electronics-engineering-ai999x — offline, verified electronics engineering solver with step-by-step answers

Desktop app & command line · Compute in your browser · Install (Python) · Guides · Evaluation · AI999X on Hugging Face · Website · Support: Buy Me a Coffee

License: Apache-2.0 · Author: Ibnul Jaif Farabi (creator and engineer) · Organization: AI999X · Version 1.0.0

Field: Engineering · Tasks: Question Answering, Sentence Similarity, Feature Extraction

electronics-engineering-ai999x is an offline, deterministic Python solver that answers electronics engineering questions with verified results from 92 certified capabilities, gives 561 documented answers identically on Windows and Linux, and refuses rather than guesses when an answer cannot be verified.

electronics-engineering-ai999x is a compact, deterministic electronics engineering engine for Python. It reads a question in plain English (or a typed call), picks a certified electronics engineering capability, computes the answer exactly, checks it, and returns the result with its steps, units, LaTeX and assumptions — or an honest refusal that says what is missing. It runs fully offline on any CPU, needs no GPU, no API key and no internet, and is built to give zero wrong answers: when it is not sure, it refuses instead of guessing.

Run it your way. electronics-engineering-ai999x comes as the AI999X desktop app and as the command-line program electronics-engineering for Windows and Linux, runs in your browser as WebAssembly, and installs as a Python package. The desktop app, the command line and the browser run the same compiled Rust program on your own device: offline, no Python needed.

Get it

Way to run it	For	Get it
AI999X desktop app 1.1.0	Windows, Linux — every AI999X model in one app: search, ask in words, forms, live figures, explanations, offline	Windows (x86-64) · Linux (x86-64)
Command line (electronics-engineering)	Windows, Linux — one program per model, offline, no Python	Windows (x86-64) · Linux (x86-64)
In your browser	any device — nothing to install	Compute on the model page
Python	Python 3.10+	pip install electronics-engineering-ai999x

Checksums: SHA256SUMS · every version: https://www.ai999x.com/downloads/desktop/latest/ · install guide: desktop & command line

Quick start

Desktop app

Download the app for your system (above), unpack it and start it; choose Electronics Engineering, type a question in words or pick a capability, fill in the form and press Compute. Figures and an explanation of the answer appear beside it.

Linux, from a terminal:

curl -LO https://www.ai999x.com/downloads/desktop/1.1.0/ai999x-1.1.0-linux-x86_64.tar.gz
curl -LO https://www.ai999x.com/downloads/desktop/1.1.0/SHA256SUMS
sha256sum -c --ignore-missing SHA256SUMS
tar xzf ai999x-1.1.0-linux-x86_64.tar.gz
./ai999x-1.1.0-linux-x86_64/ai999x

Command line

Linux:

curl -LO https://www.ai999x.com/downloads/desktop/1.1.0/ai999x-cli-1.1.0-linux-x86_64.tar.gz
curl -LO https://www.ai999x.com/downloads/desktop/1.1.0/SHA256SUMS
sha256sum -c --ignore-missing SHA256SUMS
mkdir -p ~/.local/share && tar xzf ai999x-cli-1.1.0-linux-x86_64.tar.gz -C ~/.local/share
echo 'export PATH="$HOME/.local/share/ai999x-cli-1.1.0-linux-x86_64:$PATH"' >> ~/.bashrc && source ~/.bashrc
electronics-engineering --version

Windows (PowerShell):

$F = "ai999x-cli-1.1.0-windows-x86_64.zip"
Invoke-WebRequest "https://www.ai999x.com/downloads/desktop/1.1.0/$F" -OutFile $F
Invoke-WebRequest "https://www.ai999x.com/downloads/desktop/1.1.0/SHA256SUMS" -OutFile SHA256SUMS
(Get-FileHash $F -Algorithm SHA256).Hash.ToLower() -eq ((Select-String -SimpleMatch $F SHA256SUMS).Line -split ' ')[0]
Expand-Archive $F -DestinationPath "$env:LOCALAPPDATA\AI999X" -Force
$D = "$env:LOCALAPPDATA\AI999X\ai999x-cli-1.1.0-windows-x86_64"; $env:Path += ";$D"
[Environment]::SetEnvironmentVariable("Path", [Environment]::GetEnvironmentVariable("Path", "User") + ";$D", "User")
electronics-engineering --version

Every command below was run as shown; the text under it is what the program printed.

