Lecture Notes:Modeling real systems with ideal elements

Op amp circuit example

For convenience, let $ s = i\omega $. The impedance of the capacitor is $ \frac{1}{Cs} $

Applying the golden rule: $ v_+ = v_- = 0 $

KCL at the $ V_- $ node: $ i_{in} + \frac{V_A}{R_1} + \frac{V_O}{1/Cs} = 0 $

Simplifying: $ R_1 i_{in} + V_A + R_1 C s V_O = 0 $

Solving for $ V_A $: $ V_A = -R_1 i_{in} - R_1 C s V_O \quad \quad (1) $

KCL at the $ V_A $ node: $ -\frac{V_A}{R_1} - \frac{V_A}{R_2} + \frac{V_O - V_A}{R_3} = 0 $

Simplifying: $ V_A \left ( R_2 R_3 + R_1 R_3 + R_1 R_2 \right ) = V_O \left ( R_1 R_2 \right ) $

Solving for $ V_A $: $ V_A = \frac{V_O ( R_1 R_2 )}{R_1 R_2 + R_1 R_3 + R_2 R_3} \quad \quad (2) $

Setting equations 1 and 2 equal: $ -R_1 i_{in} - R_1 C s V_O = \frac{V_O ( R_1 R_2 )}{R_1 R_2 + R_1 R_3 + R_2 R_3} $

Simplifying: $ i_in = -V_O \left [ C s + \frac{R_2}{R_1 R_2 + R_1 R_3 + R_2 R_3}\right ] = -V_O \left [ C s + \frac{R_2}{R_1 R_2 + R_1 R_3 + R_2 R_3}\right ] $

Characteristics of the low pass frequency response

Magnitude of first order low pass filter transfer function H(ω) = 1/(1 + ω) in decibels

• The magnitude of the transfer function is approximately 1 at low frequ