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Miller effect circuit

Given \$R_S=100\ k\Omega\$, \$R_1=150\ k\Omega\$, \$R_2=20\ k\Omega\$, \$R_C=7.5\ k\Omega\$, \$R_E=1\ k\Omega\$, \$C_1=C_2=C_E=0.1\ \mu F\$ and \$C_{BE}=10\ pF\$ I need to solve for \$I_C\$, \$V_B\$, \$V_E\$, and \$V_C\$. I thought I could solve for them using:

$$V_{BB}=\frac{V_{CC}R_2}{R_1+R_2}$$ $$R_B=R_1||R_2$$ $$I_C=\frac{V_{BB}-0.7\ V}{\frac{R_B}{\beta}+R_E}$$ $$V_C=V_{CC}-I_CR_C$$ $$V_E=I_ER_E=I_CR_E$$ $$V_B=V_E+0.7\ V$$

But this seems to be wrong. Why? Thank you.

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    \$\begingroup\$ Use the schematic editor on the board, and it puts component references on for you. You've used them in your text, for good reason, but left them off the schematic. VTC. What 'seems to be wrong'? What's the value of beta? What answer did you get? \$\endgroup\$ – Neil_UK Mar 31 '20 at 4:26
  • \$\begingroup\$ Is \$\text{V}_\text{BE}\$ not a given? And what are \$\text{V}_\text{s}\$ and \$\text{V}_\text{cc}\$? \$\endgroup\$ – Jan Mar 31 '20 at 13:33
  • \$\begingroup\$ The MILLER effect is a "dynamic" effect and concerne signal quantities only. However, your equations are DC relations (No capacitors in your calculations) \$\endgroup\$ – LvW Mar 31 '20 at 16:04
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Well, we have the following circuit:

schematic

simulate this circuit – Schematic created using CircuitLab

When analyzing a transistor we need to use the following relations:

  • $$\text{I}_\text{E}=\text{I}_\text{B}+\text{I}_\text{C}\tag1$$
  • Transistor gain \$\beta\$: $$\beta=\frac{\text{I}_\text{C}}{\text{I}_\text{B}}\tag2$$
  • Emitter voltage: $$\text{V}_\text{BE}=\text{V}_2-\text{V}_3\tag3$$

When we use and apply KCL, we can write the following set of equations:

$$ \begin{cases} \text{I}_3=\text{I}_1+\text{I}_2\\ \\ 0=\text{I}_2+\text{I}_4+\text{I}_9\\ \\ \text{I}_5=\text{I}_9+\text{I}_{10}\\ \\ \text{I}_\text{E}=\text{I}_4+\text{I}_5\\ \\ \text{I}_1+\text{I}_8=\text{I}_3+\text{I}_{11}\\ \\ \text{I}_{11}=\text{I}_\text{B}+\text{I}_6\\ \\ \text{I}_\text{C}=\text{I}_6+\text{I}_7\\ \\ \text{I}_{10}=\text{I}_7+\text{I}_8 \end{cases}\tag4 $$

When we use and apply Ohm's law, we can write the following set of equations:

$$ \begin{cases} \text{I}_1=\frac{\text{V}_\text{x}-\text{V}_1}{\text{R}_1}\\ \\ \text{I}_1=\frac{\text{V}_1-\text{V}_2}{\text{R}_2}\\ \\ \text{I}_3=\frac{\text{V}_2}{\text{R}_3}\\ \\ \text{I}_4=\frac{\text{V}_3}{\text{R}_4}\\ \\ \text{I}_5=\frac{\text{V}_3}{\text{R}_5}\\ \\ \text{I}_6=\frac{\text{V}_2-\text{V}_4}{\text{R}_6}\\ \\ \text{I}_7=\frac{\text{V}_\text{y}-\text{V}_4}{\text{R}_7}\\ \\ \text{I}_8=\frac{\text{V}_\text{y}-\text{V}_2}{\text{R}_8} \end{cases}\tag5 $$

Now, in your circuit we have:

  • $$\text{R}_2=\frac{1}{\text{sC}_1}\tag6$$
  • $$\text{R}_5=\frac{1}{\text{sC}_2}\tag7$$
  • $$\text{R}_6=\frac{1}{\text{sC}_3}\tag8$$

Because we use the 'complex' s-domain we also can write:

