Download A Current-Mode Square-Rooting Circuit Using Negative Feedback Technique

A Current-Mode Square-Rooting Circuit Using
Negative Feedback Technique
Ittipong Chaisayun
South-East Asia University
Bangkok 10160,THAILAND, Email:[email protected]
Abstact-In this paper, a square-rooting circuit operating
in current mode is proposed. The negative feedback technique is
used to realize the proposed circuit. This circuit uses only
fourteen CMOS transistors and requests no current source, and
moreover it can operate at low voltage supply. Simulation results
are carried out by using PSpice. Experimental results by using
CD 4007 transistor arrays are shown to confirm the operation.
II. PRINCIPLE OF OPERATION
A. Principle of The Square-Rooting Circuit
The block diagram in Fig. 1 consists of a squaring
circuit, a current-differencing amplifier (CDA) and an NMOS
transistor.
I. INTRODUCTION
A square-rooting circuit is a useful building block
used in neural networks applications, measurement and
instrumentation. For example it can be used to compute the
Euclidean distance between two vectors[1], or to calculate the
r.m.s. value of an arbitrary waveform[2]. In the past, squarerooting circuit was proposed by using operational
amplifiers(op-amp) and bipolar junction transistors[3]. This
approach provides the logarithmic principle to realize a squarerooting function. However the frequency performance is
limited by the bandwidth of an op-amp. Then, the squarerooting circuit[4] has been proposed by bipolar junction
transistors(BJT) in translinear configuration which is suitable
for implementing in monolithic integrated circuit form. After
that the voltage-mode square-rooting circuits[5-6] and the
current-mode square-rooting circuits[7-8] based on CMOS
transistors have been reported. In recent years, current-mode
analogue signal processing circuit techniques have received
wide attention due to the high accuracy, the wide signal
bandwidth and the simplicity of implementing signal
operations such as addition, subtraction and multiplication[4].
However, the current-mode square-rooting circuit is less
involved, so this paper proposes a current-mode square-rooting
circuit. The current-mode square-rooting circuit[7] is realized
by using the back gate of CMOS biased in weak inversion.
Even it uses low voltage supply and only nine transistors, but it
has more error about 20 . The current-mode square-rooting
circuit[8] is realized by a class AB configuration. Although it
achieves a wide dynamic rang and a wide-band capability, but
it uses more components (sixteen transistors and three current
sources). The novel higher precision current-mode square-root
circuit[9] has been reported recently. Although it has high
precision, but it uses more devices (twenty-six transistors). In
this paper, the current-mode square-rooting circuit is realized
by using negative feedback on current squaring circuit. It has
low error and uses only fourteen transistors and no current
source.
IA
A current-differencing amplifier
NMOS
Iin
+
Vout
A current squaring circuit
(IA)2
X2
Fig. 1. Block diagram of the proposed circuit
The operation of the block diagram can be explained as: the
current-differencing amplifier will convert difference of
I A2 and I in to be the single ended output voltage Vout , and Vout
will control I A by the NMOS transistor. Then I A is fed to the
input of the squaring circuit, and its output is I A2 which is fed
to the inverting input of CDA. Consider feedback loop on the
CDA. It is negative feedback. Effect of negative feedback on
CDA forces the current at inverting input equating the current
at noninverting input or it can be written as I A2 ! I in . In
another word, the effect of negative feedback on CDA is that
I A2 tracks I in . Assume I A is the output current and I in is the
input current. Relationship of I A to I in can be rewritten
I A ! I in
(1)
In this research, all NMOS transistors operate in saturation
region, and the drain current can be expressed as follows
ID !
K
"VGS $ VT
2
#2 ;V GS
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% VT,V DS % VGS $ VT
(2)
W
) is the transconductance, VT is the
L
threshold voltage, V DS is the drain-to-source voltage, and V GS
is the gate-to-source voltage.
Where K( ! µC ox
B. Squaring Cell
The squaring circuit[9] is shown in Fig. 2.
