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CMOS Optical Preamplifier Design Using Graphical Circuit Analysis

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5.1 <strong>Analysis</strong> of the Low-Voltage Transimpedance Amplifier 111<br />

The transimpedance gain of the circuit can be determined using Mason’s Direct<br />

Rule. The basic expression is<br />

v out<br />

--------<br />

i in<br />

where P1 is the forward transmission path from iin to vout ,<br />

P 1<br />

and loops L1 , L2 , and L3 are given by<br />

(5.5)<br />

(5.6)<br />

(5.7)<br />

(5.8)<br />

(5.9)<br />

By combining Equations (5.1) through (5.9), we can obtain the following expres-<br />

sion for the transimpedance gain:<br />

v out<br />

P1∆1 = ---------------------------------------------------------------------<br />

1 – ( L1 + L2 + L3) + L1L 3<br />

L 3<br />

( gm1 + gs1 ) ( – gm3 + Y f )<br />

= ---------------------------------------------------------------<br />

Y in( Y A + Y f ) ( Y f + Y L)<br />

where the coefficients of the denominator are given by<br />

a 3<br />

a 2<br />

L 2<br />

L 1<br />

– ( )<br />

= --------------------------------<br />

gm1 + gs1 Y in<br />

( )<br />

= ------------------------------------------<br />

– gm2 gm1 + gs1 Y in( Y A + Y f )<br />

( – gm3 + Y f )Yf ( Y A + Y f ) Y f + Y L<br />

= -----------------------------------------------------<br />

( )<br />

1 ⁄ R f + sC f – gm3 -------- ( s)<br />

iin a3s 3<br />

a2s 2<br />

= ----------------------------------------------------------<br />

+ + a1s + a0 Cin gm1 + gs1 = ----------------------------- [ C AC L + C f ( C A + CL) ]<br />

( )<br />

Cin gm1 + gs1 ⎛ C A + CL⎞ = ------------------------ ⎜gm3Cf+ --------------------- ⎟ + C AC L + C f ( C A + CL) ⎝ ⎠<br />

R f<br />

a1 =<br />

Cingm3 ------------------------------------<br />

( gm1 + gs1)Rf + gm2C L + ( gm2 + gm3)Cf C A + CL + ---------------------<br />

R f<br />

a0 =<br />

( gm2 + gm3) ⁄ R f<br />

(5.10)

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