  • Assuming a constant DC-voltage for \$\text{V}_\text{y}\$: $$\text{V}_\text{y}=\mathcal{L}_t\left[\hat{\text{v}}_\text{cc}\right]_{\left(\text{s}\right)}=\frac{\hat{\text{v}}_\text{cc}}{\text{s}}\tag9$$
  • $$\text{V}_\text{x}=\mathcal{L}_t\left[\hat{\text{v}}_\text{in}\cos\left(\omega t\right)\right]_{\left(\text{s}\right)}=\frac{\hat{\text{v}}_\text{in}\text{s}}{\text{s}^2+\omega^2}\tag{10}$$

Now, I used Mathematica to solve this (big) problem. The used code is:

FullSimplify[
 Solve[{IE == IB + IC, \[Beta] == IC/IB, VBE == V2 - V3, 
   I3 == I1 + I2, 0 == I2 + I4 + I9, I5 == I9 + I10, IE == I4 + I5, 
   I1 + I8 == I3 + I11, I11 == IB + I6, IC == I6 + I7, I10 == I7 + I8,
    I1 == (((Vin*s)/(s^2 + \[Omega]^2)) - V1)/R1, 
   I1 == (V1 - V2)/((1/(s*C1))), I3 == V2/R3, I4 == V3/R4, 
   I5 == V3/((1/(s*C2))), I6 == (V2 - V4)/((1/(s*C3))), 
   I7 == ((Vcc/s) - V4)/R7, I8 == ((Vcc/s) - V2)/R8}, {IB, IC, IE, I1,
    I2, I3, I4, I5, I6, I7, I8, I9, I10, I11, V1, V2, V3, V4}]]

Using your values in my code I got:

FullSimplify[
 Solve[{IE == IB + IC, \[Beta] == IC/IB, VBE == V2 - V3, 
   I3 == I1 + I2, 0 == I2 + I4 + I9, I5 == I9 + I10, IE == I4 + I5, 
   I1 + I8 == I3 + I11, I11 == IB + I6, IC == I6 + I7, I10 == I7 + I8,
    I1 == (((Vin*s)/(s^2 + \[Omega]^2)) - V1)/10000, 
   I1 == (V1 - V2)/((1/(s*((1/10)*10^(-6))))), I3 == V2/20000, 
   I4 == V3/1000, I5 == V3/((1/(s*((1/10)*10^(-6))))), 
   I6 == (V2 - V4)/((1/(s*((10)*10^(-12))))), 
   I7 == ((Vcc/s) - V4)/7500, I8 == ((Vcc/s) - V2)/150000}, {IB, IC, 
   IE, I1, I2, I3, I4, I5, I6, I7, I8, I9, I10, I11, V1, V2, V3, V4}]]

Using that code, it gave me:

