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Switched Capacitor Filters Franco Maloberti F. Maloberti: Switched Capacitor Filters 1 OUTLINE • Switched capacitor technique • Biquadratic SC filters • SC N-path filters • Finite gain and bandwidth effects • Layout consideration • Noise F. Maloberti: Switched Capacitor Filters 2 SWITCHED CAPACITOR TECHNIQUE • An active filter is made of op-amps, resistors and capacitors. • The accuracy of the filter is determined by the accuracy of the realized time costants since the capacitors and resitors are realized by uncorrelated technological steps δτ ----- τ 2 δR 2 δC 2 = ------- + ------ R C • In CMOS technology δR ⁄ R ≈ 40% ; δC ⁄ C ≈ 30% ; hence unacceptable for most of the applications δτ ----- ≈ 50% , τ •Hybrid realization with functional trimming •Problems for a fully integrated realization F. Maloberti: Switched Capacitor Filters 3 • Accuracy • Values of capacitors and resistors: for 70 nm oxide thickness 1 pF --> 2000 µ2; 10 pF is a large capacitance. To get τ = 10-4 sec R = 107 Ω The above problems are solved by the use of simulated resistors made of switches and capacitors. MOS technology is suitable because: •Offset free switches •Good capacitors •Satisfactory op-amps F. Maloberti: Switched Capacitor Filters 4 Simple SC structures I V1 Φ1 Φ2 C Φ1 1 I Φ2 V1 Φ1 C V2 1 T V2 Φ2 T ∆Q = C1 (V1 - V2) every ∆t = T F. Maloberti: Switched Capacitor Filters 5 I I V1 – V2 ∆Q = i∆t = -------------------T R V1 V2 T t The two SC structures are (on average) equivalent to a resistor T R eq = ------C1 If the SC structures are used to get an equivalent time constant τeq = ReqC2 it results: τ eq F. Maloberti: Switched Capacitor Filters C2 = T ------C1 6 • Its accuracy depends on the clock and on the capacitor matching accuracy • If τeq=40 T C2 = 40 C1 (acceptable spread) regardless of the value of τeq A more complex SC structure: V1 Φ1 Φ2 Φ2 Φ1 V2 ∆Q = 2C 1 ( V 1 – V 2 ) The charge is transferred twice per clock period T or we assume as clock period half of the period of phases Φ1 and Φ2. F. Maloberti: Switched Capacitor Filters 7 SC INTEGRATOR C2 R1 _ + Starting from the continuous-time circuit of the Integrator, we can obtain a SC integrator by replacing the continuous-time resistor with the equivalent resistances. F. Maloberti: Switched Capacitor Filters 8 C2 Φ2 Φ1 C1 _ Φ1 + C2 Φ2 _ Φ1 C1 Φ1 + C2 Φ2 Φ1 C Φ2 F. Maloberti: Switched Capacitor Filters _ 1 Φ1 + Φ2 Φ1 9 •We consider the samples of the input and of the output taken at the same times nT (the end of the sampling period). • Structure 1: C1 V out [ ( n + 1 )T ] = V out ( nT ) – ------- V in ( nT ) C2 taking the z-transform: C1 V out ( z ) 1 ------------------- = – ------- ⋅ -----------C2 z – 1 V in ( z ) • Structure 2: C1 V out [ ( n + 1 )T ] = V out ( nT ) – ------- V in ( n + 1 )T ] C2 taking the z-transform: F. Maloberti: Switched Capacitor Filters 10 C1 V out ( z ) z ------------------- = – ------- ⋅ -----------C2 z – 1 V in ( z ) • Structure 3: C1 V out [ ( n + 1 )T ] = V out ( nT ) – ------- { V in [ ( n + 1 )T ] + V in ( nT ) } C2 taking the z-transform: C1 z + 1 V out ( z ) ------------------- = – ------- ⋅ -----------C2 z – 1 V in ( z ) Remember that for the continuous-time integrator: V out ( s ) 1 ------------------- = – -----------------V in ( s ) sR 1 C 2 Comparing the sampled-data and continuous-time