KVL and sensitivity of a simple circuit

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Homework Statement


Textbook figure P6.20 showing the Lion circuit: a square-wave source vs(t) feeding two diodes D1 and D2, capacitors C1 (C0 - ΔC) and C2 (C0 + ΔC), two resistors R, and output node vo' to ground, plus two printed equations for vo' at Vs = +12 V and Vs = -12 V.

Printed problem text: where C1 = C0 - ΔC, C2 = C0 + ΔC, τ1 = C1R(3/2), and τ2 = C2R(3/2); assume τ1 and τ2 >> T/2 of the input square wave vs(t). Part A asks for an algebraic expression for the average v'o(t) in the SS, with R = 10^6 Ω, C0 = 30 pF, T = 10^-5 s, Vs = 12 Vpk, and the sensitivity in DC volts per ΔC/C0.


Homework Equations


Handwritten derivation of averaged output voltage: V0'+(T/2) and V0'-(T/2) expressions with ΔC/C0 terms, integrated over one period to give mean V0' equal to T/(R C0) times ΔC/C0.


The Attempt at a Solution


Note to self - simple diode concept: http://www.cs.tut.fi/kurssit/TLT-8016/Chapter3.pdf#page=5
Handwritten equivalent circuit at t=0+ with ideal diodes, showing a 12V source, capacitors C1 and C2, resistors R, an open switch and short switch, and a Thevenin equivalent circuit with 12V, 6V, R/2 and VTH, plus the expression i1(0+) = 18V/(R/2 R).


Handwritten equation: v_o'(0+) = -12 + i_L(0+)R = -12 + (18 x 2 / 3R) R = 0V, with the final 0V circled.

Unsure about this KVL (above).

Handwritten derivation of a circuit's output voltage after switching, with exponential expressions for V0 and time constants tau1 and tau2 in terms of C0, delta C, R, and a small-approximation step using e^-e ≈ 1 - e, plus a sketched triangular waveform against time t.


Handwritten derivation of average output voltage V0' = (Vs/12)(1/(RC0))(ΔC/C0) = (12V/12)(10^-5/(10^6 x 30 x 10^-12))(ΔC/C0), then V0' = 0.333 ΔC/C0 = ΔC/(3C0), labelled Sensitivity.

For the last sensitivity part, equations be falling out of the sky...
 
Attachments
  • Textbook figure P6.20 showing the Lion circuit: a square-wave source vs(t) feeding two diodes D1 and D2, capacitors C1 (C0 - ΔC) and C2 (C0 + ΔC), two resistors R, and output node vo' to ground, plus two printed equations for vo' at Vs = +12 V and Vs = -12 V.
    upload_2018-4-2_15-37-43.png
    23.2 KB · Views: 974
  • Printed problem text: where C1 = C0 - ΔC, C2 = C0 + ΔC, τ1 = C1R(3/2), and τ2 = C2R(3/2); assume τ1 and τ2 >> T/2 of the input square wave vs(t). Part A asks for an algebraic expression for the average v'o(t) in the SS, with R = 10^6 Ω, C0 = 30 pF, T = 10^-5 s, Vs = 12 Vpk, and the sensitivity in DC volts per ΔC/C0.
    upload_2018-4-2_15-38-18.png
    12 KB · Views: 770
  • Handwritten derivation of averaged output voltage from two symmetric terms, v0+ and v0-, with an integral over one period T, ending in v0 average = T/RC0 times delta C over C0.
    upload_2018-4-2_15-39-17.png
    15.4 KB · Views: 597
  • Handwritten derivation of averaged output voltage: V0'+(T/2) and V0'-(T/2) expressions with ΔC/C0 terms, integrated over one period to give mean V0' equal to T/(R C0) times ΔC/C0.
    upload_2018-4-2_15-39-19.png
    15.4 KB · Views: 854
  • Handwritten equivalent circuit at t=0+ with ideal diodes, showing a 12V source, capacitors C1 and C2, resistors R, an open switch and short switch, and a Thevenin equivalent circuit with 12V, 6V, R/2 and VTH, plus the expression i1(0+) = 18V/(R/2 R).
    upload_2018-4-2_15-42-37.png
    10.8 KB · Views: 534
  • Handwritten derivation of an initial output voltage and a sawtooth-like output waveform versus time, with equations for v0'(t) and time constant tau, and a marked sample point at t = T/2.
    upload_2018-4-2_22-54-44.png
    21.6 KB · Views: 622
  • Handwritten derivation of average output voltage V0' = (Vs/12)(1/(RC0))(ΔC/C0) = (12V/12)(10^-5/(10^6 x 30 x 10^-12))(ΔC/C0), then V0' = 0.333 ΔC/C0 = ΔC/(3C0), labelled Sensitivity.
    upload_2018-4-2_22-55-43.png
    7.3 KB · Views: 524
  • Handwritten equation: v_o'(0+) = -12 + i_L(0+)R = -12 + (18 x 2 / 3R) R = 0V, with the final 0V circled.
    upload_2018-4-2_22-58-16.png
    2.8 KB · Views: 556
  • Handwritten derivation of a circuit's output voltage after switching, with exponential expressions for V0 and time constants tau1 and tau2 in terms of C0, delta C, R, and a small-approximation step using e^-e ≈ 1 - e, plus a sketched triangular waveform against time t.
    upload_2018-4-2_22-59-5.png
    18.8 KB · Views: 510

Discussion

See if it helps with the schematic rotated 90 degrees like this.
Circuit schematic with two batteries labeled E1 and E2, resistors labeled R, and a node labeled V with a current arrow I, drawn in a rectangular loop layout.

If that doesn't quite do it,
mentally slide the batteries around to be on the left hand vertical.
 
Attachments
  • Circuit schematic with two batteries labeled E1 and E2, resistors labeled R, and a node labeled V with a current arrow I, drawn in a rectangular loop layout.
    www.physicsforums.com_upload_2018-4-2_15-42-37-png.223260.png
    2.6 KB · Views: 394

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