.EQ delim @@ .EN .sp 0.25i \fBTitle\fP .ce Brokaw: Brokaw Voltage Reference .sp 0.25i \fBDescription\fP .P This is an implementation of the Brokaw voltage reference (IEEE JSSC, Vol SC-9, pp. 388-393). At the desired DC operating point, node [3] provides a reference voltage. Another stable state is possible in which node [3] swings high almost to VCC. Finally, in the unstable state, node [3] is at approximately 9 volts. Note the \fBscaling factors\fP on Q5 and Q1, which are crucial for proper operation. .P The following describes the states of the circuit in a bit more detail. An increase in voltage v[3] above the band-gap voltage @V sub bg approx@ 1.2 volts will cause current @I sub C2@ (collector current of Q2) to increase. Because of emitter degeneration from R1, @I sub C2@ increases much faster than @I sub C1@ for a given increase in v[3]. In normal operation, the increase in @I sub C2@ is mirrored by Q4-Q5 resulting in a reduction in @V sub BE8@. Negative feedback therefore occurs since v[14] and v[3] drop. Resistors R4 and R5 are used to improve the control over the current ratio @I sub C5@/@I sub C4@, since the matching of PNP saturation currents (@I sub s@) may not be tight in some IC fabrication processes. Undesirable behavior can occur when R4 and R5 are too large. .P If the voltage v[3] is ver large, then current @I sub C2@ will produce a large voltage drop across R4 and R5. Q4 will saturate when the drop across R4 approaches .DS I .EQ ( v[3] - V sub BE2 ) R5 over R2 approx V sub BE8 . .EN .DE A further increase in v[3] will increase the drop across R5 and also increase @V sub BE8@, since @V sub CE4 approx 0@ and Q4 is no longer an inverter. Positive feedback prevails. Conceivably, the unstable state could be traversed by during the charging or discharging of a capacitive load, and the circuit may latch up into the "other" stable state, in which the output voltage is incorrect. .P The convergence of a proprietary simulator depends on the .NODESET value given for node [3]. For some values of .NODESET, the simulator will converge to one or the other of the two stable solutions. For other values of .NODESET, the simulator does not converge. .sp 0.25i \fPDescription of successful operating point computation, if available\fP .P A homotopy method has been used to identify all three DC solutions of this circuit. .bp .sp 0.25i \fBSpice File\fP .DS I \f(CW *-- Brokaw voltage reference * * This circuit has multiple DC solutions * Default models are adequate for nr100n and pr100n , * but scale parameters as indicated are important. * vcc 99 0 15.0 q1 4 3 2 nr100n 2.0 q2 6 3 1 nr100n r1 2 1 5e3 r2 1 0 15e3 q4 4 6 8 pr100n r4 99 8 5e3 q5 6 6 7 pbjt 3.0 r5 99 7 1.67e3 q8 14 4 99 pr100n q7 14 3 11 nr100n r7 11 0 45e3 q6 99 14 3 nr100n r6 3 0 10e3 r10 99 9 70e3 q10 9 9 0 nbjt q11 4 9 1 nr100n \&.end \fP .DE .bp .sp 0.25i .PS Q5:[B:box ht 1.4 wid 1.4; "\f(CWQ5\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+13.495" ljust at B.ne+(-0.5,-0.03); "+14.198" ljust at B.se+(-0.5,0.05); "+13.495" ljust at B.w; "-470.82u" ljust at B.ne+(-1.1,-0.1); "+480.43u" ljust at B.se+(-1.1,0.05); " -9.61u\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 49.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line <- from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;]; Q4:[B:box ht 1.4 wid 1.4; "\f(CWQ4\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+14.258" ljust at B.ne+(-0.5,-0.03); "+14.276" ljust at B.se+(-0.5,0.05); "+13.495" ljust at B.w; " +1.44u" ljust at B.ne+(-1.1,-0.1); "+144.77u" ljust at B.se+(-1.1,0.05); "-146.21u\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( -0.010)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (5/14); line from (1/14) to (5/14); line <- from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .w at Q5.e; Q8:[B:box ht 1.4 wid 1.4; "\f(CWQ8\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+14.969" ljust at B.ne+(-0.5,-0.03); "+15.000" ljust at B.se+(-0.5,0.05); "+14.258" ljust at B.w; "-306.63u" ljust at B.ne+(-1.1,-0.1); "+330.85u" ljust at B.se+(-1.1,0.05); "-24.22u\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 12.660)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (5/14); line from (1/14) to (5/14); line <- from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .w at Q4.e; Q6:[B:box ht 1.4 wid 1.4; "\f(CWQ6\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+15.000" ljust at B.ne+(-0.5,-0.03); "+14.208" ljust at B.se+(-0.5,0.05); "+14.969" ljust at B.w; " +1.66m" ljust at B.ne+(-1.1,-0.1); " -1.67m" ljust at B.se+(-1.1,0.05); " +8.35u\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 199.