1 and CC-CBTA2 (C1=C2=0 means the associated terminals
are open circuited). The unity coupled condition cannot be obtained by increasing one of ( Lp,
Ls) while decreasing the other, instead of both of them should coincide and become equal to
M.
A close investigation of the CMOS implementation shown in Fig. 2 implies that if the
connections of the gates of the transistors M20 and M21 are changed to p and n terminals,
respectively, then this results the change of the sign of Iz in Eq. (1) and the direction of the
dependent current source
)(
npm
VVg
−
in Fig. 1 (b). This change is also equivalent to change
the sign of gm ; hence all the results obtained for the original CC-CBTA are valid for the
modified CC-CBTA which may be called CC-CBTA+ and CC-CBTA-, respectively.
Therefore, by using these two types of components, positive and negative inductances L1, L2,
M11, M22 can be simulated [24]. Hence, it is possible to change the signs of L1, L2, and M
independently to obtain a mutual inductance with the terminal equations











±±± ±±±
=






2
1
2
1
2
1
)()(
)()(
I
I
MLM
MML
s
V
V
(10)
where, the same sign (either + or -) should be considered for all M’s. And ± signs in front of
L1 and L2 can be chosen independent of each other and that of M as well.
With the above results, it is possible to obtain positive and negative inductances
Lp = ± L1 ± ( M ) and Ls = ± L2 ± ( M ). Hence the coupling coefficients becomes
1 2
( )( )
M
kL M L M
±
=± ± ± ±
(11)
9

Hence, it is possible to achieve positive and negative coupling coefficients unlimited by the
absolute value 1, and k can be increases as large as possible but bounded by the active
component parameter limitations.
Eqs. (3) implies that normalized active and passive sensitivities of all inductance parameters,
namely; L1, L2, M11, M12, M21 and M22 are all equal to ± 1.
4. Simulation Results
To confirm the given theoretical analysis, the proposed circuit in Fig. 3 has been
simulated with the PSPICE program. The CC-CBTAs are simulated using the schematic
implementation shown in Fig. 2, with DC power supply voltages equal to VDD=-VSS=1.5 V.
The simulations are performed using PSPICE based on 0.25 μm level-7 TSMC CMOS
technology parameters.
The proposed mutually coupled circuit of Fig. 3 was used to simulate the circuit shown in
Fig. 5 with Cp=Cs=10 pF, and Rp=Rs=10 kΩ. The mutually coupled circuit of Fig. 3 is built
with C1=C2=20 pF, C3=10 pF, Rw1=Rw2=Rw3=0.5 kΩ, gm1= gm2= gm3=0.5 mS, and to obtain
Lp=Ls=30 μH and M11=M12=M21=M22=10 μH, resulting in fo=9.2 MHz, and the quality factor
Q=5.77. By using these parameters, k can be calculated as 0.33. The value of the parasitic
resistance Rw is achieved by choosing Io=60 μA. Similarly, gm value of the CC-CBTA is found
by taking the bias current IB=50 μA.
10

Figure 5. Band-pass filter example to test the presented mutually coupled circuit.
The magnitude and the phase characteristics of the filter are shown in Figs. 6a and 6b,
respectively. It appears from Figs. 6a and 6b that the theoretical and simulated results are in
good agreement. Where the numerical data yielding these figures are investigated, it is seen
that the gain error does not exceed 5% for each frequency. The error is about 2 % at most of
the frequencies, but the order of the error is higher at the peak points in the pass band. For
example, the highest error occurs at 11 MHz where the gain is supposed to be 0.5, but the
simulation yields 0.6. The gain is measured to be 0.411 instead of the ideal value 0.393 at
9.54 MHz which is the minimum gain or negative peak point. At the frequency 109.8 kHz
where the maximum gain error occurs, the gain is 0.525 V for the simulated circuit whilst the
ideal one is 0.5. The gains at the first maximum are 0.5 and 0.496 V for the ideal and
simulated cases, respectively. Considering the minimum gains in the passband, their values
are recorded to be 398 mV for the ideal case and 400 mV for the circuit shown in Fig. 6. The
same coherence between the results of the circuit shown in Fig. 6 and the ideal case appears in
Fig. 6b as well. The phase is recorded to be 74o at 3 MHz for the simulated circuit while it is
82o in the ideal case; these figures become -95o and -96o, respectively when the frequency is
10 MHz; and they are -277o and -264o at 30 MHz. It is seen that the difference between the
simulation results and the ideal results outside of the pass band is much more than that in the
pass band. These discrepancies follow from the nonidealities of the component CC-CBTA.
11