Io1
M1
The basic current-differencing amplifier makes use of a PMOS
current mirror. The ratio of the PMOS current mirror
consisting M11 and M12 is equal to 1:1, and The ratio of the
NMOS current mirror consisting M4 and M5 is also equal to
1:1. I A is input current of squaring circuit and also is output
current of square-rooting circuit, I o 1 is output current of the
squaring circuit, and I o 2 is sum of I in and I B . The CDA
converts difference of I o 1 and I o 2 to be the voltage fed to gate
terminal of M6. The effect of negative feedback on CDA
regulates the equilibrium of I o1 and I o 2
Vo1
I o1 = I o 2
IA
M2
M3
(7)
Using (6), it can be rewritten as
VSS
Fig.2. The squaring cell.
In Fig. 2., I A is the input current, VO 1 is the input voltage and
I o 1 is the output current. Relation of the drain current to input
current can be written as
Using (2), VO 1
I A ! I D 2 $ I D1
can be written as
VO 1 !
IA
2K ( $V SS $ 2VT )
(3)
&
V SS
2
,,
IA
2K * **
* 2K "$ V $ 2V
SS
T
++
V )
,
I B ! 2K *VT & SS '
2 (
+
The relation of I A to I in is
I O 1 ! I D1 & I D 3
Using (2) and (4), it obtains
2
2
,,
) ,
IA
V ) )
' & *VT & SS ' '
I o 1 ! 2K * **
* 2K "$ V $ 2V # ' +
2 ( '
SS
T (
(
++
(5)
(6)
I A ! 2K "$ VSS $ 2VT
M11
IO2
M6
IB
IO1
IA
M1
Iin
M2
M3
M4
#
I o ! 2 K "VA $ 2VT #
Where
VDD
M12
M5
Fig.3. The simple square-rooting circuit.
(9)
I in
2K
(10)
I in
2K
2
0V DD $ VTP & VTN
01
VA !
KN
1&
4K P
(11)
KN
KP
/
-.
(12)
Defing K ! K N and VT ! VTN are the transconductance and
the threshold voltage of NMOS transistor respectively, and
K P and VTP are the transconductance and the threshold voltage
of PMOS transistor respectively. Remark M1 and M8 should be
placed in separate well to eliminate body effect.
D. Input Range
Consider the circuit in Fig. 3 when I in ! 0 , it find
that I D1 !
VSS
2
It should notice that I A is square rooting of I in . To design the
circuit with no current sources and no grounded circuit node,
the simple square-rooting circuit is improved as Fig. 4. The
transistors M7-M10 is used to generate I B and the ratio of the
current mirror consisting M9 and M10 is equal to 1:2. In Fig. 4,
the relation of I o to I in is
It should notice that I o 1 is squaring of I A
C. A Square-Rooting Circuit
The simple square-rooting circuit can be realized from
the current squarer formed by M1 -M3 and two current mirrors
shown in Fig. 3
(8)
Define
(4)
The output current I o 1 can be written as
2
2
) ,
V ) )
' & *VT & SS ' ' ! I in & I B
# '( +
2 ( '
(
IB
and I A ! 0 . Then when I in % 0 , it find that
2
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VDD
M14 1 : 1
M13
2:1
1:1
M12
M10
M11
M9
M6
VA
IO
M1
M8
Iin
M2
1:1
M3
M4
1:1
M7
M5
Fig.4. The completed square-rooting circuit.
IB
and I A % 0 . Where I in ( MAX ) or maximum input
2
current, it find that I D1 ! 0 and I A(MAX ) or maximum output
40u
I D1 3
20u
current is
IB
(13)
2
Using (9) and (10), the input range can be written as
K (2VT & V SS ) 2
0 3 I in 3
(14)
8
Consider the circuit in Fig. 4. The input range can be derived
by using the same way as the circuit in Fig. 3, and it can be
expressed as
Simulated curve
I A( MAX ) !