{{IB -> ((10000 + 
        s) (s^2 (80000000000 Vcc + 
           s (-(680000000000 + s (1880171000 + 261 s)) VBE + 
              2 (40063000 + 63 s) Vcc + 
              30 (40000000 + 3 s) Vin)) + (-s (680000000000 + 
              s (1880171000 + 261 s)) VBE + 
           2 (1000 + s) (40000000 + 
              63 s) Vcc) \[Omega]^2))/(100000 s (9 s^3 (1 + \[Beta]) \
+ 4000000000000 (317 + 17 \[Beta]) + 300 s^2 (400417 + 417 \[Beta]) + 
        100000 s (15081071 + 1881071 \[Beta])) (s^2 + \[Omega]^2)), 
  IC -> ((10000 + 
        s) \[Beta] (s^2 (80000000000 Vcc + 
           s (-(680000000000 + s (1880171000 + 261 s)) VBE + 
              2 (40063000 + 63 s) Vcc + 
              30 (40000000 + 3 s) Vin)) + (-s (680000000000 + 
              s (1880171000 + 261 s)) VBE + 
           2 (1000 + s) (40000000 + 
              63 s) Vcc) \[Omega]^2))/(100000 s (9 s^3 (1 + \[Beta]) \
+ 4000000000000 (317 + 17 \[Beta]) + 300 s^2 (400417 + 417 \[Beta]) + 
        100000 s (15081071 + 1881071 \[Beta])) (s^2 + \[Omega]^2)), 
  IE -> ((10000 + 
        s) (1 + \[Beta]) (s^2 (80000000000 Vcc + 
           s (-(680000000000 + s (1880171000 + 261 s)) VBE + 
              2 (40063000 + 63 s) Vcc + 
              30 (40000000 + 3 s) Vin)) + (-s (680000000000 + 
              s (1880171000 + 261 s)) VBE + 
           2 (1000 + s) (40000000 + 
              63 s) Vcc) \[Omega]^2))/(100000 s (9 s^3 (1 + \[Beta]) \
+ 4000000000000 (317 + 17 \[Beta]) + 300 s^2 (400417 + 417 \[Beta]) + 
        100000 s (15081071 + 1881071 \[Beta])) (s^2 + \[Omega]^2)), 
  I1 -> -((s^2 (9 s^3 VBE (1 + \[Beta]) + 
           3 s^2 (-3 Vin (1 + \[Beta]) + 
              10000 VBE (4003 + 3 \[Beta])) + 
           4000000000 (2 Vcc (1 + \[Beta]) - 
              Vin (317 + 17 \[Beta])) + 
           300 s (4000000000 VBE + 42 Vcc (1 + \[Beta]) - 
              Vin (400357 + 
                 357 \[Beta]))) + (9 s^3 VBE (1 + \[Beta]) + 
           8000000000 Vcc (1 + \[Beta]) + 
           30000 s^2 VBE (4003 + 3 \[Beta]) + 
           600 s (2000000000 VBE + 
              21 Vcc (1 + \[Beta]))) \[Omega]^2)/(10000 (9 s^3 (1 + \
\[Beta]) + 4000000000000 (317 + 17 \[Beta]) + 
          300 s^2 (400417 + 417 \[Beta]) + 
          100000 s (15081071 + 1881071 \[Beta])) (s^2 + \[Omega]^2))),
   I2 -> (s^2 (27 s^4 VBE (1 + \[Beta]) + 
         8000000000000 Vcc (1 + \[Beta]) + 
         200000 s (6000000000 VBE + 120063 Vcc (1 + \[Beta]) - 
            80000 Vin (151 + \[Beta])) + 
         18 s^3 (-Vin (1 + \[Beta]) + 500 VBE (40031 + 31 \[Beta])) + 
         600 s^2 (63 Vcc (1 + \[Beta]) + 
            50000 VBE (124003 + 3 \[Beta]) - 
            2 Vin (200171 + 171 \[Beta]))) + (1000 + 
         3 s) (9 s^3 VBE (1 + \[Beta]) + 
         8000000000 Vcc (1 + \[Beta]) + 
         30000 s^2 VBE (4003 + 3 \[Beta]) + 
         600 s (2000000000 VBE + 
            21 Vcc (1 + \[Beta]))) \[Omega]^2)/(20000 s (9 s^3 (1 + \
\[Beta]) + 4000000000000 (317 + 17 \[Beta]) + 
        300 s^2 (400417 + 417 \[Beta]) + 
        100000 s (15081071 + 1881071 \[Beta])) (s^2 + \[Omega]^2)), 
  I3 -> (s^2 (9 s^4 VBE (1 + \[Beta]) + 
         8000000000000 Vcc (1 + \[Beta]) + 
         3000 s^3 VBE (40033 + 33 \[Beta]) + 
         200000 s (6000000000 VBE + (40063 Vcc + 
               600000 Vin) (1 + \[Beta])) + 
         600 s^2 (3 (7 Vcc + 5 Vin) (1 + \[Beta]) + 
            50000 VBE (44003 + 3 \[Beta]))) + (1000 + 
         s) (9 s^3 VBE (1 + \[Beta]) + 8000000000 Vcc (1 + \[Beta]) + 
         30000 s^2 VBE (4003 + 3 \[Beta]) + 
         600 s (2000000000 VBE + 
            21 Vcc (1 + \[Beta]))) \[Omega]^2)/(20000 s (9 s^3 (1 + \
\[Beta]) + 4000000000000 (317 + 17 \[Beta]) + 
        300 s^2 (400417 + 417 \[Beta]) + 
        100000 s (15081071 + 1881071 \[Beta])) (s^2 + \[Omega]^2)), 
  I4 -> ((1 + \[Beta]) (s^2 (80000000000 Vcc + 
           s (-(680000000000 + s (1880171000 + 261 s)) VBE + 
              2 (40063000 + 63 s) Vcc + 
              30 (40000000 + 3 s) Vin)) + (-s (680000000000 + 
              s (1880171000 + 261 s)) VBE + 
           2 (1000 + s) (40000000 + 
              63 s) Vcc) \[Omega]^2))/(10 s (9 s^3 (1 + \[Beta]) + 
        4000000000000 (317 + 17 \[Beta]) + 
