User:Zen-in/sandbox

Transfer function derivation
First, assigning Va = voltage at + terminal = Vs and assigning Vb = voltage at - terminal and assigning Vo = voltage at output


 * $$I_s + \frac{V_o - V_s}{R_3} = 0,$$  Kirchoff's current law


 * $$V_a = V_o + {I_s}{R_3},$$ rearranging to express $$V_a$$ in terms of $$V_o$$ and $$I_s$$


 * $$V_o = V_s - {I_s}{R_3},$$ rearranging to express $$V_o$$ in terms of $$V_s$$ and $$I_s$$


 * $$\frac{V_b}{R_1} + \frac{V_b - V_o}{R_2} = 0,$$  Kirchoff's current law


 * $$V_b =\frac{R_1 + R_2},$$ rearranging


 * $$V_o = (V_a - V_b)\alpha,$$ where $$\alpha$$= the opamp open loop gain


 * $$(V_o + {I_s}{R_3} - \frac{R_1 + R_2})\alpha = V_o,$$ substituting for $$V_a$$ and $$V_b$$


 * $$\alpha{V_o}(1 - \frac{R_1 + R_2}) + \alpha{I_s}{R_3} = V_o,$$ rearranging


 * $${V_o}(\alpha(1 - \frac{R_1 + R_2}) - 1) = -\alpha{I_s}{R_3},$$ rearranging


 * $${V_o}(1 - \frac{R_1 + R_2}) = -{I_s}{R_3},$$ cancelling out the $$\alpha$$ term and simplifying


 * $$V_o = V_s - {I_s}{R_3},$$ shown earlier


 * $$V_s(1 - \frac{R_1 + R_2}) - {I_s}{R_3}(1 - \frac{R_1 + R_2}) + {I_s}{R_3} = 0,$$ substituting for $$V_o$$


 * $$V_s(\frac{R_1 + R_2}) = - {I_s}{R_3}(\frac{R_1 + R_2})),$$


 * $$\frac{V_s}{I_s} = -\frac{R_2}$$