20.309:Recitation 092107

20.309 Fall Semester 2007 Recitation Notes for 9/21/2007

Review of ideal circuit elements

Linear passive

Independent sources

Dependent sources

Nonlinear

Modeling real components with ideal elements

Modeling a battery

Modeling an op amp

Sallen Key circuit

Approach to solving the Sallen Key circuit

1. Apply the Golden Rules
2. Apply KCL at the [math]\displaystyle{ V_x }[/math] node
3. Apply KCL at the [math]\displaystyle{ V_- }[/math] node
4. Solve one equation for [math]\displaystyle{ V_x }[/math] and substitute into the other
5. Rewrite the result in the form of a transfer function [math]\displaystyle{ V_o / V_i }[/math]

The gruesome details

=Apply the Golden Rules

In the Sallen Key circuit, a wire connects [math]\displaystyle{ V_- }[/math] to [math]\displaystyle{ V_+ }[/math]. Therefore, [math]\displaystyle{ V_- = V_+ = V_o }[/math]. This will be a useful substitution when applying KCL.

Apply KCL at the [math]\displaystyle{ V_x }[/math] node

KCL

[math]\displaystyle{ \frac{V_i-V_x}{R_1} + \frac{V_o-V_x}{R_2} + C_1 s (V_o - V_x) = 0 }[/math]      (1)

Multiply by [math]\displaystyle{ R_1 R_2 }[/math]

[math]\displaystyle{ R_2(V_i-V_x) + R_1(V_o-V_x) + R_1 R_2 C_1 s (V_o - V_x) = 0 }[/math]      (2)

Gather terms

[math]\displaystyle{ R_2 V_i + (-R_2-R_1-R_1 R_2 C_1 s) V_x + (R_1 + R_1 R_2 C_1 s)V_o = 0 }[/math]      (3)

Apply KCL at the [math]\displaystyle{ V_- }[/math] node

KCL

[math]\displaystyle{ \frac{V_x-V_o}{R_2} - V_o C_2 s = 0 }[/math]      (4)

Multiply by [math]\displaystyle{ R_2 }[/math]

[math]\displaystyle{ V_x=V_o (1 + R_2 C_2 s) }[/math]      (5)

Substitute for [math]\displaystyle{ V_x }[/math] (equation 5 into equation 3)