Temperature Independent Log Domain Filter

Surachoke Thanapitak, Phumin Kirawanich, Decha Wilairat, and Pongsathorn SedtheethornDepartment of Electrical Engineering, Faculty of Engineering, Mahidol University

25/25 Phuttamonthon 4 Rd., Nakhon Pathom 73170 ThailandPhone: +662889-2138 Ext. 6501-2, Fax: +662889-2138 Ext. 6529

Email: [surachoke.tha, phumin.kir, decha.wil and pongsathorn.sed]@mahidol.ac.th

Abstract—This paper presents the current mode log domainfilter based on the Bernoulli cell which has moderately lowsensitivity to temperature change. According to the simulationresults in a standard 0.18µm CMOS process, the sensitivity ofthe output current over temperature of this log domain circuitis 2.941pA/°C. This is archived by using the modified currentreference circuit which is directly proportion to temperature asthe bias current for the log domain filter circuit. Additionally, thismodified current reference circuit can be operated at low powersupply voltage (1.11V) while high PSSR (0.43nA/V) is maintained.The power consumption of this modified current reference is inthe sub-microwatt range (0.1µW).

Keywords—Current reference circuits, Log domain filters, Tem-perature independence

I. INTRODUCTION

For the electronics circuit, designed for the portable ap-plications (e.g., low voltage and low power consumption), theMOSFETs biased in the weak inversion region are suitablefor these certain applications. This is due to the magnitudeof the current of the weakly inverted MOSFETs which arein the order of nanoampere. According to their exponentialrelationship between gate-source voltage to drain-current, thesebiased below the threshold voltage MOSFETs are also suitablefor some certain bipolar circuit techniques such as translinearcircuits [1] and log domain filters [2]. Therefore, the circuittechniques on Bipolar Transistors (BJTs) can be implementedin CMOS technology.

However, one undesired characteristic of the MOSFETsoperated in weak inversion is temperature dependency. For ex-ample, the operation transconductance amplifier (OTA) usingsubthershold MOSFET [3] has the transconductance gain (gm)inversely proportion to thermal voltage (UT ). Another exampleis the log domain filter whose its bandwidth and current gainare partially controlled by the thermal voltage.

There is an example of using a current source which isdirectly proportional to the absolute temperature to compensatethe temperature factor on the translinear current conveyor andthe OTA [4]. In this paper, the idea of using the currentto absolute temperature circuit to diminish the temperaturedependency on the log domain filter is established. Typically,passband gain and time constant of the log domain are bothdirectly proportion to the tuned current sources and are alsoinversely proportion to the thermal voltage. If the tuned currentsources dynamically track with temperature, the temperature

M3

M1M2

M4

M6M5 M7 M8

M9

IrefAP

IrefAN

VDS3+

-

Fig. 1. All active CMOS current reference of Oguey et al. [5].

dependent terms on both tuned current source and thermalvoltage are cancelled out. The dynamically track with tempera-ture current source can be realised with the current to absolutetemperature circuit.

II. CURRENT REFERENCE WITH DIRECT PROPORTION TOABSOLUTE TEMPERATURE

The current reference circuit in the range of nanoamperewith all active elements [5] is shown in Fig.(1). To ensurethe magnitude of the current IrefAP in nanoampere range, thetransistor M1 and M2 are operated in weak inversion region.As a result, there is small voltage drop across the drain sourceterminal of M3 (VDS3) which ensures that the transistor M3biased in triode region. The operation region of the othertransistors (M4-M9) are in saturation.

The output reference current from this circuit (IrefAP ) is

IrefAP =(S3µCOX)2 · 2X · (nUT lnN)2

S4µCOX(1)

where µ is the carrier mobility. COX is the gate oxidecapacitance. S3 and S4 are the aspect ratios of the transistors

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M3

C I0

I0

Id

Iin Iout

M1M2

M4

M6M5 M7 M8

M9

M11 M12

M17

M13 M14 M20

M16

M15

M18 M19

M21 M22

IrefAP IrefBP

IrefBNIrefAN

VDS3

+

-

IrefB

VDD

Fig. 2. The modified CMOS current reference of Oguey et al. [5] with aregulated cascode stage.

M3 and M4 respectively. X is the ratio between S5 and S6 (S5

S6)

when S6 = S7 and S5 = S8. n is the subthreshold parameter.UT = KT

q is thermal voltage (K is the Boltzmann’s constant.T is the absolute temperature and q is an electron charge andN is the ratio between S2 and S1 (S2

S1).

