Frequency Response and Band Limit of Current Mirrors

Frequency response of current mirrors C2 = + + C C C C I2 1 1 2 1 GS GS DB = C C V1 2 2 GD Vout M1 M2 I1 C1 P. Bruschi Design of Mixed Signal Circuits 1

Small signal equivalent circuit i2 C2 v1 gm2v1 1/gm1 i1 C1 P. Bruschi Microelectronic System Design 2

Frequency response of the current gain AI g g C s W L = 1 C 1 m m ( ) 1 = g C V V p + C I A m OX GS t 1 2 1 = (0) A A z dB I I s g C + 1 = ) 0 ( IA 2 m fz z fp p 2 log(f) / / g g W W L L ( ) 0 ( ) IA = 2 m A 2 2 k I M 1 1 1 m g g g C g IA ( ) = ( ) 0 = 2 1 C m m A A 2 P ( ) log(f) I I + C 1 1 2 2 Z m m -90 C + ( ) = A 2 I C C -180 1 2 P. Bruschi Microelectronic System Design 3

Frequency response of current mirrors with AI<<1 I A / / g g W W L L ( ) 0 dB fp fz = 2 m A 2 2 k I M 1 1 1 m log(f) ( IA ) ( ) 0 If 1 IA ) 0 ( IA we can expect that: C + ( ) 0 ( ) = IA 2 A A I I C C log(f) 1 2 -90 -180 P. Bruschi Microelectronic System Design 4

Upper band limit (fH) of a current mirror C2 I2 Hypothesis: fp<fzthen: fH=fp V1 Vout M1 M2 I1 W L C1 ( ) C V V 1 g C 1 OX GS t p = 1 m f = p 2 2 neglecting CDB1in C1: 2 C 1 1 2 3 ( ) + + C C C C W L W L 1 1 2 1 1 2 2 gs gs OX W L 3 1 L ( ) ( ) C V V 1 V V Tf W L W L OX GS t GS t 1 2 1 = 4 1 = f 1 2 3 p 2 W L W L ( ) + 1 + 2 2 C W L W L + 1 2 2 1 1 2 2 OX 1 1 1 1 P. Bruschi Microelectronic System Design 5

Upper band limit (fH) of a current mirror f W L W L 3 1 L = = f f 1 T ( ) = f V V H p T GS t + 2 1 1 2 2 4 1 1 Fast current mirrors: Short channel length Large overdrive voltage (VGS-Vt) Precision current mirror: f f 1 T k / / W L W L W W W W L L ( ) = = = = 2 2 2 2 2 L L k p + 1 1 2 M M 1 1 1 1 1 P. Bruschi Microelectronic System Design 6

Summary 1. The frequency response of a current mirror is marked by a pole (angular frequency p) and a zero (angular frequency z). 2. The zero (sz) is positive, than it gives a phase contribution similar to that of the pole. In total, the pole and zero give an asymptotic phase variation of 180 . 3. Generally, the pole falls at lower frequencies than the zero, thus the upper band limit is given by the pole. 4. The frequency of the zero is smaller than the pole one only for mirrors designed to have a current gain << 1. 5. In all other cases, which include unity gain current mirrors, the upper band limit is given by: f f f W L W L 3 1 L = = ( ) 1 T = f V V H p T GS t 2 1 4 + 1 2 2 1 1 P. Bruschi Microelectronic System Design 7

Noise in current mirrors Simple MOSFET current mirror 2 = = = + n out i i i A i ( ) f ( ) f ( ) ( ) f S S A f S 2 1 Iout In I In 2 2 1 n I n ( ) 0 A A For f << fH I I 2 g g 2 = + ( ) f ( ) f ( ) f 2 m = + S S S ( ) f ( ) f (0) ( ) f S S A S 2 1 Iout In In 2 1 Iout In I In 1 m P. Bruschi Microelectronic System Design 8

Thermal noise Let us assume: 8 3 8 3 8 3 = ( ) f S kTg = ( ) f S kTg 1 1 In Th m In Th m g g = ( ) f S kTg ( ) 0 = = 2 m A k 2 2 In Th m I M 1 m 2 g g = + 2 m ( ) f ( ) f ( ) f 2 m S S S 8 3 8 3 g g = + 2 1 Iout In In ( ) f 2 S kTg kTg 2 1 Iout th m m 2 m 1 m kM 1 8 3 8 3 g g 8 3 ( ) = + ( ) f 2 m S kTg kTg = + ( ) f 1 S kTg k 2 2 Iout th m m 2 Iout th m M 1 m P. Bruschi Microelectronic System Design 9

