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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 ECTI-CON 2007 The 2007 ECTI International Conference ___________________________________________________________ 77 % 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 ECTI-CON 2007 The 2007 ECTI International Conference ___________________________________________________________ 78 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 ECTI-CON 2007 The 2007 ECTI International Conference ___________________________________________________________ 79 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) ECTI-CON 2007 The 2007 ECTI International Conference ___________________________________________________________ 80 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 ECTI-CON 2007 The 2007 ECTI International Conference ___________________________________________________________ 82 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 ECTI-CON 2007 The 2007 ECTI International Conference ___________________________________________________________ 83 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. ECTI-CON 2007 The 2007 ECTI International Conference ___________________________________________________________ 84