Analysis of Noise in Current Mirrors with memristive Device

1 Analysis of Noise in Current Mirrors with memristi v e De vice Nazerke Kulmukhano v a and Irina Dolzhikov a, Student Member , IEEE, Electrical and Computer Engineering Department, Nazarbaye v Univ ersity , Astana, Kazakhstan Abstract —This work pr esents an analysis of noise in a cascode current mirror with CMOS-memristive device done by compar- ison with the basic current mirr or . The analysis is completed based on THD for different frequency and channel length values by means of computer -aided design. A C and DC analyses are presented f or both balanced and unbalanced current mirrors. While the change in the channel length has similar effect in both circuits, memristor in a circuit decreases noise significantly at high fr equencies. Index T erms —Analog Cir cuits Design, Memristor , Current Mirrors, L TSpice, Noise Analysis. I . I N T RO D U C T I O N Current mirrors (CM) are widely used circuits that allow to copy a reference current level and, thus, control the current through the other device. It is widely used in operational amplifiers and integrated circuits. Current mirror eliminates a need to establish a current source with specific characteristics multiple times which is a v ery complex procedure. Instead, similar result is achiev ed by using only two transistors. It is also used in most operational amplifier circuits, and is among the building blocks of the integrated circuits. The simplest current mirror consists of two transistors as presented in the Fig. 1. It can be further improv ed to enhance the performance. Thus, there are a lot of v ariations of current mirrors. The most popular ones are Wilson and cascode current mirror , which are presented in the Fig. 2. They provide a much better performance than a basic CM with minimum increase in complexity . Figure 1. Basic CM Main noise types in current mirrors include Flicker , Burst, Thermal and Shot noises. According to Bilotti and Mariani [1], Cascode, W ilson CMs and their main variations demonstrate similar noise tolerance with slight discrepancies at low and Figure 2. Cascode (left) and W ilson (right) CMs high frequencies. In particular , Cascode CM exhibits a consis- tent performance throughout the whole range of frequencies, while W ilson CM performed better at high frequencies. An- other parameter that af fects the noise in CMs is channel length. The same study [1] suggests that when it is lower than 1.6 µ m W ilson CM is more tolerant to noise than other configurations, but there is no notable dif ference for greater channel length values. Memristor is a promising device that is gaining popularity as more and more applications of its memory effect are being discov ered [2]. The main working principle is that the oxide lev el in it can change depending on a current flowing through. When current is no longer applied its oxide layer remains on that particular lev el and resistance stays fixed at a new value. This ef fect can be used for non-volatile memory and neuromorphic circuits [3]. In software memristor models are dev eloped mostly by utilizing a concept of parallel switching channels that can hav e on and off states [4], [5]. Depending on an applied voltage lev els, channels switch ON or OFF and change the overall conductance of the de vice. This switch- based conductance change is represented as follo ws in Eq. 1. G = X R ON + 1 − X R OF F (1) Where X is the number of metastable switches in ON state, which can be calculated by: X = R ON ( R init − R OF F ) R init ( R ON − R OF F ) (2) Thus, oxide lev el in memristor is represented by a number of conducting channels in the model. This method is described in [6] and provides a suitable simulation model of memrsitor . Howe ver , memristor models av ailable do not take into account any noise types. Nev ertheless, noise in memristor was found 2 to be equiv alent to the resistor noise and, thus, have much smaller effect on a circuit than the noise in transistors, and is of no interest. In this paper , we provide a noise analysis of the cascode current mirror , and in vestigate the dependence between the noise level and frequency , transistor channel length, memristor resistance. Also, a practical case of a circuit with unbalanced values of channel length and memristor resistance is ev aluated. I I . S M A L L - S I G NA L M O D E L A N A L Y S I S A. Basic Curr ent Mirr or For a basic transistor-based current mirror as in Fig. 1, a small-signal model will be as presented in the Fig. 3. Main equations describing its operation are as follows: v 1 = h 11 i 1 + h 12 v 2 i 2 = h 21 i 1 + h 22 v 2 h-parameters can be found from the model by setting corre- sponding voltage and current values to 0; h 11 = v 2 i 2 = 1 g m 1 + g π 2 = 1 g m 1 (1 + n β o 2 ) ∼ = 1 g m 1 h 12 = 0 h 21 = i 2 i 1 = g m 2 + r π 2 1 + g m 1 r π 2 = β o 2 1 + g m 1 g m 2 β o 2 ∼ = g m 2 g m 1 ∼ = I C 2 I C 1 ∼ = n h 22 = i 2 v 2 = 1 r o 2 For MOSFET specifically the result will be: h 11 = 1 g m 1 h 12 = 0 h 21 = g m 2 g m 1 ∼ = ( W L ) 2 ( W L ) 1 ∼ = n h 22 = 1 r o 2 From the equations above, the values of input and output resistances will be 1 g m 1 and 1 r o 2 respectiv ely . It can be seen that the output/input ratio is largely determined by geometry of transistors along with the output voltage le vel. In this model many transistor parameters are assumed to be matching, thus