transfer functions we get: F. Maloberti: Switched Capacitor Filters 11 • Structure 1: T R 1 → ------C1 • FE approximation 1 (z – 1) s → --- ----------------T z BE approximation Structure 2: T R 1 → ------C1 • 1 s → --- ( z – 1 ) T Structure 3: T R 1 → ---------2C 1 2 (z – 1) s → --- ----------------T (z + 1) Bilinear approximation •It does not exist a simple SC integrator which implement the LD approximation. •Note: the cascade of a FE integrator and a BE integrator is equivalent to the cascade of two LD integrators. F. Maloberti: Switched Capacitor Filters 12 C2 C1' _ Φ1 C2' _ Φ2 C1 + Φ1 Φ2 + •The key point is to introduce a full period delay from the input to the output F. Maloberti: Switched Capacitor Filters 13 •The same result is got with: C2' C2 _ Φ1 Φ2 C1 F. Maloberti: Switched Capacitor Filters + Φ2 _ Φ1 C1' + 14 STRAY INSENSITIVE STRUCTURE The considered SC integrators are sensitive to parasitics. Toggle structure: • The top plate parasitic capacitance Ct,1 is in parallel with C1 • It is not negligible with respect to C1 and it is non linear • The top plate parasitic capacitance Ct,1 acts as a toggle structure Bilinear resistor: F. Maloberti: Switched Capacitor Filters Φ2 Φ1 C1 Ct,1 Cb,1 Φ2 C1 Φ1 Ct,1 Cb,1 15 • • • Both the parasitic capacitances Ct,1, Cb,1 act as toggle structures. Their values are different (of a factor ≈ 10) and they are non linear. Stray insensitivity can be got for the first two structures if one terminal is switched between points at the same voltage. The right-side parasitic capacitor is switched between the virtual ground and ground (note: even in DC Vv.g. must equal Vground) F. Maloberti: Switched Capacitor Filters Ct,1 Φ1 C1 Φ2 Φ2 Φ1 Cb,1 C1 Φ1 Φ2 Φ1 Φ2 Virtual ground C1 Φ1 Φ2 Φ1 Virtual ground Φ2 16 • The left side capacitor is connected, during phase 1, to a voltage (or equivalent) source. • The charge injected into virtual ground is important, not the one furnished by the input source. • Structure A is equivalent to the toggle structure, but the injected charge has opposite sign. • Equivalent negative resistance allows to implement non inverting integrators. • It is possible to easily realize a stray insensitive bilinear resistor with fully differential configuration. F. Maloberti: Switched Capacitor Filters 17 SC BIQUADRATIC FILTERS Consider a (continuous-time) biquadratic transfer function 2 p 0 + sp 1 + s p 2 H ( s ) = ---------------------------------------ω0 2 2 s + s ------- + ω 0 Q0 If the bilinear transformation is applied, it results a z-biquadratic transfer function 2 a 0 + za 1 + z a 2 H ( s ) = ---------------------------------------2 b 0 + zb 1 + z b 2 where the coefficients are: 2 4 a 0 = p 0 – ---p 1 + -----2- p 2 T T F. Maloberti: Switched Capacitor Filters 18 8 a 1 = 2p 0 – -----2- p 2 T 2 4 a 2 = p 0 + ---p 1 + -----2- p 2 T T b0 = ω0 2 2 ω0 4 – --- ------ + -----T Q T2 2 8 b 1 = 2ω 0 – -----2 T b2 = ω0 F. Maloberti: Switched Capacitor Filters 2 2 ω0 4 + --- ------ + -----T Q T2 19 All the stable z-biquadratic transfer functions are realized by the topology: E C F D 1 G B - A H + V01 + V02 I J F1 F2 Vin F. Maloberti: Switched Capacitor Filters t 20 Features: •Loop of two integrators one inverting and the other noninverting. •Damping around the loop provided by capacitor F or (and) capacitor E (usually only E or F are included in the network). •Two