        300 s^2 (400417 + 417 \[Beta]) + 
        100000 s (15081071 + 1881071 \[Beta])) (s^2 + \[Omega]^2)), 
  I5 -> ((1 + \[Beta]) (s^2 (80000000000 Vcc + 
           s (-(680000000000 + s (1880171000 + 261 s)) VBE + 
              2 (40063000 + 63 s) Vcc + 
              30 (40000000 + 3 s) Vin)) + (-s (680000000000 + 
              s (1880171000 + 261 s)) VBE + 
           2 (1000 + s) (40000000 + 
              63 s) Vcc) \[Omega]^2))/(100000 (9 s^3 (1 + \[Beta]) + 
        4000000000000 (317 + 17 \[Beta]) + 
        300 s^2 (400417 + 417 \[Beta]) + 
        100000 s (15081071 + 1881071 \[Beta])) (s^2 + \[Omega]^2)), 
  I6 -> (3 s^2 (-420000000 Vcc + 
         40 s ((1000 + s) (10000 + s) VBE - (12500 + s) Vcc + 
            1000 Vin) + 
         s (-(10000 + s) (17000 + 47 s) VBE + 2 (-19000 + s) Vcc + 
            10 (34000 + 3 s) Vin) \[Beta]) - 
      3 (-40 s (1000 + s) (10000 + s) VBE + 
         40 (10500000 + s (12500 + s)) Vcc + 
         s ((10000 + s) (17000 + 47 s) VBE - 
            2 (-19000 + 
               s) Vcc) \[Beta]) \[Omega]^2)/(100000 (9 s^3 (1 + \
\[Beta]) + 4000000000000 (317 + 17 \[Beta]) + 
        300 s^2 (400417 + 417 \[Beta]) + 
        100000 s (15081071 + 1881071 \[Beta])) (s^2 + \[Omega]^2)), 
  I7 -> (s^2 (20000000000000 Vcc \[Beta] - 3 s^4 VBE (1 + \[Beta]) + 
         500000 s (-200000 (1700 VBE - 3 Vin) \[Beta] + 
            Vcc (63 + 44063 \[Beta])) + 
         s^3 (3 Vcc (1 + \[Beta]) - 1000 VBE (33 + 47033 \[Beta])) - 
         500 s^2 (6 Vin (1 - 9999 \[Beta]) - 
            25 Vcc (3 + 163 \[Beta]) + 
            20000 VBE (3 + 
               48703 \[Beta]))) + (20000000000000 Vcc \[Beta] - 
         3 s^4 VBE (1 + \[Beta]) + 
         500000 s (-340000000 VBE \[Beta] + 
            Vcc (63 + 44063 \[Beta])) + 
         s^3 (3 Vcc (1 + \[Beta]) - 1000 VBE (33 + 47033 \[Beta])) - 
         12500 s^2 (-Vcc (3 + 163 \[Beta]) + 
            800 VBE (3 + 
               48703 \[Beta]))) \[Omega]^2)/(2500 s (9 s^3 (1 + \
\[Beta]) + 4000000000000 (317 + 17 \[Beta]) + 
        300 s^2 (400417 + 417 \[Beta]) + 
        100000 s (15081071 + 1881071 \[Beta])) (s^2 + \[Omega]^2)), 
  I8 -> (s^2 (-(40000000 + 3 s) (-10500000 Vcc + 
            s ((1000 + s) (10000 + s) VBE - (12500 + s) Vcc + 
               1000 Vin)) + (20000000000000 Vcc + 
            s (500000 (120063 Vcc - 80000 Vin) - 
               3 s ((1000 + s) (10000 + s) VBE - (12500 + s) Vcc + 
                  1000 Vin))) \[Beta]) + (-(40000000 + 
             3 s) (s (1000 + s) (10000 + s) VBE - (10500000 + 
               s (12500 + s)) Vcc) + (-3 s^2 (1000 + s) (10000 + 
               s) VBE + (20000000000000 + 
               3 s (20010500000 + 
                  s (12500 + 
                    s))) Vcc) \[Beta]) \[Omega]^2)/(50000 s (9 s^3 (1 \
+ \[Beta]) + 4000000000000 (317 + 17 \[Beta]) + 
        300 s^2 (400417 + 417 \[Beta]) + 
        100000 s (15081071 + 1881071 \[Beta])) (s^2 + \[Omega]^2)), 
  I9 -> (s^2 (-27 s^4 VBE (1 + \[Beta]) - 
         168000000000000 Vcc (1 + \[Beta]) + 
         18 s^3 (Vin + Vin \[Beta] + 500 VBE (-39973 + 27 \[Beta])) + 
         200000 s (-921323 Vcc (1 + \[Beta]) + 
            400000000 VBE (2 + 17 \[Beta]) - 
            80000 Vin (-1 + 149 \[Beta])) + 
         200 s^2 (-1449 Vcc (1 + \[Beta]) + 
            6 Vin (200021 + 21 \[Beta]) + 
            20000 VBE (10063 + 
               940063 \[Beta]))) - (27 s^4 VBE (1 + \[Beta]) + 
         168000000000000 Vcc (1 + \[Beta]) - 
         9000 s^3 VBE (-39973 + 27 \[Beta]) - 
         200000 s (-921323 Vcc (1 + \[Beta]) + 
            400000000 VBE (2 + 17 \[Beta])) - 
         200 s^2 (-1449 Vcc (1 + \[Beta]) + 
            20000 VBE (10063 + 
               940063 \[Beta]))) \[Omega]^2)/(20000 s (9 s^3 (1 + \
\[Beta]) + 4000000000000 (317 + 17 \[Beta]) + 
        300 s^2 (400417 + 417 \[Beta]) + 
        100000 s (15081071 + 1881071 \[Beta])) (s^2 + \[Omega]^2)), 
  I10 -> (s^2 (-(40000000 + 63 s) (-10500000 Vcc + 
            s ((1000 + s) (10000 + s) VBE - (12500 + s) Vcc + 
               1000 Vin)) + (420000000000000 Vcc + 
            s (-(10000 + s) (340000000000 + 
                  s (940063000 + 63 s)) VBE + (500661500000 + 