K (2VT $ V A ) 2
8
0
0A
I(rl)
5uA
SQRT(I(Iin))/135
10uA
15uA
20uA
25uA
Iin
Fig. 5. The dc transfer characteristic of circuit in Fig. 4 compares with ideal
curve ( !
I in
135
)
13
15
2.5
(15)
2
% error
0 3 I in 3
Ideal curve
1.5
1
0.5
III. RESULTS
0
1
The simulation results have been done by PSpice using
transistor model of MOSIS with level 3. The W/L ratio of
NMOS transistors M1 to M8 is equal to 34m/34m, the W/L
ratio of PMOS transistor M9 is equal to 1004m/54m, and the
W/L ratio of others is equal to 2004m/54m. The voltage
supply is set to VDD = 3 volts. The dc transfer characteristics of
the proposed circuit is demonstrated in Fig. 5 when ii is varied
from 0_A to 254A compared to ideal curve( !
shows relative error of Fig. 5. (%error=
Vi
135
3
5
7
9
11
17
19
21
23
25
the input current(mA)
Fig. 6. the relative error of the result in Fig.5 .
40uA
20uA
0A
I(rl)
20uA
). Fig. 6
Xi $ Xs
5 100 %, X i
Xi
is ideal value, X s is simulated value ) Fig. 7 shows the output
current when ii is triangular wave of peak amplitude 204A.
10uA
SEL>>
0A
0s
2.0ms
4.0ms
I(Iin)
Time
Fig. 7. Triangular wave response of circuit in Fig. 4
Upper trace output, lower trace input
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6.0ms
The average of relative error of Fig. 6 is 1.007%. It is
important to observe the no circuit node is connected to
ground, so this circuit can operate on both a single supply and
dual supplies.
The proposed circuit is tested experimentally by using
commercial CMOS transistor arrays (CD4007) and the input
current is constructed by the linear voltage to current
conversion [11]. The voltage supply is set to VDD=6.6 Volts.
Fig. 8 shows the dc transfer characteristic when I i is varied
0_A to 300µA compared to ideal curve ( ! 0.2 &
ii
).
1.7
Fig. 9 shows the output voltage waveform when I i is
triangular wave of peak amplitude 300µA at 10 kHz frequency
and it accords to simulation results in Fig. 7.
from
Note that the output current of experimental result has dc offset
current about 0.2 mA, it may occur from the mismatch problem
of devices.
IV. CONCLUSION
A new square-rooting circuit has been presented in this paper.
The realization method is based on the principle of negative
feedback. This circuit can operate under low voltage supply, so
it can be applied widely and is suitable for implementation in
CMOS integrated circuit form. Simulation results are carried
out by PSpice program. They find that the circuit can operate at
3_V voltage supply, the current input range is 204A, the
relative error is 1.007%, and the current output range is 354A.
Experimental results agree with simulated results.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
Fig. 8. The dc transfer characteristic
[9]
[10]
[11]
Fig. 9. The output waveform (upper trace) when
Ii
T. Kohonen, “Self-Organization and Associative Menory,”
Springer-Verlag, Berlin, pp.119-157, 1988.
O. E. Doebelin, “Measurement systems: Application and Design,”
New york, Mcgraw Hill.
J. Millman and A. Grabel,“Microelectronics,” New york, Mcgraw
Hill.
C. Toumazou, F.J. lidgey and D.G. Haigh, “Analogue IC Design :
the current-mode Approach,” london, UK, Peter Peregrinus.
I. M. Filanovsky, and H. P. Baltes, “Simple CMOS analog squarerooting and squaring circuits,”IEEE Trans. Circuits Syst., vol.39,
pp.312-315, 1992.
K. Dejhan, C. Soonyeekan, P. Prommee and F. Cheevasuvit, “An
MOSFET square-rooting circuit,” Proc. of ROVPIA’96, pp.597601, Malaysia, 1996.