When the ratio between S3 and S4 is unity (S3 = S4), theeq.(1) can be re-written as

IrefAP = S3µCOX · 2X · (nUT lnN)2. (2)

In eq.(2), the mobility and the thermal voltage are thermaldependent. The mobility can be described as

µ = µ0

(T

T0

)−m

(3)

where µ0 is the mobility at T0 and m is the mobility temper-ature exponent [6]. Eq.(2) can be re-written as the function ofabsolute temperature as

IrefAP = 2µ0XS3COX

(nK lnN

q

)2(1

T0

)−m

·T 2−m. (4)

The typical value of m in standard CMOS process is 1.5[7]. This means that IrefAP is direct proportion to absolutetemperature. According to the measured results of the circuitin Fig.(1) [5], the low power supply rejection ratio (PSSR) onthe output current is reported [5] with the suggested solutionby using the cascode stage on the transistors M1 and M2.However, this cascode stage requires higher power supply levelwhich is not suitable for low voltage applications.

The alternative approach to increase the PSSR of thiscurrent reference circuit, while the supply voltage can be main-tained at low level, is demonstrated in the current reference

40

30

20

10

0

Ire

fB (n

A)

43210VDD (V.)

Fig. 3. The simulation result for output current (IrefBN ) vs. supply voltageof the current reference circuit in Fig.(2).

49

48

47

46

45

44

Ire

fB (n

A)

706050403020100Temperature (ºC)

Fig. 4. The simulation result for output current (IrefBN ) vs. temperatureof the current reference circuit in Fig.(2).

circuit in Fig.(2). This current reference circuit is the modifiedversion of the Oguey et al. ’s circuit in Fig.(1). The additionaltransistors (M15, M16 and M20) are the regulated cascodeoutput stage which increases the overall output impedance ofthe current mirror (M11 and M12).

From the simulation result in a standard 0.18µm CMOSprocess (Fig.(3)), the output current (IrefBN ) begins to reg-ulate at 1.11 V. The output current sensitivity to the powersupply voltage is 0.43nA/V. The linear relationship betweenthe output current and temperature is illustrated in Fig.(4)and the output current sensitivity to temperature is 52.5pA/°C.The power consumption at 1.2V supply voltage and roomtemperature is 100.4nW. The size of each transistors is shownin Table I.

III. LOG DOMAIN FILTER AND ITS TEMPERATUREDEPENDENCE

MOSFETs operated below theirs threshold voltage (weakinversion) are the ideal approach for the low voltage and lowpower applications because the drain current of the MOSFETsin weak inversion region is in the range of nanoampere.

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TABLE I. MOSFET DIMENSION OF THE CURRENT REFERENCECIRCUIT IN FIG.(2)

MOSFET W (µm) L (µm)M11 50 0.5M12 4×50 0.5

M13, M14 10 40M15 50 0.5M16 50 10

M17, M18 0.5 35M20 5 40M21 3×10 40

M3

C I0

I0

Id

Iin Iout

M1M2

M4

M6M5 M7 M8

M9

M11 M12

M17

M13 M14 M20

M16

M15

M18 M19

M21 M22

IrefAP IrefBP

IrefBNIrefAN

VDS3

+

-

IrefB

VDD

P1 P2 P3 P4

Fig. 5. Low pass log domain filter

However, there are some drawbacks of the MOSFETs biasedin weak inversion region, for instance, the limitation on the op-erated frequency in the audio frequency range and temperaturedependency [8].

In this paper, the temperature dependency for weak in-version MOSFETs is demonstrated by the log domain filtercircuit [2] which is shown in Fig.(5). This current mode filteris proposed by Drakakis et al. as the alternative approach torealise the log domain filter circuit based on the Bernoullicell [2]. When all PMOS transistors are matched, the transferfunction of Iout(s)

Iin(s)is

Iout(s)

Iin(s)=

I0nCUT

s+ IdnCUT

(5)

where C is the filter capacitor. I0 is the gain control biascurrent and Id is the bias current which sets the time constant.The first order low pass filter is formed in eq.(5) with the filterpassband gain ( I0

nCUT) and the filter time constant ( Id

nCUT). It

can be realised that both passband gain and time constant ofthis log domain filter is temperature sensitive. This temperaturedependency is exhibited in the simulation result in Fig.(6)when the temperature is varied from 0°C to 70°C.

80

60

40

20

0

I out (

nA)

100 101 102 103 104 105 106

Frequency (Hz.)

0C 70C

Fig. 6. The output current of the log domain filter when the temperature isset at 0°C and 70°C.