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line -> from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .n at Q5.s; Q2:[B:box ht 1.4 wid 1.4; "\f(CWQ2\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+13.495" ljust at B.ne+(-0.5,-0.03); "+13.457" ljust at B.se+(-0.5,0.05); "+14.208" ljust at B.w; "+626.63u" ljust at B.ne+(-1.1,-0.1); "-874.25u" ljust at B.se+(-1.1,0.05); "+247.61u\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 2.531)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (5/14); line from (1/14) to (5/14); line -> from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .n at Q6.s; Q1:[B:box ht 1.4 wid 1.4; "\f(CWQ1\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+14.258" ljust at B.ne+(-0.5,-0.03); "+13.572" ljust at B.se+(-0.5,0.05); "+14.208" ljust at B.w; "+22.78u" ljust at B.ne+(-1.1,-0.1); "-22.90u" ljust at B.se+(-1.1,0.05); "+114.50n\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 199.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line -> from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .w at Q2.e; Q7:[B:box ht 1.4 wid 1.4; "\f(CWQ7\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+14.969" ljust at B.ne+(-0.5,-0.03); "+13.490" ljust at B.se+(-0.5,0.05); "+14.208" ljust at B.w; "+298.28u" ljust at B.ne+(-1.1,-0.1); "-299.78u" ljust at B.se+(-1.1,0.05); " +1.50u\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 199.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line -> from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .w at Q1.e; Q10:[B:box ht 1.4 wid 1.4; "\f(CWQ10\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW +0.708" ljust at B.ne+(-0.5,-0.03); " +0.000" ljust at B.se+(-0.5,0.05); " +0.708" ljust at B.w; "+203.14u" ljust at B.ne+(-1.1,-0.1); "-204.16u" ljust at B.se+(-1.1,0.05); " +1.02u\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 199.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line -> from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .n at Q2.s; Q11:[B:box ht 1.4 wid 1.4; "\f(CWQ11\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+14.258" ljust at B.ne+(-0.5,-0.03); "+13.457" ljust at B.se+(-0.5,0.05); " +0.708" ljust at B.w; "+100.00a" ljust at B.ne+(-1.1,-0.1); "~0.0~" ljust at B.se+(-1.1,0.05); "-100.50a\s+4\fP" ljust at B.w+(0.05,-0.3); line from (4/14) to (3/14); line from (1/14) to (3/14); line -> from (4/14) to (5/14); line from (1/14) to (5/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .w at Q10.e; .PE .nf \f(CW[9]: 0.70849 [11]: 13.49004 [4]: 14.96918 [7]: 14.19769 [8]: 14.27613 [1]: 13.45718 [6]: 13.49527 [2]: 13.57167 [3]: 14.20814 [4]: 14.25843 \fP .fi .ce \fBBrokaw: Solution 1; signature: 451.72798; ||rho(x*,1)|| 4.519e-17 (Stable)\fP .bp .sp 0.25i .PS Q5:[B:box ht 1.4 wid 1.4; "\f(CWQ5\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+13.532" ljust at B.ne+(-0.5,-0.03); "+14.233" ljust at B.se+(-0.5,0.05); "+13.532" ljust at B.w; "-449.85u" ljust at B.ne+(-1.1,-0.1); "+459.03u" ljust at B.se+(-1.1,0.05); " -9.18u\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 49.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line <- from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;]; Q4:[B:box ht 1.4 wid 1.4; "\f(CWQ4\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+14.291" ljust at B.ne+(-0.5,-0.03); "+14.309" ljust at B.se+(-0.5,0.05); "+13.532" ljust at B.w; "-15.97u" ljust at B.ne+(-1.1,-0.1); "+138.28u" ljust at B.se+(-1.1,0.05); "-122.31u\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 0.131)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (5/14); line from (1/14) to (5/14); line <- from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .w at Q5.e; Q8:[B:box ht 1.4 wid 1.4; "\f(CWQ8\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+10.551" ljust at B.ne+(-0.5,-0.03); "+15.000" ljust at B.se+(-0.5,0.05); "+14.291" ljust at B.w; "-206.01u" ljust at B.ne+(-1.1,-0.1); "+210.22u" ljust at B.se+(-1.1,0.05); " -4.20u\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 49.