                  s (40787500 + 63 s)) Vcc + 
               1000 (5960000000 + 
                  599937 s) Vin)) \[Beta]) + (-(40000000 + 
             63 s) (s (1000 + s) (10000 + s) VBE - (10500000 + 
               s (12500 + s)) Vcc) + (-s (10000 + s) (340000000000 + 
               s (940063000 + 63 s)) VBE + (40000000 + 
               63 s) (10500000 + 
               s (12500 + 
                  s)) Vcc) \[Beta]) \[Omega]^2)/(50000 s (9 s^3 (1 + \
\[Beta]) + 4000000000000 (317 + 17 \[Beta]) + 
        300 s^2 (400417 + 417 \[Beta]) + 
        100000 s (15081071 + 1881071 \[Beta])) (s^2 + \[Omega]^2)), 
  I11 -> (s^2 (800000000000000 Vcc + 
         s (80000000000 (11 Vcc + 150 Vin) - (10000 + s) (17000 + 
               47 s) VBE (40000000 + 3 s (1 + \[Beta])) + 
            2 s (1000 Vcc (39943 - 57 \[Beta]) + 
               3 s Vcc (1 + \[Beta]) + 45 s Vin (1 + \[Beta]) + 
               30000 Vin (20017 + 
                  17 \[Beta])))) - (-800000000000000 Vcc + 
         s (10000 + s) (17000 + 47 s) VBE (40000000 + 
            3 s (1 + \[Beta])) - 
         2 s Vcc (440000000000 + 1000 s (39943 - 57 \[Beta]) + 
            3 s^2 (1 + \[Beta]))) \[Omega]^2)/(100000 s (9 s^3 (1 + \
\[Beta]) + 4000000000000 (317 + 17 \[Beta]) + 
        300 s^2 (400417 + 417 \[Beta]) + 
        100000 s (15081071 + 1881071 \[Beta])) (s^2 + \[Omega]^2)), 
  V1 -> (9 s^5 VBE (1 + \[Beta]) + 30000 s^4 VBE (4003 + 3 \[Beta]) + 
      8000000000 Vcc (1 + \[Beta]) \[Omega]^2 + 
      10000 s^2 (800000 Vcc (1 + \[Beta]) + 
         30 Vin (800357 + 400357 \[Beta]) + 
         3 VBE (4003 + 3 \[Beta]) \[Omega]^2) + 
      200 s (20000000000 Vin (317 + 17 \[Beta]) + 
         3 (2000000000 VBE + 21 Vcc (1 + \[Beta])) \[Omega]^2) + 
      3 s^3 (600 (7 Vcc + 10 Vin) (1 + \[Beta]) + 
         VBE (400000000000 + 
            3 (1 + \[Beta]) \[Omega]^2)))/((9 s^3 (1 + \[Beta]) + 
        4000000000000 (317 + 17 \[Beta]) + 
        300 s^2 (400417 + 417 \[Beta]) + 
        100000 s (15081071 + 1881071 \[Beta])) (s^2 + \[Omega]^2)), 
  V2 -> (s^2 (9 s^4 VBE (1 + \[Beta]) + 
         8000000000000 Vcc (1 + \[Beta]) + 
         3000 s^3 VBE (40033 + 33 \[Beta]) + 
         200000 s (6000000000 VBE + (40063 Vcc + 
               600000 Vin) (1 + \[Beta])) + 
         600 s^2 (3 (7 Vcc + 5 Vin) (1 + \[Beta]) + 
            50000 VBE (44003 + 3 \[Beta]))) + (1000 + 
         s) (9 s^3 VBE (1 + \[Beta]) + 8000000000 Vcc (1 + \[Beta]) + 
         30000 s^2 VBE (4003 + 3 \[Beta]) + 
         600 s (2000000000 VBE + 
            21 Vcc (1 + \[Beta]))) \[Omega]^2)/(s (9 s^3 (1 + \
\[Beta]) + 4000000000000 (317 + 17 \[Beta]) + 
        300 s^2 (400417 + 417 \[Beta]) + 
        100000 s (15081071 + 1881071 \[Beta])) (s^2 + \[Omega]^2)), 
  V3 -> (100 (1 + \[Beta]) (s^2 (80000000000 Vcc + 
           s (-(680000000000 + s (1880171000 + 261 s)) VBE + 
              2 (40063000 + 63 s) Vcc + 
              30 (40000000 + 3 s) Vin)) + (-s (680000000000 + 
              s (1880171000 + 261 s)) VBE + 
           2 (1000 + s) (40000000 + 
              63 s) Vcc) \[Omega]^2))/(s (9 s^3 (1 + \[Beta]) + 
        4000000000000 (317 + 17 \[Beta]) + 
        300 s^2 (400417 + 417 \[Beta]) + 
        100000 s (15081071 + 1881071 \[Beta])) (s^2 + \[Omega]^2)), 
  V4 -> (s^2 (9 s^4 VBE (1 + \[Beta]) + 
         4000000000000 Vcc (317 + 2 \[Beta]) + 
         3000 s^3 VBE (33 + 47033 \[Beta]) + 
         600 s^2 (15 Vin (1 - 9999 \[Beta]) + 
            Vcc (200021 - 9979 \[Beta]) + 
            50000 VBE (3 + 48703 \[Beta])) + 
         200000 s (1500000 (1700 VBE - 3 Vin) \[Beta] + 
            Vcc (7540063 + 
               610063 \[Beta]))) + (9 s^4 VBE (1 + \[Beta]) + 
         4000000000000 Vcc (317 + 2 \[Beta]) + 
         3000 s^3 VBE (33 + 47033 \[Beta]) + 
         600 s^2 (Vcc (200021 - 9979 \[Beta]) + 
            50000 VBE (3 + 48703 \[Beta])) + 
         200000 s (2550000000 VBE \[Beta] + 
            Vcc (7540063 + 
               610063 \[Beta]))) \[Omega]^2)/(s (9 s^3 (1 + \[Beta]) \
+ 4000000000000 (317 + 17 \[Beta]) + 300 s^2 (400417 + 417 \[Beta]) + 
        100000 s (15081071 + 1881071 \[Beta])) (s^2 + \[Omega]^2))}}

