A 120dB Input Dynamic Range, Current-input Current-output CMOS Logarithmic Amplifier with 230ppm/oK Temperature Sensitivity Ming Gu iWatt Inc. Campbell, CA 95008
[email protected]
Shantanu Chakrabartty Department of Electrical and Computer Engineering Michigan State University, East Lansing, MI 48824
[email protected]
Abstract— This paper presents the design of a current-mode CMOS logarithmic amplifier, which by design is insensitive to ambient temperature variations. Its operational current ranges over 100fA - 100nA, which covers the full range of sub-threshold region. Unlike conventional logarithmic amplifiers, the proposed approach directly produces a current signal as a logarithmic function of the input current. The core of the proposed circuit is translinear ohm’s law principle, which is implemented by a floating-voltage source and a pseudo-linear resistive element within a translinear loop. The temperature sensitive parameters are reduced using a translinear-based resistive cancelation technique. Measured results from prototypes fabricated in a 0.5 µm CMOS process show that the amplifier exhibits an input dynamic range of 120dB and a temperature sensitivity of 230 ppm/o K.
paper, the functionality of the circuit is verified using measured results from prototypes fabricated in a 0.5 µm CMOS process. The paper is organized as follows. The translinear Ohm’s law principle exploited in the logarithmic amplifier is presented in section II. Section III describes the circuit implementation of the integrated programmable logarithmic amplifier. Section IV describes the measured results obtained from prototypes fabricated in a 0.5µm CMOS process and section V concludes the paper.
I. I NTRODUCTION
The concept of Translinear Ohm’s law is an extension of the celebrated translinear principle [9] which exploits the exponential relationship between voltages and currents in certain devices (diodes, BJTs and sub-threshold MOSFETs). An exemplary circuit demonstrating Translinear Ohm’s law is shown in Fig. 1. The translinear loop consists of four translinear elements diodes D1–4 , a memoryless, pseudo-linear resistive element N , and a floating-voltage source ∆V . The current flowing through N (IN ) and the voltage across N (VN ) follows an anti-symmetric non-linear function f (.) as equation (1) describes.
Logarithmic amplifiers, which generate output signals proportional to the logarithm of input signals, are widely used in biomedical applications [1]–[3]. Logarithmic amplifiers are also useful in the design of analog signal processors that use log-likelihoods (logarithm of probability scores) for computation [4]–[6], which include analog Hidden Markov Models (HMMs) processors used in speech recognition [7] and analog support vector machines (SVMs) used in pattern classification [8]. Most previously reported logarithmic amplifiers are implemented based on the exponential dependence between the current and the voltage across a p-n junction diode, a bipolar transistor [9] or a MOSFET in sub-threshold region [10]. However, this transimpedance method recurs obvious disadvantages. Firstly, the output dynamic range is limited by the supply voltage since the output of the amplifier is a voltage signal. Secondly, their performance is degraded by temperature variations severely due to the dependence on the current-tovoltage relationship. These concerns can be addressed by a current-input, currentoutput logarithmic amplifier we propose in this paper. The proposed design utilizes a Translinear Ohm’s law principle which exploits floating-voltage sources and pseudo-linear resistive elements embedded within a translinear loop. By introducing additional translinear loops, temperature compensation is achieved through a resistive cancelation technique. In this
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II. T RANSLINEAR O HM ’ S L AW BASED L OGARITHMIC C OMPUTATION
IN = f (VN ).
(1)
If the voltage drop across each diode in Fig. 1 is denoted by V1–4 then, V1 + V2 + ∆V = V3 + V4 + VN . According to the translinear diode equation I1-4 Is exp (V1-4 /UT ), equation (2) leads to I1 · I2 VN = ∆V + UT log I3 · I4 I1 · I2 IN = f ∆V + UT log I3 · I4
(2) =
(3) (4)
Here UT denotes the thermal voltage. If I1 · I2 = I3 · I4 then equation (4) leads to
521
IN = f (∆V ) .
