Download Quadrature oscillator using CDTA-based integrators

Quadrature oscillator using CDTA-based integrators
DALIBOR BIOLEK, ALI ÜMIT KESKIN*), VIERA BIOLKOVA
Depts. of Microelectronics and Radioelectronics, Brno University of Technology
Dept. of EE, University of Defence, Brno, Czech Republic
*) Department of Biomedical Engineering, Yeditepe University, Istanbul, Turkey
[email protected] http://user.unob.cz/biolek
Abstract: A current mode quadrature oscillator consisting of two CDTA (Current Differencing Transconductance
Amplifier) –based integrators is introduced. The oscillation frequency can be made adjustable by internal
transconductances of active elements without affecting the oscillation condition. Three methods of stabilizing the
oscillation amplitude are proposed. PSpice simulations are preformed, which verify the circuit performance.
Keywords: - Quadrature oscillator, CDTA, current mode
1 Introduction
Quadrature oscillators (QOs) employing various types
of active elements are reported in the literature. Current
conveyors and other active components, enabling highspeed current-mode or mixed-mode operation, have
been increasingly used. A QO in [1] consists of two
operational transresistance amplifiers (OTRAs), but six
virtually grounded passive R and C components are
required. Horng [2] reported a QO with two current
differencing buffered amplifiers (CDBAs), two
grounded capacitors, and four virtually grounded
resistors. A more economical QO in [3] uses three
current-controlled current conveyors CCCII and only
one grounded and one virtually grounded capacitor.
However, it generates relatively low-amplitude and
unequal sinusoidal output signals. Salama and Soliman
[4] proposed a pair of lossy and lossless integrators in a
feedback loop, implemented by two CDBAs and six
virtually grounded CMOS transistors, with the
possibility of digital control of oscillation frequency.
The OTA-C tunable QO in [5] contains four operational
transconductance amplifiers (OTAs), two grounded
capacitors, and special circuits for current-mode
amplitude control. The QO in [6] consists of two
allpass sections, each employing one current
differencing transconductance amplifier (CDTA) and a
virtually grounded pair of R and C components. The
frequency of generated waveforms is insensitive to
transconductances gm of CDTAs and is given by
passive R and C components.
Most of the reported oscillators provide noninteracting controls for the frequency of oscillation
(FO) and condition of oscillation (CO). There are
several methods of amplitude control, from simple
nonlinear limiters [7] to sophisticated control loops
which can represent more complicated circuitry on the
chip than the primary oscillator [5].
In this paper, a novel QO is proposed. Similarly to
[6], it employs CDTA active elements. However, the
circuitry consists of two integrators in a feedback loop.
That is why the QO is compounded of only two CDTAs
and two grounded capacitors, plus auxiliary circuitry
for automatic gain control.
2 CDTA element
The CDTA elements [8] in Fig. 1 (b) are exploited for
integration in a simple way: the difference of currents,
flowing into low-impedance p and n terminals, flows
out of the z-terminal into a load Z, causing a voltage Vz
at the z terminal. Internal transconductance gm,
controllable by an outside current source Ig, converts
this voltage into a couple of bidirectional output
currents Iout. Therefore, this active element can be
characterized by the following equations:
Vp = Vn = 0 , Iz = Ip - In , Ix+ = gmVz , Ix-= - gmVz. (1)
CDTA can be thought of as a combination of a
current differencing unit followed by a dual-output
operational transconductance amplifier, DO-OTA.
Ideally, the OTA is assumed to be an ideal voltagecontrolled current source and can be described by the
formula Ix = gmVin, where Ix is the output current, and
Vin denotes the input voltage of the OTA. Note that gm
is a function of the bias current. With dual output
available, the condition Ix+ = −Ix− is assumed.
A possible CMOS-based CDTA circuit realization
suitable for the fabrication of a monolithic IC is
displayed in Fig. 2 [6, 9]. In this circuit, transistors
from M1 through M12 perform the current differencing
operation while transistors from M13 through M24
convert the voltage at the z terminal to output currents
at the two outputs of the DO-OTA section. DO-OTA’s
transconductance (gm) is controllable via its bias current
IB3. Also, an impedance Z connected at the z terminal
can be used to adjust the gain of CDTA, while the
voltage at the input of DO-OTA is Vz = Iz. Z.