Assuming \$\beta=100\$, \$\text{V}_\text{BE}=\frac{7}{10\text{s}}\$, \$\omega=100\pi\$, \$\hat{\text{v}}_\text{cc}=15\$ and \$\hat{\text{v}}_\text{in}=5\$. That gave:

In[1]:=FullSimplify[
 Solve[{IE == IB + IC, 100 == IC/IB, (7/10)/s == V2 - V3, 
   I3 == I1 + I2, 0 == I2 + I4 + I9, I5 == I9 + I10, IE == I4 + I5, 
   I1 + I8 == I3 + I11, I11 == IB + I6, IC == I6 + I7, I10 == I7 + I8,
    I1 == (((5*s)/(s^2 + (100 Pi)^2)) - V1)/10000, 
   I1 == (V1 - V2)/((1/(s*((1/10)*10^(-6))))), I3 == V2/20000, 
   I4 == V3/1000, I5 == V3/((1/(s*((1/10)*10^(-6))))), 
   I6 == (V2 - V4)/((1/(s*((10)*10^(-12))))), 
   I7 == ((15/s) - V4)/7500, I8 == ((15/s) - V2)/150000}, {IB, IC, IE,
    I1, I2, I3, I4, I5, I6, I7, I8, I9, I10, I11, V1, V2, V3, V4}]]

