Lecture 7, ATIK. Continuous-time filters 2 Discrete-time filters
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1 Lecture 7, ATIK Continuous-time filters 2 Discrete-time filters
2 What did we do last time? Switched capacitor circuits with nonideal effects in mind What should we look out for? What is the impact on system performance, like filters. Continuous-time filters The way forward, and the background to generate filters. OTA/Gm-C and active-c evisited today
3 What will we do today? Continuous-time filters Some slides from last lecture Scaling Sensitivity Noise Discrete-time filters Simulation of the continuous-time filters Discrete-time accumulators LDI transform Bilinear transform
4 Ladder networks Use an analog reference filter, values are given by table or MATLAB-ish. Typically doubly-resistive terminated for maximum power transfer. A [db] L2 i V out 20 V in C3 C1 L What is the DC gain? 2 f [khz]
5 State-space representation Note the voltages and currents through the ladder network. Introduce a helping "" to create voltages Input current I 0= V 0 V 1 I 0 = V 0 V 1 i i Inductor current V 1 V 3 I 2= I 2= V 1 V 3 s L2 s L2 X1 1 s C3 i V 0 =V i n X 3 =V out 1 s C1 s L2 X2 i Kirchhoff I 0 =V 1 s C 3 I 2, etc. Combine all equations into a flow graph L
6 Active-C implementation eplace integrators with corresponding active components 6 7 C1 V out C3 C2 2 Vin 1 3 Inversion missing! 4 5 Identify the paths and set the values accordingly
7 Transconductance-C filters (Operational) transconductance amplifier with capacitive load V in V out g m1 V out s s C L=I out s = g m1 V i n s V out s g m1 = V i n s s C L How do we sum several inputs for the Gm-C? How do we create a gain circuit with Gm-C? CL
8 Transconductance-C filters, cont'd A first-order pole equires two transconductors! V out V in g m1 g m2 CL g m1 V i n s V out s s C L g m2 V out s =0 V out s g m1 g m1 / g m2 = = V i n s g m2 s C L s 1 g m2 /C L
9 Transconductance-C filters, cont'd Leapfrog: replacement of active-c integrators with Gm-C elements. equires more active elements, but easier to tune
10 Continuous-time filters, scaling For optimal performance, we need to scale our filters Minimize distortion, i.e., minimize your swing Maximize SN, i.e., maximize your swing What norm to use in order to scale? Different nodes might peak at different frequencies!
11 Continuous-time filters, scaling example 1) Originate from flow graph and introduce scaling coefficients. 2) All inputs scaled with K and all outputs scaled with 1/K 3) Take them "one-by-one" from input towards the output X1 K2 1 s C3 i V 0 =V i n 1 K2 K1 1/ K 2 1 K 1 K 2 K 3 V out 1 s C1 s L2 X2 K3 L
12 Continuous-time filters, sensitivity General The transfer function is a function of all values, A_0,, C, etc. We can define the sensitivity as qi H S = H qi H i Leapfrog filters are less sensitive than second-order sections.
13 Continuous-time filters, sensitivity example First-order pole with active C 8 / 4 H s = s 1 1/C 2 8 C2 4 Variation as function of C C2 H C2 1 S = = H s 8 H C2 H s 1 1 /C 2 8 H C such that s C 2 8 S = 1 s C 2 8 H C What does this imply? 8
14 Continuous-time filters, noise All OP amp noise can be reverted back to their inputs Scaled with the transfer functions to the output Use the superfunction 1 C C2 7 9 V out 4 5 Vin All resistor noise is converted to output too 6
15 Going to discrete-time domain A general transfer function is given by 2 Y z a 0 a1 z a 2 z H z = = X z b 0 b1 z b2 z 2 Identify integrators in the expression and rewrite A z = z 1 (Notice that the approach is a bit different to "digital" II/FI filter design) For filter purposes we will originate from the continuous-time filter specification eventually determines our a,b values.
16 The lossless discrete integrator transform Maps just part of the frequency range 1 1 z s=s0 [ z 0.5 z 0.5 ]=s z with the scaling factor according to T a=2 s0 sin 2 which maps the analog frequencies with the discrete-time angles. We have already decided to use high oversampling ratio, so the LDI transform is sufficiently OK.