The gate terminals of output transistors M11 and
M12 in the current-differencing section are connected
to bias-voltages to provide drain output and high
impedance at the z terminal (ideal current-controlled
current source, CCCS).
The multiple-output (MO) CDTA can be easily
implemented by copying the structure of transistor
couples M16-M21 and M20-M24. Then one can use
more I+ and I- outputs, which is advantageous in many
applications. Examples of such MO-CDTAs are given
in [10, 11]. Note that total parasitic capacitance on the
chip increases, causing lowering the bandwidth.
Bipolar CDTA structures are given in [12-14].
Ig
Ip
In
x+
p
x-
z
Vz
Table 1.Transistor W/L ratios used in CDTA model.
W / L ( µm )
8 /1
5/1
20 / 2
16 / 1
6/1
4/1
Transistor
M1 - M6
M7 - M10
M11 - M12
M13 - M14
M15 - M20
M21 - M24
3 QO basic configuration
The proposed economical oscillator is shown in Fig. 3.
It consists of one inverting and one noninverting
lossless current integrator in a feedback loop. Ideally,
such configuration represents a 2nd order system with
infinite quality factor, enabling harmonic oscillation.
g mVz
CDTA
n
All the PSpice simulations in this paper were
performed using a PSpice subcircuit based on the
structure in Fig. 2. Here, 0.5 µ MIETEC real transistor
model parameters are implemented for all transistors.
Transistor aspect ratios are indicated in Table 1.
I p − In
Z
Fig. 1: CDTA symbol indicating a possibility of gm
control and loading the z terminal.
x+
CDTA 1
p
z x-
C1
x+
CDTA 2
n
z xp
n
I out 1
I out 2
C2
Fig. 3: Proposed QO circuit employing CDTA
elements, without a gain control circuitry.
Fig. 2: CMOS-based CDTA: IB1=IB2=85 µA, IB3=(20-700)µA for controlling gm within (200-635) µS,
bandwidth=400MHz, V DD=-VSS=2.5V.
Note that the structure in Fig. 3 represents a special
case of two-CDTA KHN (Kerwin– Huelsman – Newcomb) filter, previously reported in [9,13], without
negative feedback, control-ling the quality factor.
CDTA elements in Fig. 3 seem to be employed
uneconomically because the p-terminal of CDTA1 and
the n-terminal of CDTA2 are not utilized. From this
point of view, such 2 nd-order system could be employed
by two OTAs. There are two reasons why to use
CDTAs:
1 Current outputs of internal OTAs operate into lowimpedance input terminals of CDTAs, which
increases the bandwidth of the operation,
2 Input terminals will be used by a circuitry for
amplitude stabilization as well as for prospective
external compensation of current offsets of active
elements.
In the first step, let us analyze the ideal linearized
model of QO in Fig. 3, considering frequency independent internal transconductances gm1 and gm2 as the
only parameters of CDTAs No. 1 and 2. Let us
disconnect the feedback path from the output of
CDTA2 to the n-terminal of CDTA1 and analyze the
open loop gain AOL. The current gains of CDTA
integrators are given by the ratios of transconductances
and capacitor susceptances. That is why AOL becomes
AOL = −
g m1 g m 2
.
sC1 sC 2
(2)
Setting AOL=1 shows the characteristic equation of
the harmonic oscillator with zero damping factor:
s +ω = 0,
2
2
0
(3)
where
ω0 =
g m1 g m 2
.
C1C 2
(4)
Because the harmonic signal Iout2 represents the
integral of signal Iout1, there exists a 90° phase shift
between them. For identical time constants of both
integrators, i.e. for
g m1 g m 2
(5)
=
= ω0 ,
C1
C2
the ideal model in Fig. 3 will generate two harmonic
waveforms in a quadrature, of equal magnitudes. For
practical design, we choose identical integrator
sections, i.e.
(6)
g m1 = g m 2 = g m , C1 = C 2 = C .