Out[1]={{IB -> ((10000 + 
        s) (10000 \[Pi]^2 (7240000000000 + 
           s (-1142297000 + 17073 s)) + 
        s^2 (7240000000000 + 
           s (58857703000 + 21573 s))))/(1000000 s (10000 \[Pi]^2 + 
        s^2) (8068000000000000 + 
        s (20318817100000 + 3 s (44211700 + 303 s)))), 
  IC -> ((10000 + 
        s) (10000 \[Pi]^2 (7240000000000 + 
           s (-1142297000 + 17073 s)) + 
        s^2 (7240000000000 + 
           s (58857703000 + 21573 s))))/(10000 s (10000 \[Pi]^2 + 
        s^2) (8068000000000000 + 
        s (20318817100000 + 3 s (44211700 + 303 s)))), 
  IE -> (101 (10000 + 
        s) (10000 \[Pi]^2 (7240000000000 + 
           s (-1142297000 + 17073 s)) + 
        s^2 (7240000000000 + 
           s (58857703000 + 21573 s))))/(1000000 s (10000 \[Pi]^2 + 
        s^2) (8068000000000000 + 
        s (20318817100000 + 3 s (44211700 + 303 s)))), 
  I1 -> (s^2 (273800000000000 + 5446335000 s + 39087 s^2) - 
    30000 \[Pi]^2 (43200000000000 + 7 s (52120000 + 303 s)))/(
   100000 (10000 \[Pi]^2 + s^2) (8068000000000000 + 
      s (20318817100000 + 3 s (44211700 + 303 s)))), 
  I2 -> (30000 \[Pi]^2 (1000 + 3 s) (43200000000000 + 
         7 s (52120000 + 303 s)) + 
      s^2 (129600000000000000 + 
         s (189094520000000 - 
            3 s (3248779000 + 23937 s))))/(200000 s (10000 \[Pi]^2 + 
        s^2) (8068000000000000 + 
        s (20318817100000 + 3 s (44211700 + 303 s)))), 
  I3 -> (30000 \[Pi]^2 (1000 + s) (43200000000000 + 
         7 s (52120000 + 303 s)) + 
      3 s^2 (43200000000000000 + 
         s (245564840000000 + 
            s (382111000 + 2121 s))))/(200000 s (10000 \[Pi]^2 + 
        s^2) (8068000000000000 + 
        s (20318817100000 + 3 s (44211700 + 303 s)))), 
  I4 -> (101 (10000 \[Pi]^2 (7240000000000 + 
           s (-1142297000 + 17073 s)) + 
        s^2 (7240000000000 + 
           s (58857703000 + 21573 s))))/(100 s (10000 \[Pi]^2 + 
        s^2) (8068000000000000 + 
        s (20318817100000 + 3 s (44211700 + 303 s)))), 
  I5 -> (101 (10000 \[Pi]^2 (7240000000000 + 
           s (-1142297000 + 17073 s)) + 
        s^2 (7240000000000 + 
           s (58857703000 + 21573 s))))/(1000000 (10000 \[Pi]^2 + 
        s^2) (8068000000000000 + 
        s (20318817100000 + 3 s (44211700 + 303 s)))), 
  I6 -> (-30000 \[Pi]^2 (8960000000 + s (49141000 + 431 s)) + 
    3 s^2 (-8960000000 + s (35959000 + 7069 s)))/(
   50000 (10000 \[Pi]^2 + s^2) (8068000000000000 + 
      s (20318817100000 + 3 s (44211700 + 303 s)))), 
  I7 -> (10000 \[Pi]^2 (181000000000000000 + 
         s (-10443985000000 + s (-2355206000 + 43329 s))) + 
      s^2 (181000000000000000 + 
         s (1489556015000000 + 
            s (147629644000 + 43329 s))))/(25000 s (10000 \[Pi]^2 + 
        s^2) (8068000000000000 + 
        s (20318817100000 + 3 s (44211700 + 303 s)))), 
  I8 -> (s^2 (360200000000000000 + 
         s (770376015000000 + s (6249644000 + 43329 s))) + 
      10000 \[Pi]^2 (360200000000000000 + 
         s (972376015000000 + 
            s (6264794000 + 43329 s))))/(500000 s (10000 \[Pi]^2 + 
        s^2) (8068000000000000 + 
        s (20318817100000 + 3 s (44211700 + 303 s)))), 
  I9 -> (-10000 \[Pi]^2 (1592080000000000000 + 
         s (159150526000000 + 63 s (106963000 + 303 s))) + 
      s^2 (-1592080000000000000 + 
         3 s (-4026116842000000 + 
            s (1796197000 + 23937 s))))/(200000 s (10000 \[Pi]^2 + 
        s^2) (8068000000000000 + 
        s (20318817100000 + 3 s (44211700 + 303 s)))), 
  I10 -> (10000 \[Pi]^2 (3980200000000000000 + 
         s (763496315000000 + 11 s (-3712666000 + 82719 s))) + 
      s^2 (3980200000000000000 + 
         s (30561496315000000 + 
            11 s (268985684000 + 
               82719 s))))/(500000 s (10000 \[Pi]^2 + 
        s^2) (8068000000000000 + 
        s (20318817100000 + 3 s (44211700 + 303 s)))), 
  I11 -> (-10000 \[Pi]^2 (-72400000000000000 + 
         s (4720570000000 + s (3920027000 + 8787 s))) + 
      s^2 (72400000000000000 + 
         s (595279430000000 + 
            s (61230973000 + 445713 s))))/(1000000 s (10000 \[Pi]^2 + 
        s^2) (8068000000000000 + 
        s (20318817100000 + 3 s (44211700 + 303 s)))), 
  V1 -> (30000 \[Pi]^2 (43200000000000 + 7 s (52120000 + 303 s)) + 
      s (403400000000000000 + 
         3 s (247380285000000 + 
            s (395140000 + 2121 s))))/(10 (10000 \[Pi]^2 + 
        s^2) (8068000000000000 + 
        s (20318817100000 + 3 s (44211700 + 303 s)))), 
  V2 -> (30000 \[Pi]^2 (1000 + s) (43200000000000 + 
         7 s (52120000 + 303 s)) + 
      3 s^2 (43200000000000000 + 
         s (245564840000000 + 
            s (382111000 + 2121 s))))/(10 s (10000 \[Pi]^2 + 
        s^2) (8068000000000000 + 
        s (20318817100000 + 3 s (44211700 + 303 s)))), 
  V3 -> (1010 (10000 \[Pi]^2 (7240000000000 + 
           s (-1142297000 + 17073 s)) + 
        s^2 (7240000000000 + 
           s (58857703000 + 21573 s))))/(s (10000 \[Pi]^2 + 
        s^2) (8068000000000000 + 
        s (20318817100000 + 3 s (44211700 + 303 s)))), 
  V4 -> (3 s^2 (222400000000000000 + 
         s (-473615160000000 + s (-140997889000 + 2121 s))) + 
      30000 \[Pi]^2 (222400000000000000 + 
         s (1026384840000000 + 
            s (8986961000 + 2121 s))))/(10 s (10000 \[Pi]^2 + 
        s^2) (8068000000000000 + 
        s (20318817100000 + 3 s (44211700 + 303 s))))}}