(5)
If N is a linear resistor with resistance R, then f (∆V ) = ∆V /R, and (5) turns into an equivalent Ohm’s law. If N is a pseudo-linear resistor with f (0) = 0, then f can be approximated using first-order and second-order Taylor series terms as f (∆V ) ≈ f 0 (0)∆V + 12 f 00 (0)∆V 2 . Hence equation (5) leads to 1 IN = f 0 (0)∆V + f 00 (0)∆V 2 . (6) 2 Note that f 0 (0) has units of a transconductance and equation (6) is only satisfied when the translinear condition I1 ·I2 = I3 · I4 is satisfied, hence the name “Translinear Ohm’s Law”. For the sake of simplicity and to minimize mathematical clutter in the next few sections, we will assume f 00 (0) ≈ 0.
[11]: N M OS : In = Sn ID0n e(VGn −VT n )/nn UT e−VSn /UT , P M OS : Ip = Sp ID0p e
V2 V4 IR
I4
0
I3 = n · f (0) · UT · log
V1 M6
I1
I2
M7
Iout Iin
Fig. 2.
I3
M3
M4
I1 I2
.
(11)
It consists of an input stage A1 , a reference stage A2 and a translinear stage A3 . The input stage and the reference stage is formed using the basic circuit shown in Fig. 2. Based on equation (11), the output current generated by the input stage Iin is given by n · UT Iin Ix = · log . (12) R Ib2
V2
M2 M5
Fig. 3 shows a complete implementation of a temperature insensitive logarithmic amplifier.
Ib1
M1
(9)
Thus, the output current I3 is proportional to the logarithm of the current I1 under the condition I1 > I2 which is required for the circuit to be operational.
The basic concept of Translinear Ohm’s law.
R
(8)
which leads to
R Fig. 1.
,
(7)
where the floating-voltage source VG1 − VG2 is the difference in gate voltages of transistors M1 and M2 . Using the expressions (8) for transistors M6 and M7 , the floating-gate source VG1 − VG2 can be expressed as I1 VG1 − VG2 = n · UT · log , (10) I2
I2
I3
e
Iout = f 0 (0) (VG1 − VG2 ) ,
I1
V3
VSp /UT
where Sn,p , VT n,p , nn,p , UT , VGn,p , and VSn,p are the aspect ratio, the threshold voltage, the sub-threshold slope, the thermal voltage, gate and source voltage referred to bulk potential (Vdd or gnd) for nMOS and pMOS transistor respectively. The transistor M5 serves as a feedback element which reduces the output impedance at the drain of M2 . If the sizes of the transistors are assumed to be equal, the current mirror formed by M3 and M4 ensures I1 · I2 = I3 · I4 . Then, using the Translinear Ohm’s law the output current Iout can be expressed as
V
V1
(−VGp +VT p )/np UT
I4
Ib2
For the reference stage A2 the ratio of the currents I1 and I2 are set to be 2 using the current mirrors. Thus the reference current Iref is given by
Basic current-input current-output logarithmic amplifier.
III. C IRCUIT I MPLEMENTATION OF L OGARITHMIC A MPLIFIER The basic circuit implementing Translinear Ohm’s Law is shown in Fig. 2. Transistors M1 to M4 are biased in weakinversion and form a translinear-loop. The drain-to-source voltages for all transistors are larger than 100mV, in which case the transistors satisfy the following translinear relation
Iref
=
Iref
=
n · UT · log (2) R 0.693 · n · f 0 (0) · UT .
(13) (14)
The translinear stage A3 forms a translinear loop using the gate-to-source terminals of the pMOS transistors M9 -M12 , which leads to Iout · Iref = Ib4 · Ix . Note that Ib1 in Fig. 3 is an external biasing current to establish the translinear loop and is typically set 10 times larger than Ib4 . M15 is added at the input of Ib4 to ensure M11 to be in saturation region. Thus, using equation (10) the output current Iout can be expressed
522
A1
A3
A2
Inputstage
Transl inearstage
Reference stage
Ib1
R
Ib1
R Ib1
V1
V2
M1
M2
M9
M5
M6
M10
M4
M16
M12
M17
Ix
Iref Ib4
M8
Iin
M19
Iref
Iout
Ib2
M18
M22
M15
M7 Ix
M3
M11
I1=2·I2 M14
I2
M20
M21
Vb
From DAC
From CurrentReference
Fig. 3.