Then
ω0 =
gm
.
C
(7)
For
control
current
IB3=600 µA,
the
transconductance gm is 625µS. For C1=C2=10pF, the
theoretical oscillation frequency (7) is 9.947 MHz.
Note that this frequency is valid for the small-signal
linear model, i.e. for the start-up process. In the steadystate, when the amplitude is large, the transconductance
is decreased due to the nonlinearity of the shape Ix =
f(Vz), causing harmonic distortion as well as decreasing
the oscillation frequency according to Eq. (7). The
start-up process and steady-state operation will be
analyzed in the following Chapter with the above
values of IB3, gm and C.
4 QO without amplitude stabilization
For the oscillator without the circuitry for stabilizing
the amplitude, the start-up behavior will depend on two
real parameters of the CDTA: internal resistance Rz of
the z-terminal, and the frequency dependence of gm,
which is determined by the cutoff frequency ωm0. Serial
resistances of p- and n- input terminals have no effect
due to their current excitations from the CDTA x+
terminals.
The steady-state behavior will be determined by the
nonlinear characteristics of CDTAs, namely by the
trade-off between Ix and Vz.
For the CDTA structure in Fig. 2 and IB3=600 µA,
PSpice simulation leads to the following small-signal
parameters:
gm = 625µS, Rz = 347kΩ,
(8)
fm0 = 787MHz ⇒ ωm0 = 6.945GigaRads/s.
Considering identical CDTAs in both integrators
with internal resistances Rz and transconductances gm,
the current gain AI of one integrator becomes
AI =
gm
,
sC + G z
(9)
where Gz = 1/Rz. As shown in [5], in order to analyze
the oscillator stability, the one-pole model of OTA with
the low-frequency value gm0 of the transconductance
and with the cutoff frequency ωm0 can be approximated
by the formula

s
g m ≈ g m 0 1 −
 ω m0

 .

(10)
By substitution from Eqs. (6), (9) and (10), the open
loop gain (2) becomes
2
 g (1 − s / ω m 0 ) 
AOL = −  m 0
 .
sC + G z


(11)
200u
100u
After a small arrangement, the equality AOL=1 leads
to the following characteristic equation:
s 2 + 2d x s + ω 02x = 0 ,
(12)
with the following damping factor dx and oscillation
frequency ω0x:
CG z −
dx =
C +
2
ω0x =
g m2 0 + G zě
g2
C 2 + m2 0
ω m0
= ω0
g m2 0
ω m0 ,
g m2 0
FO:
0.6u
1.2u
1.8u
1 + [ g m 0 /(ω m 0 C )] 2
. (14)
5 QO with amplitude stabilization
In order to adjust and stabilize the amplitudes of
generated waveforms and decrease the harmonic
distortion, as well as to move the frequency towards the
designed value (7), three types of methods for
stabilizing the amplitude have been proposed.
1 Connecting a nonlinear resistor RN (see Fig. 5 a) as
an amplitude limiter in parallel to C1.
(15)
IN
IN
2
3.0u
Fig. 4: PSpice simulation of starting up the oscillator
without circuitry for stabilizing the amplitude.
1 + (G z / g m 0 ) 2
1 + (G z / g m 0 ) 2
2.4u
T (Secs)
(13)
ω m2 0
1 + (ω 0 / ω m 0 )
-200u
0.0u
ω0
G
= z ,
ω m0 g m0
ω0x = ω0
-100u
ix1 (A) Ix2 (A)
Let us assume that the oscillator operates in the
frequency range below the cutoff frequency ωm0. Then
the original oscillation frequency (7) is given by the
ratio of gm0 to C. The condition of oscillation dx = 0 and
the frequency of oscillation can be rewritten as follows:
CO:
0u
≈ ω0 .
G
(16)
The squared terms in Eq. (16) are much less than 1.
That is why the oscillation frequency is practically not
affected by the finite resistance of z-terminal and the
finite cutoff frequency of internal OTA. Note that
equality (15) means a steady amplitude of oscillation
within the linear regime, whereas the smaller left side
term than the right-side term results in a growing
amplitude.
