Augmented differentiator

E_O = -R_O E_I / R_I - R_O C_I dE_I/dt = -E_I - (1/100) dE_I/dt

The circuit

The schematic, drawn by copperhead from the circuit's netlist

The schematic is drawn by copperhead’s drafting engine from this circuit’s netlist, with KiCad’s own library symbols, and it opens in KiCad as figure/augmented_differentiator.kicad_sch. The op amp is KiCad’s generic one, since the handbook’s are ideal, and each terminal is a test point named as the program names it. KiCad reads back from the sheet exactly the connections the circuit has; draw_figures.py refuses to write one that does not.

The interconnect view, fang's own projection

The interconnect view is fang’s own projection. It names the parts as the program does, so it reads against the code below.

What the program says

R_I makes an inverting amplifier of gain a_dc = -R_O/R_I = -1, C_I a differentiator of time_constant = R_O C_I = 10 ms, and the summing point adds them. In frequency terms E_O/E_I = -(1 + j 2π f × 10 ms): the two terms are equal at f_equal = 1/(2π R_I C_I) = 15.9 Hz.

R2 is bias compensation, the DC resistance the - input sees, and the program holds it to that:

require(equals(self.r_bias.resistance, parallel(r_i, r_o)))

The bench’s op amp has no bias current, so the simulation shows only that R2 does no harm. The figure gives every value, so nothing was chosen.

What the simulation found

out/simulation.txt, from the decks under out/spice/:

Run	Measured	Claimed
input_term, E_I = 1 V DC	-1	-1 (a_dc), holds
both_terms, gain at 1 Hz	1.002	|1 + j 0.063| = 1.002, holds
both_terms, gain at 15.9 Hz	1.414	|1 + j| = 1.414, holds
both_terms, phase at 15.9 Hz	-2.356 rad	-3π/4, holds
both_terms, gain at 159 Hz	10.05	|1 + j 10| = 10.05, holds
peak, where the output turns real	12.62 kHz	not a claim
peak, height	107.6 dB	not a claim

The page does not mention it, but the derivative term has no stop, so this circuit shares the plain differentiator’s peak where its rising noise gain meets the op amp’s roll-off, near √(10 MHz × 15.9 Hz) = 12.6 kHz. A practical version would need the same stop as the circuits above it.

Running it

fang check examples/ti_opamp_handbook/differentiators/augmented_differentiator/augmented_differentiator.py
python examples/regenerate.py ti_opamp_handbook/differentiators/augmented_differentiator   # needs ngspice

The whole program

examples/ti_opamp_handbook/differentiators/augmented_differentiator/augmented_differentiator.py

"""The augmented differentiator, SBOA092B page 62.
Show 22 more lines

    E_O = -(R_O/R_I) E_I - R_O C_I dE_I/dt = -E_I - (1/100) dE_I/dt

C_I and R_I sit side by side between E_I and the summing point, and R_O runs
from the output back to it. R_I makes an inverting amplifier of gain
-R_O/R_I = -1, C_I a differentiator of time constant R_O C_I = 10 ms, and the
summing point adds the two currents: "sums input and its derivative". In
frequency terms the gain is -(1 + j 2 pi f 10 ms), flat at 1 up to
1/(2 pi 10 ms) = 15.9 Hz and a derivative above it.

R2, 50 kOhm from the + input to ground, carries no signal. It is R_I in
parallel with R_O, the resistance the - input sees to ground at DC, so the
two input bias currents drop the same voltage and cancel. The program holds it
to that rule. The bench's op amp has no bias current, so the simulation cannot
show what R2 is for; it shows only that R2 does no harm.

The derivative has no stop, so this circuit shares the plain differentiator's
peak near 12.6 kHz, where its rising noise gain meets the op amp's roll-off.
The page does not mention it; the bench measures it, and does not claim it.

The figure gives every value, so nothing was chosen.
"""

import sys
from pathlib import Path

# The handbook's shared parts and bench live in the folder above the sections.
sys.path.insert(0, str(Path(__file__).resolve().parents[2]))

from decimal import Decimal

from fang.lang import Hz, Parameter, System, kOhm, ms, require, uF
from fang.parts import Capacitor, Resistor
from fang.rationale import Cites
from fang.simulation import ACSweep, OperatingPoint

from handbook import (
    Bench,
    Claim,
    Ground,
    OpAmp,
    Run,
    Terminal,
    corner,
    equals,
    negative,
    over,
    parallel,
    product,
    ratio,
    within,
)


class AugmentedDifferentiator(System):
    """E_I through C_I and R_I side by side into the summing point, R_O back."""

