A simple time domain approach to noise analysis of switched capacitor circuits Mohammad Rashtian 1a), Omid Hashemipour 2, and A.M. Afshin Hemmatyar 3 1 Department of Electrical Engineering, Science and Research Branch of Islamic Azad University, Tehran, Iran 2 Department of Electrical and Computer Engineering, Shahid Beheshti University, G. C. Tehran, Iran 3 Department of Computer Engineering, Sharif University of Technology, Tehran, Iran a) rashtian@catc.ac.ir Abstract: Thermal noise is one of the most important limiting factors on the performance of switched-capacitor (SC) circuit due to the aliasing effect of wide-band thermal noise. In this paper a new simple method for estimating the effect of thermal noise is presented. In the proposed technique only the discrete sampled noise is considered. HSPICE simulator and analytical analysis are used to estimate the sampled noise specification on each clock state. Next, using difference equations of the circuit, time domain simulation is done by MATLAB. Based on this method, a SC integrator is analyzed and results compared to the measured noise response. Keywords: thermal noise, switched-capacitor circuit Classification: Integrated circuits References [1] F. Yuan and A. Opal, Computer Methods for Analysis of Mixed-Mode Switching Circuit, Kluwer Academic, Boston, 2004. [2] C. Gobet and A. Knob, Noise analysis of switched capacitor networks, IEEE Trans. Circuits Syst., vol. 30, pp. 37 43, Jan. 1983. [3] A. R. Gregorian and G. Temes, Analog MOS Integrated Circuits for Signal Processing, John Wiley and Sons, New York, 1986. [4] R. Schreier, J. Silva, J. Steensgaard, and G. C. Temes, Design Oriented Estimation of Thermal Noise in Switched-Capacitor Circuits, IEEE Trans. Circuist Syst., vol. 52, no. 11, pp. 2358 2368, Nov. 2005. [5] E. Hegazi and N. Klemmer, Accurate modeling of noise in switched-c ΔΣ analog-to-digital converters, IEEE Trans. Circuits Syst., vol. 52, no. 11, pp. 2319 2326, Nov. 2005. 745
1 Introduction Noise is one of the most important limitations in accurate design of analog integrated circuits. The decreasing trend in supply voltage has resulted in lower signal to noise ratio. Hence; the importance of the noise analysis is more crucial. The above shortcoming is more pronounced in SC circuit due to the aliasing effect of the sampled white thermal noise. Difficulty of noise analysis in SC circuits is rooted in the discontinuous operation of these circuits. Several sophisticated methods are presented to estimate the power spectrum density (PSD) of the output noise in SC circuits [1]. Most of these methods are based on describing the aliasing effect using series expansions in the Fourier domain and contributing each of noise sources individually by the superposition theorem. However, although these techniques are appropriate for applying in the circuit simulators, they are too complicated for first hand calculation during circuit design. Another method presented in [2] is based on an approximate analytical analysis of the circuit. Later; more elaborated analytical analysis are presented in [3, 4] based on this method. In this paper a new method for estimating the PSD of thermal noise in SC circuits is presented. In the presented method only the accumulated sampled noise which is the main part of total output noise is considered. Using HSPICE and analytical analysis, the mean square (MS) value of the sampled noise voltages are estimated carefully and then discrete white noise sources with allocated MS values are supplemented to the difference equations of the circuit. Using the equations, transient analysis is done by MATLAB. Simulation results show that the presented method is a simple accurate solution compared to the reported measured data. Note that in this way, the direct noise which is fed to the output is ignored and only the discrete-time output noise is considered, therefore all spectra in this paper are periodic. The paper is organized as follows: in Section II a discussion of different noise sources is outlined. Proposed method for estimating the PSD is presented in section III. Next in section IV, effect of thermal noise in a stray-insensitive SC integrator is analyzed using proposed method. 2 Thermal noise sources The main source of noise in SC circuit is the MOSFET transistor. When the transistor acts as a switch, the noise can be represented by a voltage source. The one-sided PSD of its voltage can be expressed in the form of S sw (f) =4kθR on (V 2 /Hz). (1) Where k is the Boltzmann constant, θ is the absolute temperature of the device and R on is the equivalent resistance of the transistor in a triode region [3]. For a MOSFET operating in active region, the thermal noise can be expressed by a voltage source in series with the gate. The PSD of this equivalent voltage source is given by ( ) 2 1 S M (f) =4kθ 3 g m (V 2 /Hz) (2) 746
where g m is the transconductance of the transistor. Since in SC circuits, only operational transconductance amplifiers (OTA) are using MOSFETS in active region, the total thermal noise of the OTA can be modelled by equivalent input referred resistance (R eq )withthepsdof S OTA (f) =4kθR eq (V 2 /Hz). (3) 3 Presented method for estimation of the PSD Fig. 1a shows the ideal track and hold circuit. The switch is assumed to be a resistance of R on. When the switch is closed in ϕ 1 = 1 interval, the resistor noise affect the capacitor s voltage and during the ϕ 1 = 0 the last noise value remains on the capacitor. Fig. 1b shows equivalent of the circuit during the time of ϕ 1 = 1 in the tracking mode. Note that the input source is ignored for simplicity while the noise analysis is considered. The PSD of the associated voltage v n is 4kθR on. The power spectral density (PSD) of the output noise voltage in this figure is given by S Out (f) = 4kθR on ( ) 1+ f 2 (V 2 /Hz) (4) f SW where f is frequency and f sw, 3-dB frequency of the circuit, is 1/(2πR on C). The MS value of the output noise voltage can be calculated by integrating the PSD from dc to infinite frequency ( ) π vout 2 =4kθR on f SW = kθ 2 C (V2 ). (5) For settling the output within 0.1% of its final value, the interval time of ϕ 1 = 1 should be greater than 7R on C [3]. Assuming this interval time to be 0.4/f s where f s is the sampling frequency, it can be shown f SW > 278f s. Fig. 1. Track and hold circuit Due to this fact, the PSD of the sampled output noise will be heavily aliased and therefore the spectrum of the sampled noise signal v out (n) is nearly white [4] where v out (n) is the value of the sampled noise at the end of the previous tracking interval (nt ). The aliasing increases the PSD of the noise but the MS value of the sampled signal v out (n) in Fig. 1a remains almost the same as v out (t) in Fig. 1b. Based on the above description, the PSD of the sampled noise can be expressed by S Sampled (f) kθ C 2 f S (V 2 /Hz). (6) 747