17 The bilinear transform Maps the "whole" analog LHP on the unit disk z 1 s=s0 z 1 with the scaling factor according to T a=s 0 tan 2 which maps the analog frequencies with the discrete-time angles. Bilinear transform is accurate, but sensitivity requires us to select a low scaling factor.
18 Accumulators of different flavours See the handouts for state-space realization and implementations K_0NNN_LN_leapfrogFiltersOH1_A.pdf K_0NNN_LN_leapfrogSynthesisExtra1_A.pdf K_0NNN_LN_leapfrogSynthesisExtra2_A.pdf NTIK_0NNN_LN_switcapHandout_B.pdf
19 The steps forward 1) Analog reference filter 2) Frequency scaling of reference filter 3) Signal flow graph 4) eplacing integrators with discrete-time counterparts 5) Eliminate the z^(1/2) 6) Introduce SC network 7) Map the values 8) Scale This you will do in the lab!
20 Step 1, Analog reference filter Use a reference filter to get the order and component values A [db] L2 i V out 20 V in C3 C1 L Optimum wrt. sensitivity (exercises in lessons) Example: C 1=C 3=0.5 and L 2 =1.3333, L =1000, 0 =1000 f [khz]
21 Step 2, Denormalization of reference filter 0 is the termination load 0 L n=, L n =, C n= 0 C Example values gives =1000 L= C 1=C 3= In the lab, a MATLAB script is given
22 Step 3, Signal flow graph Example third-order leapfrog X1 1 s C3 i V 0 =V i n X 3 =V out 1 s C1 s L2 X2 i L
23 Step 4, eplacing integrators with discretetime counterparts (LDI) X1 X 3 =V out 1 z 0.5 s 0 C 3 1 z 1 i 1 z 0.5 s 0 C 1 1 z 1 z 0.5 s 0 L2 1 z 1 V 0 =V i n X2 i L
24 Step 5, Eliminate the z^(1/2) X1 V out 1 1 s 0 C 3 1 z z i 1 1 s 0 C 1 1 z 1 1 s 0 L2 1 z 1 V in X2 0.5 z i 0.5 z L
25 Step 5, Eliminate the z^(1/2), cont'd Consider the product z 0.5 = 0.5 = = 2 T T z a cos j sin a 1 j s0 2 s0 Compare with a capacitor in parallel with a resistor: C= 1 2 s0 emember that originally this resistance was in parallel with capacitor and we can reduce the C value. The change in resistance can be ignored since the value is much smaller than unity. '= 2 a 1 2 s0
26 Step 5, Eliminate the z^(1/2), cont'd X1 V out 1 1 s 0 C 3 ' 1 z 1 i 1 1 s 0 C 1 ' 1 z 1 1 s 0 L2 1 z 1 V in X2 0.5 z i Sample on counter phase. L
27 Step 6, Introduce SC network a3 a4 V L a1 c1 E a0 c3 c2 b1 b2 a2 a5 a6
28 Step 7, Map the values Consider the transfer functions from node to node Example: V i n to X 1 : 1 X 1= s 0 i C 3 ' vs. a0 c1 vs. a1 c1 etc. More variables than equations and you need to fix values: Typically choose all the opamp capacitors equal All others: "as equal as possible"
29 Step 8, Scale Follows exactly the same principles as fore continous-time filters. Introduce scaling and make sure all entry points are scaled with K and all exit points are scaled with 1/K. Left as exercise and laboratory work. emember that reference filter has a DC gain of 0.5! emember that gain is dependent on sample frequency!
30 Conclusions SC filters Today computer-aided, of course. Since it is computer aided - choose the leapfrog filters for optimum sensitivity! Use fixed templates and plug in your values emember to use reasonable values i.e., in an integrated circuit, the sizes should be in pf region. This also implies high cut-off frequencies!
31 What did we do today? Continuous-time filters Wrap-up and some more conclusions Discrete-time filters Simulation of the continuous-time filters Discrete-time accumulators LDI transform Bilinear transform
32 What will we do next time? Data converters Fundamentals DACs
33 Lecture 8, ATIK Data converters 1
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