\left.\begin{aligned} Z_{1} &= (R_{2} + R_{4}) \enspace || \enspace Z_{C_{2}} = \frac{1}{j\omega C_{2}+\frac{1}{R_{2}+R_{4}}} \\ \\ V_{+} &= \frac{R_{s}}{R_{s} + Z_{C_{3}}}V_{\text{in}} = \frac{j\omega R_{5} C_{3}}{1 + j\omega R_{5} C_{3}}V_{\text{in}} \\ \\ \\ V_{\text{out}} &= \left(1+\frac{Z_{1}}{R_{1}}\right)V_{-} = \left(1 + \frac{Z_{1}}{V_{1}}\right)V_{+} \end{aligned}\right\} \begin{aligned} \frac{V_{\text{out}}}{V_{\text{in}}} &= \left(1 + \frac{1}{R_{1}\left(j\omega R_{1}C_{2}+\frac{1}{R_{2}+R_{4}}\right)}\right)\left(\frac{j\omega R_{5} C_{3}}{1 + j\omega R_{5} C_{3}}\right) \\ &= \left[R_{5}C_{3}\frac{R_{2} + R_{4}}{R_{1}}+R_{5}C_{3}\right]\left[\frac{\left(1+j\omega\frac{R_{1}(R_{2} + R_{4})C_{2}}{R_{1}+R_{2}+R_{4}}\right)(j\omega)}{(1+j\omega [R_{2}+R_{4}]C_{2})(1+j\omega R_{5}C_{3})}\right] \end{aligned} \\ \\ K = R_{5}C_{3}\left(\frac{R_{2}+R_{4}}{R_{1}}+1\right), \enspace\enspace \omega_{1} = \frac{R_{1}+R_{2}+R_{4}}{R_{1}(R_{2}+R_{4})C_{2}}, \enspace\enspace \omega_{2} = \frac{1}{(R_{2}+R_{4})C_{2}}, \enspace\enspace \omega_{3} = \frac{1}{R_{5}C_{3}} \end{alignat}$$Operational Amplifier
An operational amplifier (op-amp) is a high-gain, direct current (DC) coupled electronic device commonly used in analog signal processing. It amplifies the difference between its two input voltages and (non-inverting and inverting), and produces an output voltage for some op-amp gain . Additionally, an op-amp contains power supply inputs , which act as voltage rails, such that , or is the op-amp’s linear dynamic range.
\usepackage{circuitikz} \begin{document} \begin{circuitikz}[american, scale=0.8] % Define the op-amp triangle \draw[very thick] (8,0) -- (1,4.5) -- (1,-4.5) -- cycle; \fill[gray, opacity = 0.1] (8,0) -- (1,4.5) -- (1,-4.5) -- cycle; % Light blue fill inside the triangle % Positive input connection \draw[thick] (0, 2) node[left]{$V_{+}$} to [short, o-, i=$i_{+}$] (3, 2); % Negative input connection \draw[thick] (0, -2) node[left]{$V_{-}$} to [short, o-, i=$i_{-}$] (3, -2); % Input resistance Ri \draw[thick] (3,2) to[R, R=$R_i$, v_=\footnotesize $(V_{+}-V_{-})$, i] (3,-2); % Voltage-controlled voltage source \draw[thick] (4.5, -1.5) -- (4.5, -1.5) node[ground]{} to[cV, v_=\footnotesize$A(V_{+} - V_{-})$, invert] (4.5, 0) to[R, R=$R_o$] (8, 0) to[short, -o, i=$i_o$] (9, 0) node[below]{$V_o$} node[above]{+}; \end{circuitikz} \end{document}An ideal operational amplifier follows the following “golden” rules:
- The current of both differential inputs is zero, such that .
- The voltages of both differential inputs are equal to one another, such that .
- This is all caused by setting . Additionally, , and the op-amp gain .