Task 1: RC band-pass analysis

Component values

X1=1, X2=8, Y1=2, Y2=5. V1=160 V (AC magnitude, phase 0 degrees); R1=1 kOhm; R2=80 kOhm; C1=200 nF; C2=5 uF.

Part 1: Analytical calculation

VP1 is the input, VP2 is the C2/R1/R2 junction, and VP3 is the R1/C1 output. All voltages are measured relative to ground.

Nodal equations: (VP2-VP1)sC2 + VP2/R2 + (VP2-VP3)/R1 = 0 (VP3-VP2)/R1 + sC1 VP3 = 0

H3(s)=VP3/VP1=R2 C2 s/[1+(R1 C1+R2 C1+R2 C2)s+R1 R2 C1 C2 s^2] =0.4s/(1+0.4162s+0.00008s^2).

H2(s)=VP2/VP1=(1+0.0002s)H3(s); H1=1.

For w=2pif: |H3(jw)|^2=0.16w^2/[(1-0.00008w^2)^2+(0.4162w)^2]. For exactly -3 dB relative to VP1, q=10^(-3/10). Setting x=w^2 gives: a^2 x^2+(b^2-2a-c^2/q)x+1=0, where a=0.00008, b=0.4162 and c=0.4.

The two positive frequency solutions are fL=0.4162717 Hz and fH=760.6298 Hz. At both frequencies VP3=113.2713 V, or 41.0824 dBV, while VP1=160 V or 44.0824 dBV.

The conventional half-power (-3.0103 dB) frequencies are 0.4151966 Hz and 762.5995 Hz. These cutoffs are relative to the INPUT, not the passband peak. Peak frequency=17.7941 Hz and peak gain=-0.344842 dB.

Part 2: CircuitSim verification

AC logarithmic sweep: 0.01 Hz to 100000 Hz, 200 points per decade. Plot magnitude (dBV) and phase (degrees) for VP1, VP2 and VP3. VP1 green, VP2 blue, VP3 red. The phase of VP1 is 0 degrees.

Actual simulation table readings (4 decimal places):

Frequency (Hz)	VP1 (dBV)	VP3 (dBV)	VP3/VP1 (dB)
0.4120975	44.0824	41.0421	-3.0403
0.4168694	44.0824	41.0881	-2.9943
758.5776	44.0824	41.0931	-2.9893
767.3615	44.0824	41.0472	-3.0352

Interpolating dB against log10(f) between the two rows bracketing each crossing gives 0.4162751 Hz and 760.6162 Hz. Relative differences from calculation are +0.000813% and -0.001789%, respectively. Both agree within 0.002%; precision is limited by sweep spacing and displayed rounding.

Phase expressions: phase(H3)=90 degrees-atan2(0.4162w,1-0.00008w^2), and phase(H2)=phase(H3)+atan(0.0002w), with angles expressed in degrees.

The loaded two-capacitor network is band-pass: VP3 falls at low and high frequencies. The two simulated crossings verify the complete-network calculation.

A graphical ground-routing warning remains, but the generated SPICE netlist connects the source, R2 and C1 to node 0 and the AC simulation completes.