Figure 6. (a) Gain characteristics, (b) Phase characteristics of the theoretical and simulated
band-pass filter in Fig. 5.
The results for the coupling coefficients greater than one in magnitude are also obtained
when k=-2 (M=-40 μH, L1=L2=20 μH) and k=2 (M=40 μH, L1=L2= -20 μH). The component
values of the mutually coupled circuit is changed while keeping the same component values
for Rp, Cp, Rs and Cs in Fig. 5. To obtain a negative inductance, the gate terminal of the
transistor M21 in the CMOS implementation given in Fig. 2 is connected to the n-output; in a
similar manner, the gate of the transistor M20 is connected to the p-output [24]. The results
obtained by using the simulated circuit in Fig. 4 and the ideal case are presented in Fig. 7.
12

Figure 7. Gain characteristics of the theoretical and simulated band-pass filter in Fig. 5,
a) k=-2, b) k=2
A sinusoidal signal having 1 V amplitude and 9 MHz frequency is applied at the input of the
filter in order to investigate its time domain characteristics. The results are shown in Fig. 8. It
is observed that the ideal results and the simulation results are in good agreement. The total
harmonic distortion (THD) in the time domain response obtained by simulation for the
nonideal case is measured to be smaller than 3 %. A non-zero THD is expected owing to the
nonlinearity of the resistor (Rw) realized using active devices in addition to the other
nonlinearities of the active devices themselves. The total power consumption of the circuit is
measured to be 6 mW.
Figure 8. The input and output waveforms of the proposed band-pass filter
output of Fig. 5 for 9 MHz sinusoidal input voltage of 2 V (peak to peak)
13

Due to the non-idealities of the CC-CBTA, some discrepancies exhibit between
theoretical and simulation results as shown in Figs. 6, 7, and 8. In order to find operating point
and non-idealities of CC-CBTA, the PSPICE simulations are also done by using above
transistor model. Therefore, corner frequencies are ωαp=2765, ωαn=3075, ωgm=3770 and
ωµ=5650 Mrad/s and errors of these gains are εαp=-0.08, εαn=-0.03, εgm=-0.044 and εµ=0.024.
For low-frequency application αp, αn, gm and µw can be assumed to be the constants with
values 1-εαp=1.08, 1-εαn=1.03, 1-εgm=1.044 and 1-εµ=0.976, respectively. As a result, the
maximum operating frequency of the CBTA can be found as follows fmax=min{fαp, fαn, fgm,
fμ}≈440 MHz. The CC-CBTA has parasitic resistances and capacitances as shown in Fig. 9.
The parasitic resistances and capacitances values of the CC-CBTA are given in Table 3.
Figure 9. Parasitic resistance and capacitance of the CBTA.
TABLE 3. PARASITIC IMPEDANCES OF THE CC-CBTA
Parasitic Impedances Values
Rp90 kΩ
Rn98 kΩ
Rz370 kΩ
Cp28 fF
Cn30 fF
Cz105 fF
5. Conclusion
In this work, a new mutually coupled circuit is presented. The proposed mutually coupled
circuit uses only three CC-CBTA, and three grounded capacitors which is more suitable for
14

IC fabrication. The primary self-inductance, the secondary self-inductance, and the mutual
inductance can be independently controlled and can be tuned electronically by changing the
biasing current of the CC-CBTA. It has a good sensitivity performance with respect to
tracking errors. The workability of the proposed circuits is demonstrated by PSPICE
simulations and experimental results. In addition to the simulation of the real transformer with
the positive and negative coupling, the proposed circuit realizes the synthetic transformers
with coupling coefficients greater than 1 in magnitude. Further, the symmetry condition is
hardly spoiled due to the nonidealities since the ratios αn1/αp1 and αn2/αp2 are almost equal for
the identical active elements CC-CBTA1 and CC-CBTA2 are used in the design. The effects of
the nonidealities of the individual active elements are in cancelling forms due to the ratios
αpi/αni or αni/αp, i= 1,2,3 not only in the mutual inductance but in and as well.
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Author Affiliations
Muhammet Koksal is with the Department of Electronics and Communications Engineering, Halic University,
Büyükdere Cad. No:101, Mecidiyeköy, 34394, Istanbul, Turkey. (e-mail: muhammetkoksal@yildiz.edu.tr).
Umut Engin Ayten is with the Department of Electronics and Communications Engineering, Yildiz Technical
University, Yıldız 34349, Istanbul, Turkey. (e-mail: ayten@yildiz.edu.tr).
Mehmet Sagbas is with the Department of Engineering, Maltepe University, Maltepe 34857, Istanbul, Turkey.
(corresponding author to provide phone: +90(216)6261050/1409;e-mail: sagbas@maltepe.edu.tr).
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