Macromodelo Linear de um Amplificador Operacional

Dedução Matemática

\[ i_x + i_y + i_z = 0 \] \[ A_1(V_p - V_N) + \frac{V_2}{R} + C\frac{dV_2}{dt} = 0 \]

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\[ -V_o = A_2 \cdot V_2 \cdot R_o \] \[ V_2 = -\frac{V_o}{A_2 \cdot R_o} \]

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\[ i_1 = i_2 \] \[ \frac{V_a - V_p}{R_1} = \frac{V_p - V_o}{R_2} \] \[ i_3 = i_4 \] \[ \frac{V_b - V_N}{R_3} = \frac{V_N - V_o}{R_4} \] \[ V_p\!\left(\frac{1}{R_1} + \frac{1}{R_2}\right) = \frac{V_a}{R_1} + \frac{V_o}{R_2} \] \[ V_p\!\left(\frac{R_1 + R_2}{R_1 R_2}\right) = \frac{V_a}{R_1} + \frac{V_o}{R_2} \] \[ V_p = V_a\!\left(\frac{R_2}{R_1+R_2}\right) + V_o\!\left(\frac{R_1}{R_1+R_2}\right) \] \[ V_N = V_b\!\left(\frac{R_4}{R_3+R_4}\right) + V_o\!\left(\frac{R_3}{R_3+R_4}\right) \]

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\[ V_p = C_p + V_o\, \delta^{+} \] \[ V_N = C_N + V_o\, \delta^{-} \]

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\[ -\frac{RC}{A_2 R_o}\frac{dV_o}{dt} - \left(\frac{1}{A_2 R_o}\right)V_o + R A_1(V_p - V_N) = 0 \] \[ \left(\frac{RC}{A_2 R_o}\right)\frac{dV_o}{dt} + \left(\frac{1}{A_2 R_o}\right)V_o + R A_1(V_N - V_p) = 0 \] \[ \left(\frac{RC}{A_2 R_o}\right)\frac{dV_o}{dt} + \left(\frac{1}{A_2 R_o}\right)V_o + R A_1\!\left(C_N - C_p + V_o(\delta^{-} - \delta^{+})\right) = 0 \] \[ \left(\frac{RC}{A_2 R_o}\right)\frac{dV_o}{dt} + \left(\frac{1}{A_2 R_o} + R A_1(\delta^{-} - \delta^{+})\right)V_o + R A_1\!\left(V_b\frac{R_4}{R_3+R_4} - V_a\frac{R_2}{R_1+R_2}\right) = 0 \]

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\[ Ax' + Bx + C = 0 \]

EDO 1ª ordem  →

\[ f(t) = K\,e^{\displaystyle-\frac{B}{A}\,t} + \frac{C}{B} \]