Note that the MS value of the sampled noise is independent of the R on and the sampling frequency. Based on the previous description, the sampled thermal noise in SC circuits is almost white and the MS noise voltage value of the sampled signal v nci (n) and the main signal v nci (t) are nearly the same where v nci (n) is the discrete representation of the i th capacitor s voltage. Unlike the simple track and hold circuit, the estimation of the MS noise value in complicated circuits is a tedious work. To alleviate this problem, the HSPICE simulator is utilized in order to estimate the MS values in different time intervals using equivalent circuits. In other word, noise analysis of the equivalent circuits is done in two states of clock separately by HSPICE to find accurate MS values of noise signal. Based on the previous discussion, the MS value of the sampled noise is almost the same as the MS value of the main signal and it is reasonable to use the calculated values for simulating the circuit in discrete time domain. Next, appropriate white noise sources calculated previously are added to the difference equations of SC circuit in time domain. Transient analysis is done by MATLAB to estimate the PSD of the output using previous difference equations. The total time interval in transient analysis has an important role for accurate estimation of the PSD. Due to the inherent property of noise signals, and the complexity of the circuit, the total time of simulation should be long enough for stationary estimation of PSD. 4 Noise analysis of an integrator Noise effects in a stray-insensitive SC integrator depicted in Fig. 2a is analyzed in this section. Clock frequency and effective resistance of switches (R on ) are assumed to be 10 khz and 3.5 kω according to the previous work in [2]. Opamp s input referred noise voltage is assumed 0.16 μv/ Hz where its unity-gain frequency is 700 khz [2]. Value of both capacitors C 1 and C 2 is 10 pf. Equivalent circuit for calculating the MS value of voltage noise on C 1 during ϕ 1 = 1 and ϕ 2 = 1 are illustrated in Fig. 2b and Fig. 2c receptively where v in = 0 is assumed for simplicity. The MS value of sampled noise on C 1 during ϕ 1 = 1 due to witches M 1 and M 3 is (kθ/c 1 ) as explained in [4]. This sampled noise voltage v nc1,1 (n) is delivered to C 2 in the next phase during ϕ 2 = 1. During the interval time ϕ 2 = 1, three noise sources contribute together as shown in Fig. 2c. Due to the fact that these noise sources are uncorrelated, the total MS value of sampled noise on C 1 can be calculated by summing the effect of every one of them. Different approximate analytical solutions are presented in [3, 4, 5] to estimate the MS value of the C 1 noise under this condition. Here, circuit shown in Fig. 2c is simulated by HSPICE. The accurate MS value of the C 1 voltage noise v nc1,2 (t) can be obtained using ac analysis of the HSPICE. The MS value is obtained from simulation result about (111.08 μv) 2. Fig. 2c also shows the simple equivalent circuit used as a opamp in this 748
Fig. 2. Stray insensitive SC integrator paper. The input referred noise v nop shown in this figure is replaced by a R eq =1.55MΩ =v 2 nop/4kθ during ac simulation with HSPICE. Based on previous description, the MS value of the sampled voltage noise on C 1 in the end of ϕ 2 = 1 (v nc1,2 (n)) can be assumed the same value obtained by HSPICE. Applying the equivalent noise sources to the difference equation of the circuit results in v o (nt )=v o (n 1)T + C 1 C 2 v i where ( n 1 2 ) T + C 1 C 2 (v nc1,1 (n)+v nc1,2 (n)) (7) vnc1,1 2 kθ (n) =(20.34 μv) 2 and vnc1,2 2 C (n) (111.08 μv)2. (8) 1 The PSD of the output voltage is shown in Fig. 3 where the total time 749
of simulation is mentioned 3.2768 seconds (2 15 sampling periods). Measured data extracted from [2] is also illustrated in this figure which is very close to the simulated results. Note that the output of the integrator also contains continues noise mounted in sampled noise. However the effect of this continues noise is considered in the presented method where the output of the circuit is sampled by the next stage like SC filters. Fig. 3. Output noise power of SC integrator ( Measurment, This paper) 5 Conclusion The effects of thermal noise generated by the switches and op-amp in the SC circuits were analysed. It was assumed that the MS value of the sampled thermal noise is nearly equal to the MS value of the main signal. This assumption allowed a simple discrete noise analysis in time domain using calculated MS values of main sampled capacitors voltage. The MS value in every state of clock was calculated using HSPICE simulator separately. Using HSPICE was a good solution to find an accurate estimate of the MS values in the complicated circuits. Time domain simulation was done by MATLAB using obtained MS values. Due to the fact that only discrete noise analysis was considered, estimated PSD was limited to the half of clock frequency. A SC integrator is analysed based on the presented method. The close agreement between estimated and reported measured noise response shows the validity of the presented technique. 750