What a capability needs — The equation or method, every input and output with its SI unit. Read this before solve: it tells you the input names.

$ electronics-engineering info elx_bjt_voltage_divider_bias
BJT voltage-divider bias: exact Q-point
  VTH = VCC R2/(R1 + R2); RTH = R1 || R2; IB = (VTH - VBE)/(RTH + (beta + 1) RE); VCE = VCC - IC RC - IE RE
  in   VCC      supply voltage [V]
  in   R1       upper divider resistor (supply to base) [ohm]
  in   R2       lower divider resistor (base to ground) [ohm]
  in   RC       collector resistor [ohm]
  in   RE       emitter resistor [ohm]
  in   beta     current gain [1]
  in   VBE      base-emitter voltage [V]
  out  VTH      Thevenin voltage of the divider [V]
  out  RTH      Thevenin resistance of the divider [ohm]
  out  IB       base current [A]
  out  IC       collector current [A]
  out  IE       emitter current [A]
  out  VB       base voltage [V]
  out  VE       emitter voltage [V]
  out  VC       collector voltage [V]
  out  VCE      collector-emitter voltage [V]

Solve — Inputs as name=value, each with any unit of the right kind. The answer lists every output, the method used, each input converted to SI, and the conditions the answer assumes.

$ electronics-engineering solve elx_bjt_voltage_divider_bias "VCC=12 V" "R1=47 kohm" "R2=10 kohm" "RC=2.2 kohm" "RE=470 ohm" beta=150.0 "VBE=0.7 V"
VTH = 2.105263158 V
RTH = 8245.614035 ohm
IB = 1.773972436e-05 A
IC = 0.002660958654 A
IE = 0.002678698378 A
VB = 1.958988238 V
VE = 1.258988238 V
VC = 6.145890962 V
VCE = 4.886902724 V
  by: BJT voltage-divider bias: exact Q-point  (VTH = VCC R2/(R1 + R2); RTH = R1 || R2; IB = (VTH - VBE)/(RTH + (beta + 1) RE); VCE = VCC - IC RC - IE RE)
  VCC = 12 V
  R1 = 47000 ohm
  R2 = 10000 ohm
  RC = 2200 ohm
  RE = 470 ohm
  beta = 150
  VBE = 0.7 V
  assumes: VTH > VBE; active region: VCE >= 0.2 V

Many questions at once — One JSON request per line in, one JSON answer per line out, in order: for scripts, notebooks and agents (the program starts once). In PowerShell: Get-Content cases.jsonl | electronics-engineering batch

cases.jsonl:

{"key": "elx_bjt_voltage_divider_bias", "inputs": {"VCC": "12 V", "R1": "47 kohm", "R2": "10 kohm", "RC": "2.2 kohm", "RE": "470 ohm", "beta": 150.0, "VBE": "0.7 V"}}
{"key": "elx_sallen_key_lowpass", "inputs": {"R1": "10 kohm", "R2": "10 kohm", "C1": "20 nF", "C2": "10 nF", "K": 1.0}}

