1 Leapfrog Filter Synthesis Extra info

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1 TST80, ATI eapfog Filte Synthesis xta info Page (7 eapfog Filte Synthesis xta info. Use the filte specification. Pefom needed tansfomatioms (BP/HP/P/BS. A min. Identify the filte ode and component values fo the passive filte implementation. Find the ladde stuctue. Unnomalize the component values.. Intoduce cuents and voltages in the ladde netwok. A max 0 ω c ω s ω I 0 I I I N- IN Y i Y Y N N- Z Z Z N Y. Find the equations detemining the elation between the I i and i. I 0 = Y i (, = Z ( I 0 I,... I k = Y k ( k k, k = Z k ( I k I ( k,... I N = Y ( N 0, N = (dependent on the filte ode 5. Nomalize all equations with the constant to achieve voltage-unit vaiables Z I 0 = Y i (, = ( I,... 0 I Z k I k = Y k ( k k, k = ( I,... k I ( k I N = Y ( N 0, N = (dependent on the filte ode. eate the signal flow gaph descibing the equation system. xample fo a thid ode filte: / i /s /s /s / - - I 0 - I I - J. Jacob Wikne Depatment of lectical ngineeing, inköpings univesitet

2 TST80, ATI eapfog Filte Synthesis xta info Page (7 Modify the gaph by popagating the potentials, e.g., : - s though the net and notify with negative = -/s -/s / i / /s I 0 / i -I I In the filte, we have thee amplifying and summating integatos: s, s, and s. A summating integato can be implemented using active components. The tansfe function fo this integato becomes: N N /s N = s N This is a inveting integato. This implies that the integato s has to use an inveting buffe to achieve a pope signal path. This gives the following stuctue: 0-7 Z 8 = 5 5 inveting buffe X -I Y 9 0 Now, the component values have to be identified. This is done by compaing the (oiginal signal flow gaph with the active netwok. Fo example: J. Jacob Wikne Depatment of lectical ngineeing, inköpings univesitet

3 TST80, ATI eapfog Filte Synthesis xta info Page (7 Active eapfog Signal flowgaph esult [ = [ s = = i s s i = i [ [ I = [ s 5 s = I [ I ---- = = i s s i 5 = i I s = [ I ( = [ I s 5 7 = ( s 5 7 = ( [ I = [ I s 5 = s 5 8 = [ I s = I [ I 9 I s 9 = [ = [ s = s = s 0 = In ode to achieve good matching popeties, we choose the capacitances to be equally lage. Fo example: = 5 = = 0nF (This may be somewhat too lage in an integated cicuit. Fom the equations above, we see that and i.nf kω = 5 = = =.kω och 0nF 0 = =.kω = 9 = , 7 = 8 = Fo symmety and hence pope matching, we choose = 7 = 8 = 9, and Finally = = = kΩ 5 = 7 = 8 = 9 =..kω.7kω The esistance values of the inveting buffe,, also have to be detemined. hoose (fo matching fo example = =.7kΩ Note that can be chosen so that the dc gain is. This is done by educing to half its value (scaling of the entie filte. J. Jacob Wikne Depatment of lectical ngineeing, inköpings univesitet

4 TST80, ATI eapfog Filte Synthesis xta info Page (7 Special case: aue Filtes xample: onside the aue filte. I I 0 I I i The equations fo the voltages and cuents ae (unnomalized I 0 = , I,, = ( i I = = ( I, and. s 0 I = ( I s I Fo voltage-unit vaiables, we have nomalized with : I 0 = ---- (,,, I i s ( s = ( I = = ( I, and. s 0 I = ( I s I The inductance and capacitance in paallel intoduces some complexity to the signal flow gaph. The capacitance can be eliminated by intoducing voltage dependent souces to the cicuit. i /( /( xamin the help cuent, I'. Now we can ewite the equations as I = I ' s ( and I ' = (. s This gives J. Jacob Wikne Depatment of lectical ngineeing, inköpings univesitet

5 TST80, ATI eapfog Filte Synthesis xta info Page 5 (7 = ( I s 0 I ' s ( = s ( ( I I ' And simila = s ( ( I ' I The coesponding signal flow gaph becomes: - - /( /( /s( / i /s( - /s - / The s ae eliminated: - = - /( - /( / i -/s( /s -/s( / / i -I J. Jacob Wikne Depatment of lectical ngineeing, inköpings univesitet

6 TST80, ATI eapfog Filte Synthesis xta info Page (7 The active filte implementation becomes: Note the use of 7 and 8 as summation elements in the integatos. omponent values ae identified as in the pevious case. Gm- implementasttion (Non-elliptic The same signal-flow gaph is used, but now we inset the coesponding integating and summating elements descibed by Gm- building blocks. g m g m5 - g m 5 g m7 g m -I g m g m J. Jacob Wikne Depatment of lectical ngineeing, inköpings univesitet

7 TST80, ATI eapfog Filte Synthesis xta info Page 7 (7 Gm- Signal flow gaph esult g m [ = [ s = = i s s i = g m i [ g m = s [ = = s i s i = g m i [ I [ I g m = ( I s [ I g m = s [ I 5 I s = g m = ( s 5 = g m g m5 [ I = s [ I = s 5 = g m5 [ I g m Special case: aue Filtes = ( I s [ I I s = g m g m7 [ = s [ = s = s = g m7 g m g m5 - g m8 g m g m9 gm0 g m g m g m 5 g m7 -I g m g m g m g m9 With: = = and g m0 α g m = g m8 J. Jacob Wikne Depatment of lectical ngineeing, inköpings univesitet