outputs available V0,1 V0,2. •Denominator of the transfer function determined by the capacitors along the loop (A, B, C, D, E, F). •Transmission zeros (numerator) realized by the capacitors (G, H, I, J). •Input signal sampled during Φ1 and held for a full clock period •Charge injected into the virtual ground during Φ1. F. Maloberti: Switched Capacitor Filters 21 Charge conservation equations: DV0,1(n+1) = DV0,1(n) - GVin(n+1) + HVin(n) - CV0,2(n+1) - E[V0,2(n+1) - V0,2(n)] (B + F)V0,2(n+1) = BV0,2(n) + AV0,1(n) - IVin(n+1) + JVin(n) Taking the z-transform and solving, it results: 2 V 0, 1 ( IC + IE – GF – GB )z + ( FH + BH + BG – JC – JE – IE )z + ( EJ – BH ) H 1 = ----------- = ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------2 V in ( DB + DF )z + ( AC + AE – 2DB – DF )z + ( DB – AE ) 2 V 0, 2 DIz + ( AG – DI – DJ )z + ( DJ – AH ) H 2 = ----------- = ------------------------------------------------------------------------------------------------------------------------------------------2 V in ( DB + DF )z + ( AC + AE – 2DB – DF )z + ( DB – AE ) • 10 Capacitors • 6 Equations a0, a1, a2, b0, b1, b2 • Dynamic range optimization F. Maloberti: Switched Capacitor Filters 23 • Scaling for minimum total capacitance in the groups of capacitors connected to the virtual ground of the op-amp1 and the op-amp2. • Since there are 9 conditions, one capacitor can be set equal to zero E=0 “F type” F=0 “E type” Firstly the 6 equations are satisfied. Later capacitors D and A are adjusted in order to optimize the dynamic range. Finally all the capacitor connected to the virtual ground of the op-amp are normalized to the smaller of the group. F. Maloberti: Switched Capacitor Filters 24 Scaling for minimum total capacitance Cn C1 _ C2 + C3 C4 Assume that C3 is the smallest capacitance of the group. In order to make minimum the total capacitance C3 must be reduced to the smallest value allowed by the technology (Cmin) • Multiply all the capacitors of the group by C min k = -----------C3 F. Maloberti: Switched Capacitor Filters 25 SC LADDER FILTERS Orchard’s observation Doubly-terminated LC ladder network that are designed to effect maximum power transfer from source to load over the filter passband feature very low sensitivities to value component variation. Syntesis of SC Ladder Filters: Symple approach • Replace every resistance Ri in an active ladder structure with a switched capacitor Ci = T/Ri. • Use a full clock period delay along all the two integrator loop (it results automatically verified in single ended schemes). It results an LD equivalent, except for the terminations. F. Maloberti: Switched Capacitor Filters 27 Quasi LD transformation: Attenuation DESIRED SPECIFICATION Asb Apb wpb wsb w Attenuation PREWARPED SPECIFICATION Asb Apb sin( w pb T/2) w sin( w sb T/2) Prewarp the specifications using sin(ωT/2) F. Maloberti: Switched Capacitor Filters 28 Effect of the terminations: C3 R C2 _R 1 _ C1 + C2 _ + R3 C1 H DI ( s ) = --------------------------------------- if R1 = T/ C1 and R3 = T/C3 we get: H DI ( s ) = ---------------------------sC 2 R 1 R 3 + R 1 sTC 2 + C 3 V out ( n + 1 ) ( C 2 + C 3 ) = V out ( n )C 2 + C 1 V in ( n ) F. Maloberti: Switched Capacitor Filters 29 Taking the z-transform we get: zV out ( C 2 + C 3 ) = C 2 V out + C 1 V in –1 ⁄ 2 C1 C1 z H DI ( z ) = ----------------------------------------- = --------------------------------------------------------------------1⁄2 –1 ⁄ 2 1⁄2 C 2 ( z – 1 ) + zC 3 C2 ( z –z ) + z C3 along the unity circle z=ejωT – j ωT ⁄ 2 H DI ( e jωT – j ωT ⁄ 2 C1 e C1 e ) = ------------------------------------------------------------------------------------- = --------------------------------------------------------------------------------jωT ⁄ 2 – j ωT ⁄ 2 jωT ⁄ 2 ωT ωT C2 ( e –e )+e C3 2j ( C 2 + C 3 ) sin -------- + C 3 cos -------2 2 The half clock period delay will be used in the cascaded integrator in order to get the LD transformation • The termination is complex and frequency dependent. • The integrating capacitor C2 must be replaced by C2 + C3/2. F. Maloberti: Switched Capacitor Filters 30 Complex termination: C3 C1 C2 _ F1 + Note: the output voltage changes during Φ 2 C2 V out ( n + 1 )C 2 = V out ( n ) -------------------- + C 1 V in ( n ) C2 + C3 Taking the z-transform: C2 C3 zV out C 2 = V out C 2 – -------------------- + C 1 V in C 2 + C 3 F. Maloberti: Switched Capacitor Filters 31 –1 ⁄ 2 C1 z C1 H DI ( z ) = ---------------------------------------------------- = ------------------------------------------------------------------------------------C2 C3 1⁄2 –1 ⁄ 2 –1 ⁄ 2 C2 C3 C 2 ( z – 1 ) + -------------------------------------C2 ( z –z )+z C2 + C3 C2 + C3 along the unity circle z=ejωT – j ωT ⁄ 2 C1 e jωT H DI ( e ) = ---------------------------------------------------------------------------------------------------------------C2 C3 ωT 1 C2 C3 ωT 2j C 2 – --- -------------------- sin -------- + -------------------- cos ------- 2 C2 + C3 2 C 2 + C 3 2 • • The imaginary part of the contribution of the termination is negative The integrating capacitor F. Maloberti: Switched Capacitor Filters C2 must be replaced by 1 C2 C3 C 2 – --- -------------------2 C2 + C3 32 Example: 5th order filter RS IS L2 L4 V1 V3 Vin I2 Passive prototype R6 _ V2 + 1 F. Maloberti: Switched Capacitor Filters 1 _ R/R 6 + τ5 T τ4 T 1 6 1 τ3 T 1 _V + 1 τ2 T + - 1 V4 + + τ1 T _1/s τ5 1 1 implementa- _ 1/s τ 4 _ + + + _ 1/s τ 3 _ V5 - Vs _1/s τ 2 _ + _1/s τ1 R/Rs _ - + V3 _ _ C5 + _ V1 _ C3 - Vin SC tion I6 I4 C1 Flow diagram Vout V5 1 1 33 FINITE GAIN AND BANDWIDTH EFFECT C2 C1 _ + If the op-amp has finite gain A0 the “virtual ground” voltage is V0/A0 V0 ( n + 1 ) 1 1 C 2 V 0 ( n + 1 ) 1 + ------ = C 2 V 0 ( n ) 1 + ------ – C 1 V in ( n + 1 ) + ------------------------ A 0 A 0 A0 z-transforming: C1 z Vo ( z ) H ( z ) = --------------- = – ---------------------------------------------------------------V in ( z ) C1 1 C 2 1 + ------ ( z – 1 ) + ------- z A0 A 0 F. Maloberti: Switched Capacitor Filters 53 Comparing H(z) with the transfer function with A0 → ∞ C1 z H id ( z ) = – -----------------------C2 ( z – 1 ) H id ( z ) H id ( z ) H id ( z ) H ( z ) = ---------------------------------------------------------- = ----------------------------------------------------------------------------------- = ---------------------------------------------------------------------------------------C1 C1 C1 C1 z + 1 z 1 1 1 1 1 1 1 1 + -----1 + ------- + --------------- ------------ + --- + --- + --------------- ----------- 1 + ------- + --- --------------- + ------------------- ----------- A 0 C 2 A 0 z – 1 2 2 A 0 C 2 A 0 z – 1 A 0 2 C 2 A 0 2C 2 A 0 z – 1 Substituting z = esT, on the imaginary