Flicker Noise Let us assume: N 1 f = 2 m ( ) f S g 1 1 In F N WL f W L f 1 f = 2 m 1 1 ( ) f S g In F N 1 f = 2 m ( ) f S g 1 2 In F W L f 2 2 2 N N 2 m g g 1 1 g g f f = + = + 2 m 2 m ( ) f 2 S g g ( ) f ( ) f ( ) f 2 m S S S 2 1 Iout F 2 1 Iout In In 2 m W L f W L f 2 2 1 1 1 1 m N N 1 1 N 1 f W L W L f f = + 2 m 2 m ( ) f S g g f = + 2 m ( ) f 1 S g 2 2 2 2 Iout F W L f W L f 2 Iout F W L 2 2 1 1 2 2 1 1 P. Bruschi Microelectronic System Design 10

Parameters that affect the output noise 8 3 ( ) = + ( ) f 1 S kTg k Thermal noise: 2 Iout th m M N 1 f W L W L f = + 2 m ( ) f 1 S g 2 2 Flicker noise: In F 2 W L 2 2 1 1 8 3 I The higher current ID2, the higher the output current PSD High values of VTE reduce noise ( ) = + ( ) f 1 S kT 2 k D Iout th M V I 2 TE N = g D Using: 2 m 1 f I W L W L V f = + ( ) f 1 S 2 2 2 D TE In F V W L 2 2 2 1 1 TE P. Bruschi Microelectronic System Design 11

Dynamic range of a current mirror = = + ( ) t I I out i out out Q = + 1( ) i t I I out FS i i 1 1 Q = DR out n dc bias = out i I signal FS 2 out Q Iout For simplicity, let us consider a sinusoidal stimulus iout-FS = Iout-Q 2 2 I I peak out Q Ipeak =2 only if we allow the minimum value of Ioutto be zero -> distortion t P. Bruschi Microelectronic System Design 12

DR of a current mirror ( ) 2 = out i I FS out Q out FS i i = DR ( ) f f = = H 4 4 np i i S df out n p n rms Iout f L 2 2 out Q I Thermal 16 S B 2 2 out Q I IBB S ( ) 2 = DR f ( ) f df H 2 2 out Q 16 S I Iout f Flicker L f f 16 ln k H F L P. Bruschi Microelectronic System Design 13

DR of a current mirror 2 8 3 I N 1 f I W L W L ( ) = + ( ) f 1 S kT 2 k D f = + ( ) f 1 2 2 2 S D In F Iout th M V V W L 2 2 2 1 1 TE 2 TE SIBB = I I kf 2 D out Q Thermal: 2 V I 3 2 2 out Q I ( ) 2 TE out Q k + = 2 = DR ( ) I 128 1 kT B 8 3 ( ) out Q V + 16 1 kT k B S M M S 2 TE Flicker: 2 2 out Q 2 2 I V W L W L W L ( ) 2 = = 2 2 2 TE DR 2 I N W L W L f f + out Q V f + 16 1 ln 16 1 ln( / ) N f f 2 2 H 2 2 f H L W L 2 2 2 1 1 TE L 1 1 P. Bruschi Microelectronic System Design 14

Examples Thermal 2 V I 3 2 TE out Q k + = DR ( ) 128 1 kT B S M DR (88 dB, 14 bit) 3 25.7 10 = = = = 1 A I out Q V k B 100 mV 1, 1kHz 2 TE = 1.5 M S P. Bruschi Microelectronic System Design 15

Examples Flicker 2 2 V W L W L W L = DR 2 2 2 TE + 16 1 ln( / ) N f f 2 2 f H L 1 1 DR 319 = = = = 1 A I out Q V k B = = 1 m 2 WL (50 dB, 8 bit) 100 mV 1, 1kHz 2 TE 6 10 = 10 2 2 m N N V = 1.5 f fn M Flicker noise dominates This is the total DR = 0.01 Hz f L S P. Bruschi Microelectronic System Design 16