M. Vander Gevel and J.C. Kuenen, “ x circuit based on a
novel,back-gate using multiplier,” Electronics letters, vol.30,
pp.183-184,1994.
V. Riewruja, K. Anutahirunrat and W. Surakampontorn, “A class
AB CMOS square-rooting circuit,” Int. J. Electronics, vol. 85, no. 1
pp. 55-60, 1998.
K. Bult and H. Wallinga. “A class of analog CMOS circuits based
on the square-law characteristic of an MOS transistor in saturation,”
IEEE J. Solid-State Circuits. Vol. SC-22. No. 3, pp. 357-365, 1987.
S. Menekay, R.C Tarcan, H. Kuntman, “a novel higher precision
current-mode square-root circuit,” Proc. of 2006 IEEE 14th Signal
Processing and Communications Applications, pp.1-4, 2006
R. Mancini, Edition in Chief, “Op-Amp for Everyone ; Design
Reference”, Texas Instrument, 2nd edition, 2003.
is triangular wave (lower
trace)
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1
TABLE I
LONGITUDINAL VELOCITY OF MATERIAL
Material
0.5
Longitudinal Velocity (c)
(mm / s)
6.350
4.430
4.700
2.680
5.610
5.940
1.494
Aluminum
Brass
Copper
Plexiglas
Stainless
Steel
Water
0
1
0.5
B. Ultrasonic Signal Simulation
Ultrasonic signal can be modeled by gaussian function in
Figure 2 and sine wave in Figure 3. e(t) may be expressed in
(2). F0 is transducer center frequency. t is time. !2 is variance
of gaussian function. t0 is position of scatters in time domain.
Figure 4 shows e(t) at t0 =2 s. h(t) is an impulse response of a
scatter function used to convolute with e(t) as described in (3).
e ( t ) $ exp[-2# 2! 2 ( t - t 0 ) 2 ]sin(2# F0 t )
s ( t ) $ e ( t ) % h( t )
0
4 t ( s)
1
2
3
Figure 3. Sine wave at 5 MHz
1
0
(2)
(3)
-1
1
For a medium of scatters, the echo signal s(t) can be modeled
in frequency domain as follows:
s( t ) $ e ( t ) % h( t ) $ ' e ( t ) % ha2 ( M , t ) % hd ( M , t )
4 t ( s)
2
Figure 2. Gaussian function at !2=0.709
(4)
4 t ( s)
2
3
Figure 4. e(t) function
Amplitude (V)
1
M
where h(t) is impulse response of scatters that relate to (5) and
2
(6) in frequency domain. ha is the two ways impulse response of
2
attenuation of a scatter in position M and H a ( M , f ) is frequency
2
domain of h . 2Z is the total distance traveled by the ultrasound
a
wave. hd(M,t) is a set of impulses in time domain that relate to
position M in frequency domain as shown in (8). " is
attenuation coefficient, which is estimated from FCS.
S( f )
(5)
E( f )! H
$
M
(6)
2
a
(M , f )! H d (M , f )
#2 Z " f
H (M , f ) e
H d (M , f ) % (Z )
(7)
(8)
hd is the scattering impulse function simulated by Dirac
function. The position Z of a scatter is determined with
function of object modeled. In this paper, two positions of
Dirac function are utilized according to experimental system.
hd ( f , Z )
% (Z )
2
4
6
8
10
12
14
t ( s)
Figure 5. Ultrasonic simulation signal
S( f ) E( f )! H ( f )
2
a
-1
(9)
TABLE II
ATTENUATION OF TISSUE AT 1 MHZ [5]
Tissue
Blood
Spleen
Liver
Fat
Brain
Muscle
Bone
Lung
(Water)
Characteristic
Impedance &0c
(106Ns/m3)
1-62
1-6
1-65
1-38
1-60
1.65-1.74
3.2-7.4
0.26-0.46
1.49
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Attenuation
at 1 MHz
(dB/cm)
0.2
0.4
0.7
0.8
0.8
1.5-2.5
11
40
0.002
1
III. FOURIER CENTROID SHIFT METHOD
Incident wave
FCS is one of strategies in Fourier statistics method that
depends on the spectral moment estimation. The spectral
centroid fc(&i) at spectral amplitude 'S(f)' as expressed in (10).
f c (&
i
m 1 (& i )
m 0 (& i )
)
(10)
mj is moment of order j defined by (11). &i is position of the
window on the echo line.