So, the output voltage \$\text{V}_4\$, given by:

(3 s^2 (222400000000000000 + 
      s (-473615160000000 + s (-140997889000 + 2121 s))) + 
   30000 \[Pi]^2 (222400000000000000 + 
      s (1026384840000000 + 
         s (8986961000 + 2121 s))))/(10 s (10000 \[Pi]^2 + 
     s^2) (8068000000000000 + 
     s (20318817100000 + 3 s (44211700 + 303 s))))

Which gives when plotted:

enter image description here

\$\endgroup\$
3
  • \$\begingroup\$ Frequency analysis with and without Cbc capacitor will give much more inside information in this case. \$\endgroup\$ – G36 Mar 31 '20 at 15:12
  • \$\begingroup\$ And treat the rest of the capacitors as a short circuits. \$\endgroup\$ – G36 Mar 31 '20 at 15:14
  • \$\begingroup\$ Jan....where I can see the results of the MILLER effect? \$\endgroup\$ – LvW Mar 31 '20 at 16:35
0
\$\begingroup\$

Mark - what do you want: A "demonstration" (calculation) or an explanation of this effect?

  • Let me start with an explanation: The MILLER effect concerns the input impedance of your circuit at the base node. Here, we see two pathes: (a) into the base and (b) into the 10pF cap. The input impedance is defined as the ratio of the input signal voltage divided by the current going through the node carrying this voltage.

However, as far as the current through the cap is concerned, there are two voltages which determine the current through the capacitor: The input voltage at the base (Vb) and the (much larger) voltage at the other side (collector voltage Vc=-|A|*Vb with A=gain).

Therefore, the current through the cap is much larger in comparison to the case where the other side of the cap would be at a fixed potential (ground or any DC value). Hence, the input resistance of this path appears correspondingly lower. That is the content of the MILLER effect.

  • The calculation is rather simple:

The voltage across the cap is (Vb-Vc) with Vc=-|A|*Vb resulting in

(Vb-Vc)=Vb(1+|A|),

and the current through the capacitor is Ic=(Vb-Vc)/Xc=Vb(1+|A|)/Xc.

That means the capacitive reactance Xcc- as seen from the input only - appears much lower than Xc=1/wC with:

Xcc=Vb/Ic=Xc/(1+|A|)

  • In conclusion, the MILLER effect reduces the input impedance at the base node.

  • Final comment: Of course, the total input impedance contains in parallel the input resistance as seen into the transistor.

\$\endgroup\$

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