    figure = Cites(
        "E_O = -R_O E_I / R_I - R_O C_I dE_I/dt; E_O = -E_I - (1/100) dE_I/dt. "
        "Sums input and its derivative",
        document="SBOA092B, Handbook of Operational Amplifier Applications",
        locator="page 62, Augmented Differentiator",
    )

    a_dc = Parameter("1", default=-1 * ratio, description="-R_O / R_I, the E_I term")
    time_constant = Parameter(
        "s", default=10 * ms, description="R_O C_I, the dE_I/dt term: 1/100 s"
    )
    f_equal = Parameter(
        "Hz",
        default=Decimal("15.9155") * Hz,
        description="1/(2 pi R_I C_I): where the two terms are equal",
    )

    e_in = Terminal()
    common = Terminal()
    e_out = Terminal()
    c_in = Capacitor(capacitance=Decimal("0.1") * uF)
    r_in = Resistor(resistance=100 * kOhm)
    r_out = Resistor(resistance=100 * kOhm)
    r_bias = Resistor(resistance=50 * kOhm)
    amp = OpAmp()
    ground = Ground()

    def architecture(self):
        self.e_in.probe >> self.c_in.p1
        self.e_in.probe >> self.r_in.p1
        self.c_in.p2 >> self.amp.inverting.signal
        self.r_in.p2 >> self.amp.inverting.signal
        self.amp.inverting.signal >> self.r_out.p1
        self.r_out.p2 >> self.amp.output.signal
        self.amp.output.signal >> self.e_out.probe
        self.amp.non_inverting.signal >> self.r_bias.p1
        self.r_bias.p2 >> self.ground.node
        self.common.probe >> self.ground.node

    def constraints(self):
        r_i, c_i, r_o = self.r_in.resistance, self.c_in.capacitance, self.r_out.resistance
        require(equals(self.a_dc, negative(over(r_o, r_i))))
        require(equals(self.time_constant, product(r_o, c_i)))
        # The two terms meet where C_I's reactance equals R_I.
        require(within(self.f_equal, corner(r_i, c_i), 0.00001))
        # Bias compensation: the + input sees what the - input sees at DC.
        require(equals(self.r_bias.resistance, parallel(r_i, r_o)))


BENCH = Bench(
    page=62,
    title="Augmented Differentiator",
    runs=[
        Run(
            "input_term",
            OperatingPoint(),
            drive={"e_in": "DC 1"},
            measure={"gain": "v({e_out.1}) / v({e_in.1})"},
            claims=[Claim("gain", "a_dc", within=0.001,
                          note="At DC the derivative is zero and only -E_I is left")],
        ),
        Run(
            "both_terms",
            ACSweep(points=100, start="1", stop="1k"),
            drive={"e_in": "DC 0 AC 1"},
            measure={
                "gain_equal": "find vm({e_out.1}) at=15.9155",
                "phase_equal": "find vp({e_out.1}) at=15.9155",
                "gain_159": "find vm({e_out.1}) at=159.155",
                "gain_1": "find vm({e_out.1}) at=1",
            },
            claims=[
                Claim("gain_equal", 1.41421, within=0.001,
                      note="|1 + j| at f_equal: the two terms equal and 90 degrees apart"),
                Claim("phase_equal", -2.35619, within=0.001,
                      note="-3 pi/4 radians: -(1 + j), inverted and 45 degrees ahead"),
                Claim("gain_159", 10.0499, within=0.001,
                      note="|1 + j 10| a decade up, where the derivative dominates"),
                Claim("gain_1", 1.00197, within=0.001,
                      note="|1 + j 2 pi 1 Hz 10 ms|: at 1 Hz, nearly E_I alone"),
            ],
            note=(
                "The page's E_O = -E_I - (1/100) dE_I/dt, in frequency terms: "
                "E_O/E_I = -(1 + j 2 pi f x 0.01 s)."
            ),
        ),
        Run(
            "peak",
            ACSweep(variation="lin", points=2401, start="12k", stop="13.2k"),
            drive={"e_in": "DC 0 AC 1"},
            measure={
                "f_peak": "when vi({e_out.1})=0",
                "gain_peak_db": "max vdb({e_out.1})",
            },
            units={"f_peak": "Hz", "gain_peak_db": "dB"},
            note=(
                "Not a claim of the page. The derivative has no stop, so the "
                "noise gain rises like the plain differentiator's and meets the "
                "10 MHz op amp's roll-off near sqrt(GBW x 15.9 Hz) = 12.6 kHz."
            ),
        ),
    ],
)

The files it writes

The parts, then the nets and the pads on them.

out/netlist.txt

C1     0.1 uF           -
GND1   Ground           -
R1     50 kOhm          -
R2     100 kOhm         -
R3     100 kOhm         -
TP1    Terminal         -
TP2    Terminal         -
TP3    Terminal         -
U1     OpAmp            -
Net-(C1-Pad1)            C1.1 R2.1 TP2.1
Net-(C1-Pad2)            C1.2 R2.2 R3.1 U1.IN-
Net-(GND1-Pad1)          GND1.1 R1.2 TP1.1
Net-(R1-Pad1)            R1.1 U1.IN+
Net-(R3-Pad2)            R3.2 TP3.1 U1.OUT

Every check that ran, and every one left undecided.

out/checks.txt

4 checks, 0 failed, 0 undecided

What the elaborated graph contains, by entity kind.

out/graph.txt

    1  block
    9  component
   20  connection
    4  constraint
    1  evidence
    3  interface
   15  pin
   15  port
   68  total
snapshot sha256:8f80c31ca51cdfe192801f0aeabced58d61cbe27cb274ce9d1985451258e4fd7

All of it, including the KiCad netlist, is in examples/ti_opamp_handbook/differentiators/augmented_differentiator/out/. Rebuild it with:

fang build examples/ti_opamp_handbook/differentiators/augmented_differentiator/augmented_differentiator.py