80

60

40

20

0

I out (

nA)

100 101 102 103 104 105 106

Frequency (Hz.)

0C 70C

Fig. 7. The output current of the log domain filter combined with the currentreference circuit in Fig.(2) when the temperature is set at 0°C and 70°C.

IV. LOG DOMAIN FILTER INSENSITIVE TO TEMPERATURECHANGE

To obtain the log domain filter which is insensitive totemperature change, the gain control bias current (I0) and thetime constant bias current (Id) are implemented by the currentreference circuit in Fig.(2). In this case, the filter passbandgain and the filter time constant of this modified log domainfilter are:

Passband gain :I0

nCUT=IrefBP

nCUT(6)

Time constant :Id

nCUT=IrefBP

nCUT. (7)

With this implementation, the current I0 and Id are di-rectly proportion to temperature. The thermal dependent term(thermal voltage) on both the passband gain and the timeconstant is cancelled out by these temperature tracking currentsources. Therefore, the passband gain and the time constant, ineq.(6) and (7) respectively, are temperature independent. Thevalidity of this thermal independence of the log domain filteris confirmed with the simulation result in Fig.(7).

259

When the temperature is swept from 0°C to 70°C, theoutput current (Iout) sensitivity to temperature (the simulationresult in Fig.(7)) is 205.87pA per 70 degree change of temper-ature (2.941pA/°C). However, this output current sensitivityto temperature of the uncompensated log domain filter (thesimulation result in Fig.(6)) is 4.394nA per 70 degree changeof temperature (62.771pA/°C). With these two figures, it canbe concluded that the output current sensitivity to temperatureof the log domain filter is decreased by 21.34 times by usingthe modified current reference circuit in Fig.(2).

However, there is slight thermal deviation on the bandwidth(time constant) of this log domain filter. This is because ofthe mismatch on the current magnitude between the currentsource (I0 at the source terminal of the transistor P3) and thecurrent sink (I0 at the drain terminal of the transistor P3).This mismatch on the current sink and the current source, asa result, deviates the magnitude of the drain current of P2.

V. SUMMARY

The modified current reference circuit of Oguey et al.with higher PSSR is introduced in this paper. This higherPSSR (0.43nA/V) is maintained while the supply voltage canbe kept at the low level (1.11V.) by the regulated cascodetechnique. Additionally, this modified current reference circuitis further employed to minimise the temperature dependencyof the log domain filter which is validated by the simulationresult. Both the modified current reference circuit and thelog domain filter are simulated in a standard 0.18µm CMOSprocess.

REFERENCES

[1] B. Gilbert, “Translinear circuits: a proposed classification,” ElectronicsLetters, vol. 11, no. 1, pp. 14–16, 1975.

[2] E. M. Drakakis, A. J. Payne, and C. Toumazou, “Log-domain state-space: a systematic transistor-level approach for log-domain filtering,“Circuits and Systems II: Analog and Digital Signal Processing, IEEETransactions on, vol. 46, no. 3, pp. 290–305, 1999.

[3] R. Sarpeshkar, R. F. Lyon, and C. Mead, “A low-power wide-linear-range transconductance amplifier,” Analog Integrated Circuits and SignalProcessing, vol. 13, no. 1-2, pp. 123–151, 1997.

[4] W. Surakampontorn, V. Riewruja, K. Kumwachara, and C. Fongsamut,“Temperature compensation of translinear current conveyor and OTA,”Electronics Letters, vol. 34, no. 8, pp. 707–709, 1998.

[5] H. J. Oguey and D. Aebischer, “CMOS current reference withoutresistance,” Solid-State Circuits, IEEE Journal of, vol. 32, no. 7, pp.1132–1135, 1997.

[6] I. M. Filanovsky and A. Allam, “Mutual compensation of mobilityand threshold voltage temperature effects with applications in CMOScircuits,” Circuits and Systems I: Fundamental Theory and Applications,IEEE Transactions on, vol. 48, no. 7, pp. 876–884, 2001.

[7] K. Ueno, T. Hirose, T. Asai, and Y. Amemiya, “A 300 nW, 15 ppm/c, 20 ppm/v CMOS voltage reference circuit consisting of subthresholdMOSFETs,” Solid-State Circuits, IEEE Journal of, vol. 44, no. 7, pp.2047–2054, 2009.

[8] M. Nishida and H. Ohyabu, “Temperature dependence of MOSFETcharacteristics in weak inversion,” Electron Devices, IEEE Transactionson, vol. 24, no. 10, pp. 1245–1248, 1977.

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