9. Switched Capacitor Filters. Electronic Circuits. Prof. Dr. Qiuting Huang Integrated Systems Laboratory
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1 9. Switched Capacitor Filters Electronic Circuits Prof. Dr. Qiuting Huang Integrated Systems Laboratory
2 Motivation Transmission of voice signals requires an active RC low-pass filter with very low ff cutoff = 3.4 khz. ff cutoff = 1 2ππππππ RR = 1 CCCCCff cutoff CC=10 pf 4.7 MΩ Such a resistor may occupy a large area when realized on an integrated circuit. How can we build this filter without resistor? 2
3 Switched Capacitor Operating Principle Transfer charge ΔQQ from potential VV 1 to potential VV 2 at a fixed rate ff c = 1 TT c Phase 1 (Φ 1 closed, Φ 2 open): QQ 1 = CCVV 1 Phase 2 (Φ 1 open, Φ 2 closed): QQ 2 = CCVV 2 Transferred charge per time TT cc : ΔQQ = CC VV 1 VV 2 Average current II 2,avg = ΔQQ = CC VV 1 VV 2 TT c TT c Equivalent resistor: RR eq = TT c = 1 CC ff c CC 3
4 Inverting Integrator Using Switched Capacitors CC 2 VV out CC 2 VV out CC 1 VV in CC 1 VV in Initial condition: VV out = 0 Phase 1: CC 1 is charged to VV in. Same amount of charge is moved to CC 2. Phase 2: CC 1 is discharged. CC 2 VV 1 out VV in 2TT 3TT nnnn c c = CC CC 22 VV 1 VV 1out in VV in TT c TTnn c = 01 CC 1 TT VV c in 2TT CC 1 c VV in nntt CC 1 VV cin 3TT c 4
5 Inverting Integrator Using Switched Capacitors CC 2 VV out CC 2 VV out CC 1 VV in CC 1 VV in Initial condition: VV out = 0 Phase 1: CC 1 is charged to VV in. Same amount of charge is moved to CC 2. Phase 2: CC 1 is discharged. CC 2 VV 1 oooooo out VV iiii nnnn 2TT 3TT cc ccc = CC CC 22 VV 1 VV oooooo 1out iiii VV iiii TT cc TTnn cc = 1 0 CCTT 1c VV iiii 2TT CC 1 cc VV in nntt CC 1c VV iiii 3TT cc Output signal VV out looks like a continuous-time signal for sufficiently small TT c. 5
6 Inverting Integrator Using Switched Capacitors Initial condition: VV out = 0 CC 2 VV out nnnn c = CC 2 VV out nn 1 TT c CC 1 VV in nntt c nn 1 VV out nntt c = VV out nn 1 TT c CC 1 CC 2 VV in nntt c = CC 1 CC 2 small TT c VV out nntt c = CC 1 TT c CC 2 lim TTc 0 nn 1 kk=0 kk=0 VV in [(nn kk)tt c ] VV in (nn kk)tt c TT c = CC nntt cvvin 1 tt dtt TT c CC 2 0 Note: differentiation would require an input-output relation: VV in nntt c = const kk VV out [ nn kk TT c ] Transform for time-discrete signals is needed in order to solve difference equation and calculate transfer function 6
7 Z-Transform Definition ZZ xx nntt c = XX zz = xx kktt c zz kk kk= Time delay Z xx nn kk TT c = zz kk XX(zz) Integration Differentiation TT c 1 zz 1 1 zz 1 TT c Mapping to Laplace domain ss = zz 1 TT c or ss = 1 zz 1 TT c (forward or backward Euler transform) Mapping to jjjj-axis zz = ee jjjjtt c = ee jj2ππππ ff c Solve difference equation of SC inverting integrator Difference equation: CC 2 VV out nntt c = CC 2 VV out (nn 1)TT c CC 1 VV in nntt c Apply Z-Transform: CC 2 VV out zz = zz 1 CC 2 VV out zz CC 1 VV in zz Transfer function: TT zz = VV out(zz) = CC 1 = CC 1 1 VV in (zz) CC 2 CC 2 zz 1 CC 2 1 zz 1 7
8 Example: SC Low-pass Filter First order low-pass filter with unity gain and ff cutoff = 3.4 khz. ττ = RRRR = 1 2ππff cutoff 47μμs CC = 10 pf RR = 4.7 MΩ SC realization (RR TT c ): ττ = CC TT CC RR CC c RR Ratio of capacitors can be realized more accurately than absolute values of RR and CC. ff c = 100 khz 3.4 khz CC RR = CC 2.1 pf ττff c 8
9 Non-Inverting SC Integrator Exactly the same circuit can be operated as non-inverting integrator only by changing the switching schedule. Phase 1: CC 1 is charged to VV in. Phase 2: Charge is transferred to CC 2. Charge on CC 2 is inverse compared to inverting integrator. VV out zz CC 2 VV 2out VV out nnnn nnnn c c = = CC 2 CC VV out 2 VV out nn nn1 TT c 1 + TT c CC 1 VV in CC 1 VVnnTT inc nntt c VV in (zz) = CC 1 1 CC 2 1 zz 1 CC 1 VV in zz = CC 2 1 zz 1 VV out zz 9
10 Switched Capacitor Tow-Thomas Biquad All resistors are replaced by switched capacitors. Non-inverting integrator can be realized with only one stage. 10
11 Switched Capacitor Tow-Thomas Biquad (2) TT zz = VV out zz VV in (zz) = CC RR4 CC RR 3 zz 2 +zz 1 2 CC RR1 CC1 CC RR 2 CC RR3 CC1CC2 +1+CC RR1 CC1 +CC RR2 CC RR3 CC1CC2 Design equations: CC RR4 = kk CC RR3 CC CC RR3 2 RR2 = CC RR = 3 = ωω 1 CC 0 ωωtt c CC 0 TT c 2 2 CC RR1 CC RR3 = = ωω QQ 0TT c CC RR1 QQ General form of continuous-time low-pass filter is transformed to discrete-time filter by backward Euler transform ss = 1 zz 1 TT ss = kkωω 0 2 ss 2 + ωω 0 QQ ss + ωω TT zz 2 0 zz 2 + zz 1 2 ωω 0TT cc QQ kk ωω 0 TT cc ωω 0TT cc QQ CC 1 TT c : CC 1 + ωω 0 TT cc 2 11
12 SC Ladder Filter Ladder filter can be realized without inductors and without resistors. All RR nn are replaced by corresponding CC RRnn. 12
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