Active filter calculations - t-shaw374/209_group7_2026 GitHub Wiki

Overview

Purpose: Subtract the two input signals (Vvs and Vis), apply gain, and low-pass filter the result before measurement. Find the frequency response (Bode plot) and choose component values.

Input: 500 Hz source signals, Vis (inverting) and Vvs (non-inverting)
Output: filtered difference signal, offset by the 2.1 V reference
Supply: 5 V and GND (U5B)

Circuit

Item	Description
U5B	Op-amp, 5 V supply
Vis	Signal input, through R20 to the inverting input
Vvs	Signal input, through R22 to the non-inverting input
R19 \|\| C14	Feedback from output to inverting input
R23, C16	Divider and filter at the non-inverting input
Vref	2.1 V reference at the bottom of R23
D7, D9	Input clamp diodes

Assumptions

Ideal op-amp: infinite gain, no input current, so V- = V+
D7 and D9 are open circuits for small-signal AC, so they are ignored
Vref is DC only. By superposition it only sets the output offset, so set Vref = 0 for the Bode plot
Notation: w = 2pif, and j is the imaginary unit

Initial: General transfer function

Step 1: Non-inverting node

KCL at V+ (no current into the op-amp):

(Vvs - V+)/R22 + (Vref - V+)/R23 = V+ * jwC16

Multiply through by R22*R23 and solve:

        Vvs*R23 + Vref*R22
V+ = --------------------------------
      R22*R23*jwC16 + R22 + R23

Step 2: Op-amp stage

Feedback impedance (R19 in parallel with C14):

        R19
Zf = --------------
      1 + jw*R19*C14

Output:

Vout = -(Zf/R20) * Vis + (1 + Zf/R20) * V+

In components:

               R19                          R19 + R20*(1 + jw*R19*C14)
Vout = - ------------------------ * Vis + ----------------------------- * V+
          R20 + jw*R20*R19*C14               R20 + jw*R20*R19*C14

Step 3: Combine

Substituting V+ from Step 1:

               R19                          R19 + R20*(1 + jw*R19*C14)     Vvs*R23 + Vref*R22
Vout = - ------------------------ * Vis + ----------------------------- * -------------------------------
          R20 + jw*R20*R19*C14               R20 + jw*R20*R19*C14           R22*R23*jwC16 + R22 + R23

Issue: This has two poles and one zero. The two inputs see different frequency responses, so the circuit is not a true difference amplifier and common-mode rejection is poor.

Matching the two sides

The circuit is a true difference amplifier when the V+ network matches the feedback network:

R23/R22 = R19/R20     and     R22*C16 = R20*C14

Simplest way to meet both:

R23 = R19,   R22 = R20,   C16 = C14

More general family (scales the Vvs network):

R22 = k*R20,   R23 = k*R19,   C16 = C14/k

Simplification

With the matched values, the denominator of V+ becomes:

R22*R23*jwC16 + R22 + R23  ->  R19 + R20 + jw*R19*R20*C14

This is identical to the numerator of the non-inverting gain, so they cancel:

R19 + R20*(1 + jw*R19*C14) = R19 + R20 + jw*R19*R20*C14

Result:

        (R19/R20) * (Vvs - Vis)          Vref
Vout = ------------------------- + ----------------
          1 + jw*R19*C14            1 + jw*R19*C14

Difference gain: R19/R20
One low-pass pole at w = 1/(R19*C14)
Vref passes through with unity DC gain, so it sets the output offset

Bode plot

With Vref = 0:

                 Vout           R19/R20
H(jw) = ------------- = ----------------
          Vvs - Vis       1 + jw*R19*C14

Magnitude

|H| = (R19/R20) / sqrt(1 + (w*R19*C14)^2)

|H|dB = 20*log10(R19/R20) - 10*log10(1 + (w*R19*C14)^2)

Phase

angle(H) = -atan(w*R19*C14)

For the Vis input alone, add 180 degrees (inverting input).

Corner frequency

fc = 1 / (2*pi*R19*C14)

Region	Magnitude	Phase
f << fc	flat at 20*log10(R19/R20) dB	about 0 deg
f = fc	3 dB below passband	-45 deg
f >> fc	falls at -20 dB/decade	tends to -90 deg