Complete schematic of the temperature compensated logarithmic amplifier.
TABLE I C IRCUIT PARAMETERS USED FOR DESIGNING THE LOGARITHMIC
as Ix Iref
Iout
=
Ib4 ·
Iout
=
1.44 · Ib4 · log
AMPLIFIER
Iin Ib2
.
Fabrication Process Die Size M1-4, 6-7 (refer to Fig. 2) M5 (refer to Fig. 2) R (refer to Fig. 2)
(15)
If the currents Ib4 and Ib2 are assumed to be invariant to temperature, Iout is also theoretically invariant with respect to temperature.
Standard CMOS 0.5µm 3000µm × 3000µm 10µm/5µm 30µm/1.5µm 10MΩ
IV. M EASUREMENT R ESULTS The proposed logarithmic amplifier has been prototyped in a 0.5µm standard CMOS process and Fig.4 shows the micrograph of the prototype. Table I summarizes the sizes of the circuit elements used for designing the amplifier.
4
x!10
9
Ib4=40pA Ib4=100pA
Iout(A)
3
Logarithmic Amplifier
Ib4=250pA
2
1
0 10
100µm
10
10
9
10
8
10
7
Ii n(A)
Input DAC
Fig. 5. Measurement DC response showing logarithmic relationship between the input and output current.
Fig. 4.
Die microphotograph of the logarithmic amplifier.
A. DC Response of the logarithmic amplifier The first set of experiments measured the DC characteristics of the logarithmic amplifier. The input current was varied from 100pA to 40nA (within the weak-inversion region of the
transistor) at different settings of Ib4 . The measured results are shown in Fig. 5, which is plotted in a log-linear scale. The results show that the output current is a log-linear function of the input current and closely approximates the equation (15). The result also shows that the current Ib4 can be used to control the gain of the logarithmic amplifier and hence the amplifiers output range.
523
2.1
B. Temperature Measurement and Compensation
2.16 2.095
Iout (T ) = 1.44Ib4 (T ) · (log(Iin ) − log(Ib2 (T ))) .
CurrentRati o
The principle of the Translinear Ohm’s law intrinsically leads to a temperature compensated operation. According to equation (15), the temperature coefficient of Iout is determined by the temperature characteristics of Ib4 , Ib2 , and Iin and can be expressed as
2.12
2.09 2.085 40
45
50
55
2.1 2.08
(16)
where each current reference is assumed to be a function of temperature T. Note that the temperature variation is mainly due to the temperature coefficient of the resistor in floating-gate current reference. However, the objective of the experiments presented in this section is to verify that the logarithmic function can be implemented in a temperature invariant manner. Therefore, we use an external temperature compensated current source for the input current Iin , where as the temperature dependent terms in equation (16) is eliminated by using a bilinear measurement technique. If Iout1 (T ), Iout2 (T ) and Iout3 (T ) are the output currents corresponding to three different values of the input currents Iin1 , Iin2 and Iin3 , measured at a fixed temperature T , then the following bilinear transformation is obtained:
2.14
230 ppm/K
30
Fig. 6.
40 50 o Temperature ( C)
Measured response showing
temperatures.
60
Iout1 (T ) − Iout2 (T ) under different Iout1 (T ) − Iout3 (T )
translinear loops and programmable current references. The current-input, current-output topology allows the proposed circuit to exhibit a large output dynamic range by using different biasing parameters, which is superior to current-input voltageoutput alternatives. Also, the proposed amplifier is temperature compensated where as temperature compensation of currentinput voltage-output topology is generally considered to be difficult.