$ electronics-engineering batch < cases.jsonl
{"outcome":"SOLVED","message":"","value":0.00266095865382,"unit":"A","solved_for":"IC","si":[["VCC",12.0],["R1",47000.0],["R2",10000.0],["RC",2200.0],["RE",470.0],["beta",150.0],["VBE",0.7]],"values":[["VTH",2.10526315789,"V"],["RTH",8245.61403509,"ohm"],["IB",0.0000177397243588,"A"],["IC",0.00266095865382,"A"],["IE",0.00267869837818,"A"],["VB",1.95898823774,"V"],["VE",1.25898823774,"V"],["VC",6.1458909616,"V"],["VCE",4.88690272386,"V"]],"target":"IC","phrase":"Thevenin voltage of the divider","assumptions":["VTH > VBE","active region: VCE >= 0.2 V"],"name":"BJT voltage-divider bias: exact Q-point","equation":"VTH = VCC R2/(R1 + R2); RTH = R1 || R2; IB = (VTH - VBE)/(RTH + (beta + 1) RE); VCE = VCC - IC RC - IE RE"}
{"outcome":"SOLVED","message":"","value":0.707106781187,"unit":"","solved_for":"Q","si":[["R1",10000.0],["R2",10000.0],["C1",2e-8],["C2",1e-8],["K",1.0]],"values":[["f0",1125.3953952,"Hz"],["Q",0.707106781187,""],["H0",1.0,""],["peak_gain",0.707106781187,""]],"target":"Q","phrase":"natural frequency","assumptions":["stable: R1 C2 + R2 C2 + R1 C1 (1 - K) > 0","ideal amplifier"],"name":"Sallen-Key low-pass filter: f0, Q and gain","equation":"w0 = 1/sqrt(R1 R2 C1 C2); Q = sqrt(R1 R2 C1 C2)/(R1 C2 + R2 C2 + R1 C1 (1 - K))"}

In your browser

Open Electronics Engineering on ai999x.com, choose a capability and press Compute: the model runs on your own device as WebAssembly; nothing is sent anywhere. Change any input (any unit of the right kind) and the answer, its explanation and the figures follow.

In your own page: electronics_engineering_ai999x.wasm and the loader ax-wasm.js (https://www.ai999x.com/compute/rust/ax-wasm.js).

Python

pip install electronics-engineering-ai999x

Worked examples in every form

The same worked example from the command line, from Python and in the browser. Each command-line output is what the program printed.

Butterworth filter: Q of each second-order section

The quality factors of the cascaded second-order sections of an n-th order Butterworth filter (all at the cutoff frequency), and whether a first-order section is needed.

Command line:

$ electronics-engineering solve elx_butterworth_section_q n=4
Q_max = 1.306562965
Q = [0.541196100146,1.30656296488]
sections = 2
first_order = 0
  by: Butterworth filter: Q of each second-order section  (Q_k = 1/(2 sin((2k - 1) pi/(2n))), k = 1 .. floor(n/2))
  assumes: 2 <= n <= 20

Python:

import electronics_engineering_ai999x as m

a = m.solve("elx_butterworth_section_q", n="4")
print(a.values or (a.value, a.unit))

Browser: open this capability on the model page and press Compute in your browser.

Butterworth or Chebyshev low-pass filter order

The minimum order of a Butterworth or Chebyshev type I low-pass filter meeting a passband edge with at most Ap dB attenuation and a stopband edge with at least As dB.

Command line:

$ electronics-engineering solve elx_filter_order ftype=Butterworth "fp=1000 Hz" "fs=2500 Hz" Ap=1 As=40
n = 6
  by: Butterworth or Chebyshev low-pass filter order  (Butterworth: n >= log10(D)/(2 log10(fs/fp)); Chebyshev: n >= acosh(sqrt D)/acosh(fs/fp), D = (10^(As/10) - 1)/(10^(Ap/10) - 1))
  fp = 1000 Hz
  fs = 2500 Hz
  Ap = 1
  As = 40
  ftype = Butterworth
  assumes: fs > fp; As > Ap > 0

Python:

import electronics_engineering_ai999x as m

a = m.solve("elx_filter_order", ftype="Butterworth", fp="1000 Hz", fs="2500 Hz", Ap="1", As="40")
print(a.values or (a.value, a.unit))

Browser: open this capability on the model page and press Compute in your browser.

Chebyshev type I filter: section Q and frequencies

The quality factors and natural frequencies (relative to the passband edge) of the second-order sections of an n-th order Chebyshev type I low-pass filter with a given passband ripple, and the real pole of an odd order.