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line <- from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .w at Q4.e; Q6:[B:box ht 1.4 wid 1.4; "\f(CWQ6\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+15.000" ljust at B.ne+(-0.5,-0.03); " +9.803" ljust at B.se+(-0.5,0.05); "+10.551" ljust at B.w; "+979.38u" ljust at B.ne+(-1.1,-0.1); "-984.31u" ljust at B.se+(-1.1,0.05); " +4.92u\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 199.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line -> from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .n at Q5.s; Q2:[B:box ht 1.4 wid 1.4; "\f(CWQ2\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+13.532" ljust at B.ne+(-0.5,-0.03); " +9.068" ljust at B.se+(-0.5,0.05); " +9.803" ljust at B.w; "+581.34u" ljust at B.ne+(-1.1,-0.1); "-584.26u" ljust at B.se+(-1.1,0.05); " +2.92u\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 199.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line -> from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .n at Q6.s; Q1:[B:box ht 1.4 wid 1.4; "\f(CWQ1\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+14.291" ljust at B.ne+(-0.5,-0.03); " +9.169" ljust at B.se+(-0.5,0.05); " +9.803" ljust at B.w; "+20.17u" ljust at B.ne+(-1.1,-0.1); "-20.27u" ljust at B.se+(-1.1,0.05); "+101.36n\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 199.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line -> from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .w at Q2.e; Q7:[B:box ht 1.4 wid 1.4; "\f(CWQ7\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+10.551" ljust at B.ne+(-0.5,-0.03); " +9.094" ljust at B.se+(-0.5,0.05); " +9.803" ljust at B.w; "+201.09u" ljust at B.ne+(-1.1,-0.1); "-202.10u" ljust at B.se+(-1.1,0.05); " +1.01u\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 199.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line -> from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .w at Q1.e; Q10:[B:box ht 1.4 wid 1.4; "\f(CWQ10\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW +0.708" ljust at B.ne+(-0.5,-0.03); " +0.000" ljust at B.se+(-0.5,0.05); " +0.708" ljust at B.w; "+203.14u" ljust at B.ne+(-1.1,-0.1); "-204.16u" ljust at B.se+(-1.1,0.05); " +1.02u\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 199.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line -> from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .n at Q2.s; Q11:[B:box ht 1.4 wid 1.4; "\f(CWQ11\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+14.291" ljust at B.ne+(-0.5,-0.03); " +9.068" ljust at B.se+(-0.5,0.05); " +0.708" ljust at B.w; "+100.00a" ljust at B.ne+(-1.1,-0.1); "~0.0~" ljust at B.se+(-1.1,0.05); "-100.50a\s+4\fP" ljust at B.w+(0.05,-0.3); line from (4/14) to (3/14); line from (1/14) to (3/14); line -> from (4/14) to (5/14); line from (1/14) to (5/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .w at Q10.e; .PE .nf \f(CW[9]: 0.70849 [11]: 9.09449 [4]: 10.55055 [7]: 14.23343 [8]: 14.30862 [1]: 9.06795 [6]: 13.53214 [2]: 9.16931 [3]: 9.80273 [4]: 14.29078 \fP .fi .ce \fBBrokaw: Solution 2; signature: 379.21495; ||rho(x*,1)|| 3.325e-17 (Unstable!)\fP .bp .sp 0.25i .PS Q5:[B:box ht 1.4 wid 1.4; "\f(CWQ5\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+14.326" ljust at B.ne+(-0.5,-0.03); "+14.956" ljust at B.se+(-0.5,0.05); "+14.326" ljust at B.w; "-25.86u" ljust at B.ne+(-1.1,-0.1); "+26.39u" ljust at B.se+(-1.1,0.05); "-527.71n\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 49.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line <- from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;]; Q4:[B:box ht 1.4 wid 1.4; "\f(CWQ4\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+14.360" ljust at B.ne+(-0.5,-0.03); "+14.956" ljust at B.se+(-0.5,0.05); "+14.326" ljust at B.w; " -8.63u" ljust at B.ne+(-1.1,-0.1); " +8.81u" ljust at B.se+(-1.1,0.05); "-176.13n\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 49.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line <- from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .w at Q5.e; Q8:[B:box ht 1.4 wid 1.4; "\f(CWQ8\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW +1.887" ljust at B.ne+(-0.5,-0.03); "+15.000" ljust at B.se+(-0.5,0.05); "+14.360" ljust at B.w; "-12.84u" ljust at B.ne+(-1.1,-0.1); "+13.11u" ljust at B.se+(-1.1,0.05); "-262.11n\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 49.