Numerical values (8) and an oscillation frequency of
9.947MHz give
ω0
G
ω
=& 0.0126 , z =& 0.00461 , 0 x =& 0.99993 .
ω0
ω m0
g m0
The first term is smaller than the second one, which
indicates soft-start oscillations. The amplitude is
growing, effective gm is decreasing due to the nonlinear
curve Ix versus Vz, and the second term is continuously
decreasing. In the steady-state, Eq. (15) is dynamically
fulfilled. The results of PSpice simulation in Fig. 4
confirm the above. In the steady state, the oscillating
frequency is about 6.5MHz with large THD=23%.
C1
-E
VN
RN
(a)
G
E
VN
(b)
Fig. 5: (a) Nonlinear resistor as an amplitude limiter,
(b) its I/V characteristic [7].
According to the I/V characteristic in Fig. 5 (b), if
the amplitude of a voltage across C1 does not exceed
the value E, or alternatively, if the output current of the
CDTA does not exceed the value of Egm, this resistance
does not have any effect. The peaks above this value
will cause an additional current IN which will cause
conductance losses. Two examples of circuit
implementation are proposed in [7].
Computer simulations indicated some imperfections
of this method of controlling the amplitude for the
oscillator in Fig. 3, consisting in an unacceptable
nonlinear distortion of output waveforms. For the
current-mode oscillator, it is more advantageous to use
current limiters.
2 Using a current-to-current feedback limiter
The THD of generated waveforms is 3.48% for Ix1
and 1.64% for Ix2. The repeating frequency is 9.59MHz.
Its decrease below the theoretical value 9.947MHz is
due to the decrease of the effective value of
transconductance.
IFB
x+
CDTA 1
p
z x-
n
b
-IE
IX
IE
b
IX
(b)
C1
IX
IFB
(a)
R
IFB
(c)
Fig. 6: (a) Current-to-current limiter in the feedback
loop, (b) its I/I characteristic, c) an example of simple
implementation.
Fig. 6 (a) demonstrates a more effective method for
amplitude stabilization. In contrast to the above
method, this nonlinear negative feedback does not
create a DC path from the z-terminal to the ground,
which is the main source of nonlinear distortion in the
case of the method from Fig. 5 (a). A simple
implementation of current-to-current limiter is in Fig. 6
(c). For a relatively low current Ix, when the voltage
drop on the resistance R is below the threshold voltage
VTH≈0.6V, the diodes are in the OFF state and IFB = 0.
If the absolute value of Ix is increased above the value
of IE=VTH/R, the diode is switched to the ON state and
IFB is increased or decreased according to the slope
b=gDR/(1+gDR). Here gD is the small-signal
conductance of the diode. A more sophisticated current
limiter can be implemented by means of a current
comparator [15].
The PSpice simulation in Fig. 7 has been
accomplished for limiter parameters IE = 50µA and b =
1.
80u
As shown in Fig. 7, the waveforms generated exhibit
a current offset which is caused by the offset and other
DC imperfections of CDTA structure. These
imperfections can be compensated by combining the
auxiliary DC current sources Icp, Icn, I1+, and I1- in Fig.
8. It is obvious that the sources I1+ and I1- compensate
the individual offsets of oscillator output signals
without affecting the internal state of oscillator. As
shown in [13], the transfer functions from Icp to Iout1 and
from Icn to Iout2 are of bandpass character whereas the
transfer functions from Icp to Iout2 and from Icn to Iout1
represent lowpass filters with unity DC gains. That is
why DC current Icp has no effect on the output current
terminals of CDTA1 but it is reflected as an additional
offset current at the output terminals of CDTA2 (as a
positive offset value at x+ and a negative value at x-).
Similarly, Icn can independently control the offset at the
output terminals of CDTA1.
x+
CDTA 1
p
z x-
Icp
x+
CDTA 2
n
z xp
n
I out 1
I1-
Icn
C1
I out 2
I2-
C2
I out 1
Fig. 8: The principle of independent offset control via
auxiliary current sources.