not af fecting the performance. Howe ver , in more realistic model parameters such as oxide layer capaci- tance, of fset threshold and channel length modulation come into picture. B. Curr ent Mirr or with Memristor In the cascode CM two transistors is substituted by mem- ristors. This causes a circuit to be simplified into a basic current mirror with, essentially , non-linear resistors as in Fig. Figure 3. Basic Current Mirror small-signal model Figure 4. The CM with memristors 4. Memristors in the small-signal model are represented as additional resistances in series with existing ones. Thus, the ov erall operation of the circuit will not change. Input and output resistances become as follows: r in = 1 g m 1 + r mem 1 r out = 1 r o 2 + r mem 2 Where R mem 1 and R mem 2 are memristor resistance values at input and output sides of current mirrors respectively . When these memristors are identical their resistance expressions are the same. Mathematically these resistances can be expressed through a state variable w(t) with range from 0 to 1 as presented by Z. Biolek [7]: V ( t ) = [ R ON w ( t ) D + R OF F (1 − w ( t ) D )] I ( t ) v D = dw ( t ) dt = µ D R ON D I ( t ) w ( t ) = µ D R ON D q ( t ) Where v D is a drift velocity of oxygen deficiencies. A win- dowing function can be added to ensure that the value of w(t) does not exceed the range from 0 to 1. Then, by equating w ( t ) D 3 to x(t) : x ( t ) = w ( t ) D dx dt = µ D R ON D 2 I ( t ) F ( x ( t )) a windowing function will be F ( x ( t )) = 1 − (2 x ( t ) − 1) 2 p When value of the state variable is 0 memristor is operating at the lo west resistance R ON , and 1 stands for the maximum resistance state R OF F . Parameter p defines a sharpness of a windo w boundary - lower v alues will pro vide a smoother transition, while higher values will give an almost rectangu- lar function. This mathematical representation was used in memristor Spice model proposed by HP lab [6], and all other models are largely based on it. I I I . S I M U L A T I O N P A R A M E T E R S In the simulation of the circuit presented in the Fig. 4 the BSIM 3.1 MOSFET model provided by MOSIS [8] is used. The Length of the channel is 180nm. A memristor model added in the circuit w as de veloped by Knowm or ganization [4]. Its initial major parameters are R on 500 Ω , R of f 1500 Ω and threshold voltage 0.27V . For an A C analysis because the .noise command in L TSPice does not tak e into account the ef fect of memristor in the circuit, a Fourier coefficients are ev aluated instead. A comparison is made on the basis of total harmonic distortion. A major problem of the Fourier analysis is that its accuracy is heavily dependent on the resolution of a data sample and its size. Howe ver , eliminating this problem by setting a resolution and length of the simulation high is not feasible as it takes too long for the software to e valuate it. So, this calculation error is instead kept constant throughout the simulation by providing the same number of ev aluation points and periods for estimation. Thus, for dif ferent frequencies appropriate total time and maximum time step are set. For the DC analysis cases of imbalanced transistor lengths and imbalanced memristor resistances are considered. The channel lengths of both transistors M1 and M2, from the Fig. 4, are varied to observe the dependence between the current and channel length difference. The parameter LINT in the model file was changed for that [9].For the case of imbalanced memristor resistance values, the memristor in series with the output is varied. The v alues considered are in the range of 10% from the original value. I V . R E S U LT S A. F requency Dependence In the simulation THD for Six dif ferent frequencies from 1Hz to 10GHz with logarithmic step of 100 in circuits with and without memristor was observed. The circuit without memristor shows approximately constant noise for the whole range with abrupt fall at 0.1GHz which is caused by a decrease in noise of the main transistor and not yet coming into picture effect of the other transistor . In contrast, the circuit with memristor exhibits a clear decline in noise at higher frequencies. In particular, it becomes 0.03% when frequency is lower than 10KHz, 0.009% at 100MHz and 0.000122% at 10GHz as can be seen in the Fig. 6. Figure 5. THD in a CM without memristor for different channel lengths Figure 6. THD in a CM with memristor for different channel lengths B. Channel length dependence The same procedure is done for different channel length values. The v alues considered were 1 . 7 · 10 − 8 , 2 . 0 · 10 − 8 , 2 . 5 · 10 − 8 and 3 . 0 · 10 − 8 m. The resultant change in the total harmonic distortion is very small, in the order of 10 − 6 . This change is irrespective of frequency as can be seen in the Fig. 7 and 8. The av erage slope for six frequencies of the CM with memristor is calculated to be 6.27 1 . 7 · 10 − 8 per µ m and for the CM without it 1.95e-5 per µ m. C. Channel length dependence for the DC source When the channel length of the transistor which is in series with the output is changed the reference current does not change, but some discrepanc y occurs between it and the output current that previously was not observed. Thus, for the source voltage amplitude equal to 3V , the current changes by 74.9 µ A per nm change of the length; for 5V source this v alue becomes 146 µ A per nm; and 229 µ A per nm for 8V supply v oltage. For the basic current mirror without memristors, the current on channel length dependence is higher , namely 86 µ A per nm change of the length for a 3V source as shown in Fig. 9. 