ee541F10FinalSolution

ee541F10FinalSolution - EE 541 University of Southern...

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EE 541 University of Southern California Viterbi School of Engineering J. Choma. Final Examination Solutions 1 Fall Semester, 2010 U niversity of S outhern C alifornia USC Viterbi School of Engineering Ming Hsieh Department of Electrical Engineering EE 541: Final Examination 14 December 2010 (SOLUTIONS) 11:00 -to- 1:00 Problem #1: Solution (25%) The circuit shown in Figure (E1) is a simplified model of a MOSFET technology amplifier that combines shunt peaking via the inductance L 1 with series peaking through induc- tance L 2 to achieve a broadbanded network. The two inductances utilized in the circuit are un- coupled. The input signal is the voltage, V s , while the output response to this signal is voltage V o . It can be demonstrated that the input/output voltage transfer function, H n (p) , normalized to the voltage gain realized at zero frequency is of the form, 2 n 22 1k Qp H( p ) , 1pQp + = ++ where “p” is complex frequency “s” normalized to the uncompensated circuit bandwidth, say B u . Specifically, B u represents the radial 3-dB bandwidth that is realized when L 1 = L 2 = 0 . gV ms L 1 L 2 R L C L V o Figure (E1) (a). Determine parameter “k” in terms of the two inductances, L 1 and L 2 . The input/output (I/O) transfer function, H(s) = V o /V s , is easily confirmed to be () 1 L1 L o L mm L 2 s LL 1 2 L L12 L 1 sL Rs L 1 sC V R H(s) g g R , 1 V 1s RC s L LC Ls L ⎡⎤ ⎛⎞ + + ⎢⎥ ⎜⎟ ⎝⎠ == = + +++ ⎣⎦ (E1-1) where the zero frequency value, say H(0) , of the I/O voltage gain is clearly seen to be –(g m R L ) . Moreover, we note that when L 1 = L 2 = 0 , the transfer relationship in (E1-1) reduces to 12 LL0 L L H(0) , RC = + (E1-2) which clearly projects an uncompensated 3-dB bandwidth, B u , (i.e. the bandwidth of the circuit with the inductors replaced by short circuits) of
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EE 541 University of Southern California Viterbi School of Engineering J. Choma. Final Examination Solutions 2 Fall Semester, 2010 u L L 1 B . R C = (E1-3) By inspection of the denominator of the transfer function in (E1-1), the undamped natural fre- quency of oscillation (also known as the “self-resonant frequency”), ω n , is () n 12 L 1 ω , LL C = + (E1-4) while the quality factor, Q , of the circuit derives from nu 11 R C Q ω B == . (E1-5) Clearly, u n B Q ω = . (E1-6) It now follows that uLL uu ss sRC BRC 1 p, BB ⎛⎞ ⎜⎟ ⎝⎠ = (E1-7) and, appealing to (E1-4) and (E1-6), 22 2 2 u L 2 un n B 2 2 s C Q p B ω ω += = = . (E1-7) Finally, in the numerator of the transfer relationship L 2 u 1 1 u 2 1 2 L 1 2 u 1 2 n C B sL sL L L s B pQ , RR L L C L L B L L ω + = = ++ + (E1-8) where once again, (E1-4) and (E1-6) have been exploited. It can now be seen that parameter k in the given transfer relationship is 1 L k = + .
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ee541F10FinalSolution - EE 541 University of Southern...

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