,+
m j (& i )
#
f
j ! S (&
i,
f ) ! df
(11)
+
With the classical assumption of linear with frequency
attenuation, ((f) is attenuation expressed in dB/cm and " is the
attenuation coefficient expressed in dB/cmMHz. The relation
between ((f) and " can be described in (12).
((f) = " ! f
(12)
An estimation of attenuation coefficient can be obtained by
(13). )2 is variance of spectrum that can be calculated by (14).
Additionally, the moment order 0 to 2 is used to obtain )2. C is
ultrasonic longitudinal velocity. Equation (15) expresses unit
conversion from Neper to dB.
"
#8.68
df c
C) .& / d&
2
Reflected wave
!
m 2 (& i ) 0 m1 (& i ) 1
) .& i /
#
m 0 (& i ) '3 m 0 (& i ) 24
(13)
0
4
5
6 f (MHz)
Figure 7. Spectrum of incident wave and reflected wave
From Figure 7, the attenuation coefficient is solved to verify
the FCS that described in previous section. The result of " in
simulation is 0.50 dB/cmMHz same as [6]. Additionally, other
" value at 1 MHz and 5 MHz are tested as shown in Table IV.
TABLE IV
ATTENUATION COEFFICIENTS COMPUTED BY SIMULATED SIGNAL
"
(dB/cmMHz)
0.1
0.3
0.5
0.7
1.0
1.5
2.0
3.0
Fourier Centroid Shift Method
1 MHz
5 MHz
0.1124
0.1001
0.3032
0.2998
0.5004
0.4997
0.7000
0.6995
1.0000
0.9993
1.5000
1.4989
2.0000
1.9986
3.0000
2.9978
TABLE V
ERROR OF ATTENUATION COEFFICIENTS COMPUTED BY SIMULATED SIGNAL
"
2
2
(14)
" [dB/cmMHz] = 8.68! " [Neper/cmMHz]
(15)
IV. NUMERICAL SIMULATION
According to ultrasonic transducer, 5 MHz of center
frequency is selected to simulate ultrasonic signal with
attenuation coefficient of brass = 0.50dB/cmMHz [6] as shown
in Figure (6). The difference of time corresponds to Z=2.5 cm
at c=4.430 mm/*s as described in Table I.
(dB/cmMHz)
0.1
0.3
0.5
0.7
1.0
1.5
2.0
3.0
Error (%)
1 MHz
12.4000
1.0667
0.0800
0.0000
0.0000
0.0000
0.0000
0.0000
Error (%)
1V
5 MHz
0.1000
0.0667
0.0600
0.0714
0.0700
0.0733
0.0700
0.0733
" theory # " estimation
!100
" theory
V. EXPERIMENTAL VALIDATION
TABLE III
INSTRUMENT AND DEVICE USED IN EXPERIMENTATION
0
-1V
0
10
20 time (*s)
Figure 6. Simulation of Ultrasonic signal at 5 MHz
Instrument/Device
Pulse Generator
Digital Scope
Ultrasonic Transducer
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Model
Panamatrics: Model500PR
Tektronix: TDS3012
Panamatrics: Model V309
(15)
Transducer
Z
Brass
Water
Figure 8. Ultrasonic simulation signal
The experimentation is built up as shown in Figure 8. The
metal, brass, is immersed in Plexiglas water tank. Furthermore,
the ultrasonic transducer is located at the top of water tank and
linked to digital oscilloscope. The instruments and devices as
shown in Table III are setup to obtain an ultrasonic signal.
