ee541F10HWSolutions06

ee541F10HWSolutions06 - U niversity of S outhern C...

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U niversity of S outhern C alifornia USC Viterbi School of Engineering Ming Hsieh Department of Electrical Engineering EE 541: Solutions, Homework #06 Fall, 2010 Due: 11/02/2010 Choma Solutions Problem #26: Five operational transconductors (OTAs) and several capacitors are interconnected as shown in the schematic diagram of Figure (P26). The filter realizes a voltage transfer func- tion, H(s) = V o /V i , that assumes the biquadratic form, ( ) 2 o o 2 i nn H 1 as bs V H(s) . V ss 1 Q ωω ++ == ⎛⎞ ⎜⎟ ⎝⎠ + + G 1 G 4 + G 2 2C x 2C x + G 3 2C y 2C y + V o + G 5 V c1 V c4 V C2 c3 V c5 2C z 2C z V i + Figure (P26) (a). Using the modeling guidelines documented on Page #134 of Lecture Aid #3 , derive the ex- pression for the network transfer function. Specifically, derive relationships, in terms of network parameters, for the zero frequency gain, H o , quality factor Q , self-resonant fre- quency ω n , and parameters a and b . Figure (26.1) displays a roadmap of node voltages and branch currents that are used to compile the relevant KVL and KCL equations of equilibrium for the subject biquadratic filter. For the current, I y , conducted by the capacitances, (2C y ) , in the filter diagram,
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EE 541 University of Southern California Viterbi School of Engineering Choma Solutions, Homework #06 69 Fall Semester, 2010 y o y 2I 0 2sC =− V , (P26-1) + + G 1 G 4 + G 2 2C x 2C x + G 3 2C y 2C y + V o + G 5 V c1 V c4 V C2 c3 V c5 2C z 2C z V i + + + V o V o GV 3o 0 0 + V o 1o 4i 22 5i + V 2 I y I y + V i I z I z I z I z I z I z I GV z5 i i Figure (P26.1) which promulgates yy Is C V = o . (P26-2) For the electrical loop embracing voltage V i , voltage V o , and capacitances (2C z ) , z i z V 2sC =+ o V , ) . (P26-3) which implies, ( zz i o C V V (P26-4) Note that this current must ultimately be supplied by the signal source, which suggests that the Thévenin resistance of said signal source should be small to ensure that most of the Thévenin sig- nal voltage is applied directly to the input port of the fourth transconductor. The last of the pre- liminary disclosures derives from KVL applied to the capacitive branches containing capacitances (2C x ) . Specifically, () 2 x 2GV 0 2sC = V , + (P26-5) which delivers 2 x V sC = . . (P26-6) KCL applied to the node at which capacitance (2C y ) is incident now delivers i 2 2y3 o IG V G VIG V −= −+ + (P26-7) If variables I z , V 2 , and I y in this relationship are eliminated through use of (P26-4), (P26-6), and (P26-2), the voltage transfer function, H(s) = V o /V i , can be formulated. After some algebraic gym- nastics, this transfer function is seen to be
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EE 541 University of Southern California Viterbi School of Engineering Choma Solutions, Homework #06 70 Fall Semester, 2010 () 2 x5 xz 24 o4 i1 xy z 2 x3 12 CG CC 1s s GG VG H(s) . CC C s ⎧⎫ ⎛⎞ ⎪⎪ −+ ⎜⎟ ⎝⎠ == ⎨⎬ + ++ ⎩⎭ (P26-8) A comparison of this disclosure with the originally provided form of the voltage transfer function leads to the conclusions, nn ab .
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ee541F10HWSolutions06 - U niversity of S outhern C...

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