60u
40u
3 Using a linear inertial feedback control
20u
0u
An auxiliary circuitry, represented in Fig. 9 by a
controlled current source I FB = ßIX, stabilizes the
amplitude of generating waveform and dynamically
fulfills the CO by local feedback from the output to
positive input of CDTA1.
-20u
-40u
-60u
-80u
0n
ix1 (A)
120n
ix2 (A)
240n
360n
480n
600n
T (Secs)
Fig. 7: PSpice simulation of steady-state oscillation
with amplitude stabilization by means of current-tocurrent limiter.
For ideal model of the oscillator, the condition of
oscillation is independent on the frequency of
oscillation (4):
CO:
β = 0.
(17)
x+
CDTA 1
p
z x-
n
β IX
IX
IX
β
C1
I amplitude
amplitude
p
x+
I adjust
z
p
x+
A
x-
IFB
IX
B
n
x-
z
n
Fig. 9: Linear current-controlled current source in the
feedback loop for inertia-type amplitude control; β
depends on the amplitude of generated waveforms.
Fig. 10: Proposed circuit for automatic gain control.
The current gain ß is controlled according to the
formula
(18)
β = I amplitude (t ) / I adjust − 1 ,
where Iamplitude is instantaneous amplitude of generated
waveforms, sensed by rectifying, Four-Phase MAX [5]
or other method, and Iadjust is adjustable value of the
required amplitude. In the initial state, the value of ß =
-1 provides flat soft-start of the oscillation. In steadystate operation, amplitude is dynamically perturbed
around the nominal value Iadjust, causing changes of ß
around zero which subsequently stabilize the
amplitude.
For well-matched control laws gm(Ig) of both the
CDTAs, ß = 0 when Iamplitude = Iadjust. For the simulation,
Iadjust has been set to 50µA. Amplitude sensing was
simulated by a block which performed squared root of
the sum of powers of quadrature output currents,
followed by a simple RC filter. PSpice simulation led
to the similar results as in Fig. 7. The THD of
generated waveforms was about 2.5%. It can be
decreased by eliminating the offset of output
waveforms, because it influences the algorithm of
amplitude sensing. Finally, the THD has been
minimized below 1%. It can be further decreased by
improving the loop gain stabilization circuitry.
For the nonideal case, the simplified CO and FO are
as follows:
4 Conclusions
CO:
β=
2ω 0
ω m0 −
FO:
ω0x = ω0
Gz
C
≈ 0,
1 − ( 2 − β )G z / g m 0
≈ ω0 .
1 + (2 − β )ω 0 / ω m 0
(19)
(20)
For concrete values of Gz, gm0 and ω0, we get
practically values, corresponding to the ideal case.
The proposed circuit solution of linear currentcontrolled current source from Fig. 9 is shown in Fig.
10. It consists of two CDTAs. The element “B” utilizes
internal OTA for simulating the gmB conductance at the
z-terminal. The CDTA “A” is connected as current
amplifier which current gain is given by a ratio of
gmA/gmB. The current source connected at the x+ output
of CDTA “A” simulates subtraction of this current
from the output current of the current amplifier. This
source can be implemented by a current mirror or by
simple applying the x-terminal of multiple-output
CDTA [11]. The current gain ß is then given by the
relation
β = g mA / g mB − 1 .
(21)
The quadrature oscillator circuit proposed has the
following features: (i) use of two CDTAs for providing
direct current-to-current integration; (ii) use of only
two grounded capacitors; (iii) dynamical fulfilling the
CO by simple auxiliary circuitry; (iv) electronic gmcontrol of FO, which does not affect the CO. All these
features have not been achieved simultaneously in any
of the previous QOs published in [1 – 7] including
those cited therein. As follows from (i) and (ii), the new
QO also represents an economical circuit solution
because its core consists of only two active elements
and two grounded capacitors. One can choose a
circuitry for stabilizing the amplitude from simple
nonlinear limiter to complicated systems of loop gain
control.
Acknowledgments
This work has been supported by the Grant Agency of
the Czech Republic under grants No. 102/04/0442 and
102/05/0277, and by the research programmes of Brno
University of Technology and University of Defence
Brno.
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