Ultrasonic pulse generator and transducer are selected in same
manufacturer, Panamatrics. The reflection mode is selected to
measure the ultrasonic signal. The echo signal from transducer
is digitized with digital oscilloscope and sent as a digital data
set to personal computer by GPIB cable. The ultrasonic signal,
s(t), is digitized with a sampling interval 5t = 1/Fs (Fs is the
sampling frequency = 500 MHz), in 10,000 samples. Figure 9
displays a captured signal. The digitized data is computed to
obtain " by FCS algorithm. The peaks of spectrum at 5 MHz
correspond to transducer center frequency used. The spectrum
of measured signal is displayed in Figure 10. So, " of
measured signal is 0.4415 dB/cmMHz. The error calculated by
(15) of " between simulation and experimentation is
approximately 11%.
Figure 9.Captrued signal from brass at transducer center frequency=5MHz
dB
0
-10
-20
-30
-40
Incident wave
-50
Reflected wave
4
5
6
Figure 10. Normalized spectrum of measured signal
VI. CONCLUSION
A FCS method for estimation of ultrasonic attenuation
coefficients has been developed. The FCS algorithm is tested
by ultrasonic simulation signal. The method has been
established on the basis of spectrum in frequency domain that
is used to calculate the echoes from the front and back surfaces
of the immersed metal. To validate the model, the
experimentation has been setup. The 5MHz ultrasonic
transducer and brass are selected and immersed in Plexiglas
water tank. The data is acquired and computed by FCS
algorithm. The result show that " from simulated signal is 0.50
dB/cmMHz while " from experimentation is 0.4415
dB/cmMHz. The error is around 11% because this work omits
the ultrasonic diffraction phenomenon. However, the result in
simulation and experimentation is satisfied. To reduce an error,
the Lommel diffraction correction method [7] will be
compensated in future work.
REFERENCES
[1]
[2]
[3]
[4]
[5]
[6]
ACKNOWLEDGEMENT
This work was sponsored by NSTDA and TRF funding
contract no. F-31-206-22-02 and MRG4980072 respectively.
f (MHz)
[7]
Ping Wu, Tadeusz Stepinski, “Quantitative estimation of ultrasonic
attenuation in a solid in the immersion case with correction of diffraction
effects,” Ultrasonics, Elsevier,38, pp.481-485, 2000.
T. Baldeweck, P. Laugier, A. Herment, G. Berger, “Application of
Autoregressive Spectral Analysis for Ultrasound Attenuation Estimation
Interest in Highly Attenuating Medium,” IEEE Transactions on
Ferroelectrics and frequency control, 42, pp.99-110, 1995.
Celine Fournier, S. Lori Bridal, Alain Coron, Pascal Laugier, “Optimization
of Attenuation Estimation in Reflection for In Vivo Human Dermis
Characterization at 20 MHz,” IEEE Transactions on Ferroelectrics and
frequency control, 50, pp.408-418, 2003.
Valery Roberjot, S. Lori Bridal, Pascal Laugier, Genevieve Berger,
“Absolute Backscatter Coefficient over a Wide Range for Frequencies in
a Tissue-Mimicking Phantom Containing Two Populations of Scatters,”
IEEE Transactions on Ferroelectrics and frequency control, 43, pp.970978, 1996.
Heinrich Kuttruff., “Ultrasonics Fundamentals and Applications,”
ELSEVIER Applied Science, 1991.
V. R. Singh and Ashok Kumar (1995): “Development of A Focused
Ultrasonic Transducer with Increased Efficiency,” Proceedings of IEEEEMBS, pp.4.47-4.48, 1995.
S. Boonsang and R J Dewhurst, “Pulsed Photoacoustic signal
characterization incorporating near- and far-field diffraction effects,”Meas.
Sci. Technol., vol 16, pp. 885-899, 2005.
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