4 Figure 7. THD in a CM without memristor for different frequencies Figure 8. THD in a CM with memristor for different frequencies Figure 9. Current dependence on the channel length of M2 for 3V When the length of another transistor , M1, is increased it affects both current v alues. In particular , the reference current increases, and the output current decreases, though much slower . Notably , the discrepancy between two currents in both cases is the same. Thus, for the 3V source voltage the difference between reference current and output current increased by 74.9 µ A per nm change in the length of M1 as well. D. Memristor r esistance dependence for the DC source Similar procedure is performed for the ON resistance of memristor . The v ariation is set to 10% of the initial v alue. Figure 10. Current dependence on the channel length of M2 for 3V in a basic current mirror Figure 11. Current dependence on the channel length of M1 for 3V As expected, the current is decreasing as the resistance value increases linearly . For the source voltage value equal to 3V the difference between the reference and the output currents rises by 13 µ A/ Ω , for 5V by 1.14 µ A/ Ω and for 8V by 2.67 µ A/ Ω . Figure 12. Current dependence on the memristor resistance for 3V 5 V . D I S C U S S I O N The A C analysis results suggest that for a wide range of frequencies the CM with memristor has higher THD lev el than the basic CM. Howe ver , at high frequencies the THD v alue of the circuit with memristor goes to almost zero. In addition, the change in transistor channel length af fects the proposed circuit less, which can be useful in minimizing the effect of inaccuracies that frequently occur in manufacturing process. The DC analysis of the circuit also shows that the discrep- ancy that occurs because of the channel length imbalance is less for the CM with memristor when compared to the basic CM. Also, when memristor resistance variations are considered and changed from 500 Ω to 550 Ω the discrepancy between the output current and the reference current is much smaller for memristor resistance variations when compared to the transistor channel length imbalance. This analysis is limited to the comparison of the basic CMs with and without memristor . Similar comparison can be drawn between the cascode CM and a basic CM with memristor . Also, a basic CM with resistors instead of memristors could be considered. Memristor in this circuit does not really utilize its memory ef fect, so the main reason for its introduction into the circuit is to sa ve area. Howe ver , in this paper the case when the voltage level of the source is close to the threshold v oltage of memristor were not considered. So, other possible advantageous properties could be observed in that case. Moreover , a more complete analysis could be performed by introducing the noise of memristor and using a more practical transistor models. V I . C O N C L U S I O N In conclusion, the analysis of noise in the CM with mem- ristor is presented in this paper . Both A C and DC cases are considered for transistor channel length and memristor resis- tance variations. The results show that memristor in the basic current mirror makes its noise lev el less at high frequencies, and also decreases the dependence on channel length variations significantly . Small area of memristor makes the proposed circuit a viable option when high transistor variation tolerance or lo w noise at high frequencies is required. The analysis, howe ver , is limited to the comparison with the basic CM. It can be further e xtended by comparing the circuit with the other configurations. In addition, transistor and memristor models do not exhibit real-life properties of the devices, which puts a limitation to the viability of results obtained. R E F E R E N C E S [1] Alberto Bilotti and Eduardo Mariani. ”Noise Characteristics of Current Mirror Sinks/Sources, ” IEEE Journal of Solid-State Circuits 10(6):516- 524, January 1975. [2] A. James, Memristor and Memristive Neural Networks , Intech, 2018, DOI: 10.5772/6653. [3] A. Irmano va and A. P . James, Neur on inspir ed data encoding memristive multi-level memory cell , Analog Integrated Circuits and Signal Process- ing, Springer , p.1-6, 2018. [4] Molter, Tim, ’The Generalized Metastable Switch Memristor Model Knowm.or g’, 2015. [Online]. A vailable: https://kno wm.org/the- generalized-metastable-switch-memristor-model/ [Accessed: 15- Feb- 2018]. [5] A. Irmanova and A. P . James, Multi-level Memristive Memory with Resistive Networks , arXiv preprint arXi v:1709.04149, 2017. [6] Chris Y akopcic, T arek M. T aha, Guru Subramanyam, and Robinson E. Pino. ”Memristor SPICE Modeling, ” in Advances in neuromorphic memristor science and applications , Springer, 2012. [7] Zdenk Biolek, Dalibor Biolek and V iera Biolkov a. ”SPICE Model of Memristor with Nonlinear Dopant Drift, ” Radioengineering , p.210-214. [8] MOSIS W afer Acceptance test. http://www .ece.ucsb.edu [9] W eidong Liu, Xiaodong Jin, Xuemei Xi, James Chen, Min-Chie Jeng, Zhihong Liu, Y uhua Cheng, Kai Chen, Mansun Chan, Kelvin Hui, Jianhui Huang, Robert T u, Ping K. K o and Chenming Hu. BSIM3v3.3 MOSFET Model User’ s manual.