axis jωT jωT H id ( e ) H id ( e ) jωT H ( e ) = ----------------------------------------------------------------------------------------------------- = ------------------------------------------C1 C1 1 – m ( ω ) – jθ ( ω ) 1 1 + ------- + ------------------- – j --------------------------------------------------A0 2 C2 A 0 2C 2 A 0 tan ( ωT ⁄ 2 ) Magnitude error Phase error C1 1 m ( ω ) = – ------ 1 + ----------- A0 2 C 2 C1 C1 θ ( ω ) = ------------------------------------------------ ≅ ----------------------2C 2 A 0 tan ( ωT ⁄ 2 ) C 2 A 0 ωT F. Maloberti: Switched Capacitor Filters 54 For the noninverting integrator C2 C1 _ + V0 ( n + 1 ) 1 1 C 2 V 0 ( n + 1 ) 1 + ------ = C 2 V 0 ( n ) 1 + ------ + C 1 V in ( n ) + ----------------------- A 0 A 0 A0 z-transforming and solving C1 Vo ( z ) H ( z ) = ---------------- = ---------------------------------------------------------------V in ( z ) C1 1 C 2 1 + ------ ( z – 1 ) + ------- z A0 A 0 Same magnitude and phase error result F. Maloberti: Switched Capacitor Filters 55 FULLY DIFFERENTIAL CIRCUITS •Fully differential configurations reduce the clock feedthrough noise and increase the dynamic range. •They allow an increase design flexibility C2 C1 (Φ2) _ Φ2 Φ1 Φ2 Φ1 (Φ1) + Φ2 Simple integrator (inverting and non inverting) F. Maloberti: Switched Capacitor Filters 66 Immediate sampling (inverting and non inverting) integrator: Vin -Vin Φ1 Φ2 Φ1 Φ2 _ Φ1 -Vin Vin + Φ1 Φ2 Φ1 Φ1 Φ2 Delayed sampling (inverting and non inverting) integrator: Vin -Vin Φ1 Φ2 Φ1 Φ2 _ Φ2 -Vin Vin F. Maloberti: Switched Capacitor Filters + Φ1 Φ2 Φ1 Φ2 Φ2 67 •It is possible to reduce the op-amp finite bandwidth dependence by the use of delayed sampling inverting and non inverting integrators along a second order loop. Φ1 Φ1 Φ2 Φ2 Φ1 Φ1 Φ2 Φ2 _ + + _ Φ2 Φ2 Φ1 Φ2 F. Maloberti: Switched Capacitor Filters Φ1 Φ1 Φ1 Φ2 68 •The peaking in the frequency response due to the phase error is strongly reduced •It is easy to realize bilinear integrators Vin C1 C2 Φ1 Φ1 Φ2 Φ2 _ C1 _V in Φ2 + Φ1 Φ1 Φ2 C2 F. Maloberti: Switched Capacitor Filters 69 NOISE IN SC CIRCUITS The noise sources in a SC network are: • Clock feedthrough noise • Noise coupled from power supply lines and substrate • kT/C noise • Noise generators of the op-amp The first two sources are the same as in mixed analog-digital circuits. kT/C noise: F. Maloberti: Switched Capacitor Filters 74 Consider the simple network: vin S1 In the “on” state the switch can be modeled with a noisy resitor C Noise equivalent circuit: Ron S1 C 4kTRon f The white spectrum of the “on” resistance is shaped by the low pass F. Maloberti: Switched Capacitor Filters 75 action of the RonC filter. The noise voltage across the capacitor C has spectrum: 4kTR on ∆f 2 2 S n,c = v n, c = 4kTR on H ( f ) ∆f = ---------------------------------------2 1 + ( 2πfR on C ) When the switch is turned “off” the noise voltage vn,c is sampled and held onto C S F. Maloberti: Switched Capacitor Filters f 76 The folding of the spectrum in band-base gives a white spectrum. * v n,c f CK/2 f It power (the dashed area) is equal to the integral of Sn,c 2 v n, c ∞ 4kTR on ∆f 4kT ∞ kT = ∫ ----------------------------------------2- df = ----------- ( atan x ) 0 = ------2πC C 0 1 + ( 2πfR on C ) Procedure for the noise calculation in SC networks: F. Maloberti: Switched Capacitor Filters 77