Iin1 Ib4 (T0 ) Iin1 R EFERENCES log · log Iout1 (T0 ) − Iout2 (T0 ) log 2 I I in2 = in2 [1] Y. Liu, M. Gu, E. C. Alocilja, and S. Chakrabartty, “Co-detection: = Ib4 (T0 ) Iin1 Iin1 Iout1 (T0 ) − Iout3 (T0 ) Ultra-reliable nanoparticle-based electrical detection of biomolecules · log log in the presence of large background interference,” Biosensors and log 2 Iin3 Iin3 Bioelectronics, vol. 26, no. 3, pp. 1087–1092, Sep. 2010. (17) [2] M. Gu, Y. Liu, and S. Chakrabartty, “FAST: A simulation framework for solving large-scale probabilistic inverse problems in nano-biomolecular It can be seen from the equation (17), that the bilinear circuits,” in Proc. IEEE Int. Symp. Circuits and Systems, May 2010, pp. transformation is only a function of the logarithmic function 3160–3163. [3] M. Gu and S. Chakrabartty, “FAST: A framework for simulaof the input, where as all the temperature dependent terms tion and analysis of large-scale protein-silicon biosensor circuits,” due to the current references are canceled. Thus, if the IEEE Transactions on Biomedical Circuits and Systems, Dec. 2012, proposed amplifier is indeed temperature compensated, then DOI:10.1109/TBCAS.2012.2222403. [4] ——, “A 100pJ/bit, (32,8) CMOS analog low-density parity-check the bilinear measurement should be in principle be temperature decoder based on margin propagation,” IEEE J. Solid-State Circuits, compensated. vol. 46, no. 6, pp. 1433–1442, Jun. 2011. For the next set of experiments, the device under test was [5] M. Gu, K. Misra, H. Radha, and S. Chakrabartty, “Sparse decoding of low density parity check codes using margin propagation,” in Proc. of placed in a programmable temperature chamber and the temIEEE Globecom, Nov. 2009. o o perature was varied from 27 C to 57 C. For each temperature [6] M. Gu and S. Chakrabartty, “An adaptive analog low-density paritysetting, the input current is set to three pre-set values and the check decoder based on margin propagation,” in Proc. IEEE Int. Symp. Circuits and Systems, May 2011, pp. 1315–1318. corresponding output current and the bilinear expression is [7] J. Lazzaro, J. Wawrzynek, and R. P. Lippmann, “A micropower analog computed according to equation (17). The experimental results circuit implementation of hidden markov model state decoding,” IEEE are shown in Fig. 6 which shows that the bilinear expression Journal of Solid-State Circuits, vol. 32, no. 8, pp. 1200–1209, Aug. 1997. is temperature compensated. Based on this measurement, the [8] S. Chakrabartty and G. Cauwenberghs, “Sub-microwatt analog VLSI temperature sensitivity was calculated to be 230 ppmo K as trainable pattern classifier,” IEEE Journal of Solid-State Circuits, vol. 42, shown in the Fig. 6(inset). no. 5, pp. 1169–1179, May 2007. V. C ONCLUSIONS In this paper we have presented the design of a currentinput, current-output logarithmic amplifier circuit for lowpower sensory signal processing applications. The circuit is based on a Translinear Ohm’s law principle which requires embedding a floating voltage source and a pseudo-linear resistive element within a translinear loop. Temperature dependence and circuit non-linearity is compensated using additional
[9] B. Gilbert, “Translinear circuits: A proposed classification,” Electron. Lett., vol. 11, no. 1, pp. 14–16, 1975. [10] A. Basu, R. W. Robucci, and P. E. Hasler, “A low-power, compact, adaptive logarithmic transimpedance amplifier operating over seven decades of current,” IEEE Trans. Circuits Syst. I, vol. 54, no. 10, pp. 2167–2177, Oct. 2007. [11] E. Vittoz and J. Fellrath, “CMOS analog integrated circuits based on weak inversion operations,” IEEE J. Solid-State Circuits, vol. 12, no. 3, pp. 224–231, Jun. 1977.
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