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line <- from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .w at Q4.e; Q6:[B:box ht 1.4 wid 1.4; "\f(CWQ6\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+15.000" ljust at B.ne+(-0.5,-0.03); " +1.192" ljust at B.se+(-0.5,0.05); " +1.887" ljust at B.w; "+118.85u" ljust at B.ne+(-1.1,-0.1); "-119.45u" ljust at B.se+(-1.1,0.05); "+597.26n\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 199.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line -> from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .n at Q5.s; Q2:[B:box ht 1.4 wid 1.4; "\f(CWQ2\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+14.326" ljust at B.ne+(-0.5,-0.03); " +0.534" ljust at B.se+(-0.5,0.05); " +1.192" ljust at B.w; "+26.56u" ljust at B.ne+(-1.1,-0.1); "-26.70u" ljust at B.se+(-1.1,0.05); "+133.48n\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 199.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line -> from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .n at Q6.s; Q1:[B:box ht 1.4 wid 1.4; "\f(CWQ1\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+14.360" ljust at B.ne+(-0.5,-0.03); " +0.579" ljust at B.se+(-0.5,0.05); " +1.192" ljust at B.w; " +8.89u" ljust at B.ne+(-1.1,-0.1); " -8.94u" ljust at B.se+(-1.1,0.05); "+44.69n\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 199.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line -> from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .w at Q2.e; Q7:[B:box ht 1.4 wid 1.4; "\f(CWQ7\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW +1.887" ljust at B.ne+(-0.5,-0.03); " +0.554" ljust at B.se+(-0.5,0.05); " +1.192" ljust at B.w; "+12.25u" ljust at B.ne+(-1.1,-0.1); "-12.31u" ljust at B.se+(-1.1,0.05); "+61.54n\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 199.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line -> from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .w at Q1.e; Q10:[B:box ht 1.4 wid 1.4; "\f(CWQ10\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW +0.708" ljust at B.ne+(-0.5,-0.03); " +0.000" ljust at B.se+(-0.5,0.05); " +0.708" ljust at B.w; "+203.14u" ljust at B.ne+(-1.1,-0.1); "-204.16u" ljust at B.se+(-1.1,0.05); " +1.02u\s+4\fP" ljust at B.w+(0.05,-0.3); "\s-4( 199.000)\s+4" ljust at B.c+(0.1,0); line from (4/14) to (3/14); line from (1/14) to (3/14); line -> from (4/14) to (3/14); line from (1/14) to (3/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .n at Q2.s; Q11:[B:box ht 1.4 wid 1.4; "\f(CWQ11\fP" at B.nw+(0.05,-0.3) ljust; C:(12/14); E:(12/14); line from (3/14) to (11/14); line from (5/14) to B.c; "\s-4\f(CW+14.360" ljust at B.ne+(-0.5,-0.03); " +0.534" ljust at B.se+(-0.5,0.05); " +0.708" ljust at B.w; "+105.50f" ljust at B.ne+(-1.1,-0.1); "-105.93f" ljust at B.se+(-1.1,0.05); "+429.13a\s+4\fP" ljust at B.w+(0.05,-0.3); line from (4/14) to (3/14); line from (1/14) to (3/14); line -> from (4/14) to (5/14); line from (1/14) to (5/14); box invis ht 1.5 wid 1.5 with .c at B.c;] with .w at Q10.e; .PE .nf \f(CW[9]: 0.70849 [11]: 0.55384 [4]: 1.88721 [7]: 14.95594 [8]: 14.95597 [1]: 0.53449 [6]: 14.32606 [2]: 0.57917 [3]: 1.19212 [4]: 14.36015 \fP .fi .ce \fBBrokaw: Solution 3; signature: 243.21344; ||rho(x*,1)|| 2.511e-18 (Stable; "correct" operation)\fP \&-------- PostScript ------------ %!PS-Adobe-1.0 %%Creator: HPF version 3.4 SUN4-P2 28 Feb 1990 %%CreationDate: Sun Nov 4 20:23:13 1990 %%Pages: 1 %%EndComments save /a{arc}def /c{0 360 arc}def /l{lineto}def /m{moveto}def /n{newpath}def /s{stroke}def /w{setlinewidth}def 0.240000 0.240000 scale 90 rotate 1290.000000 -2400.000000 translate %%EndProlog %%Page: x0y0 1 gsave n 764 1499 m [] 0 setdash 2 w 719 1499 l s n 767 1529 m 787 1526 l s n 764 1529 m 781 1547 l s n 763 1544 m 4 w 763 1454 l s n 764 1544 m 764 1454 l s n 764 1529 m 2 w 809 1544 l 809 1559 l s n 764 1469 m 809 1454 l 809 1439 l s n 464 1499 m 419 1499 l s n 467 1529 m 487 1526 l s n 464 1529 m 481 1547 l s n 463 1544 m 4 w 463 1454 l s n 464 1544 m 464 1454 l s n 464 1529 m 2 w 509 1544 l 509 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