Command line:

$ electronics-engineering solve elx_chebyshev_sections n=4 ripple=0.5
Q_max = 2.940554174
Q = [0.705110236803,2.94055417439]
f0_ratio = [0.59700239512,1.03127040148]
sections = 2
real_pole = 0
  by: Chebyshev type I filter: section Q and frequencies  (eps = sqrt(10^(r/10) - 1); a = asinh(1/eps)/n; poles -sinh a sin th_k + j cosh a cos th_k; Q = |p|/(2 |Re p|))
  ripple = 0.5
  assumes: 2 <= n <= 16; 0.001 dB <= ripple <= 6 dB

Python:

import electronics_engineering_ai999x as m

a = m.solve("elx_chebyshev_sections", n="4", ripple="0.5")
print(a.values or (a.value, a.unit))

Browser: open this capability on the model page and press Compute in your browser.

Equal-component Sallen-Key low-pass design

The resistor value and amplifier gain of an equal-component (R1 = R2 = R, C1 = C2 = C) Sallen-Key low-pass filter for a cutoff frequency and Q, with the non-inverting gain ratio Rf/Rg = K - 1.

Command line:

$ electronics-engineering solve elx_sallen_key_equal_component_design "fc=1000 Hz" Q=0.7071 "C=10 nF"
R = 15915.49431 ohm
K = 1.585772875
Rf_over_Rg = 0.5857728751
  by: Equal-component Sallen-Key low-pass design  (R = 1/(2 pi fc C); K = 3 - 1/Q; Rf/Rg = K - 1)
  fc = 1000 Hz
  Q = 0.7071
  C = 1e-08 F
  assumes: 0.5 <= Q <= 20

Python:

import electronics_engineering_ai999x as m

a = m.solve("elx_sallen_key_equal_component_design", fc="1000 Hz", Q="0.7071", C="10 nF")
print(a.values or (a.value, a.unit))

Browser: open this capability on the model page and press Compute in your browser.

Documentation

• Rust user guide — the command line, every capability with a command and what it printed: PDF · Markdown
• Python guide — every capability with a Python example: PDF · Markdown
• Desktop app & command line — downloads, install, ask and search, figures, offline use, checksums: https://www.ai999x.com/desktop-cli/
• Model page — compute in your browser, every capability explained: https://www.ai999x.com/models/electronics-engineering-ai999x/

Highlights

• Verified, not generated. Every answer is computed by a certified electronics engineering capability and checked (substituted back, dimension-checked or compared with an independent computation) before it is returned.
• Zero wrong answers by design. Seven explicit outcomes — SOLVED, NEEDS_INPUT, AMBIGUOUS, INVALID_INPUT, OUT_OF_SCOPE, UNRECOGNIZED_FORMAT, INTERNAL_ERROR — never a silent guess. Every one of its 561 documented answers is reproduced exactly on every tested platform.
• Plain-English questions and a guaranteed typed interface. ask() reads documented question forms; solve(capability_id, **inputs) always solves a supported task with valid inputs.
• Explains itself. Steps, formulas in LaTeX, units, assumptions and the model used, ready for notebooks, tutoring apps and homework checkers.
• Native and in the browser. Every capability also computes natively in Rust: the desktop app and the electronics-engineering command line for Windows and Linux, and WebAssembly on the model page — no Python, no network.
• Small and portable. Pure Python (sympy, mpmath); identical answers on Windows and Linux (Python 3.10, 3.11, 3.12, 3.13, 3.14).
• Coverage: semiconductors, diode and transistor circuits, amplifiers, active filters, oscillators, data converters, timing and power supplies.

Model overview

Property	Value
Package	electronics-engineering-ai999x 1.0.0 (import electronics_engineering_ai999x)
Type	Deterministic neuro-symbolic electronics engineering engine (rule-based reading + certified symbolic capabilities)
Capabilities	92 certified procedures
Knowledge pack	version 1.0.0
Golden answers	561 documented answers, identical on every tested platform
Platforms tested	Windows (Python 3.13), Linux x86-64 (Python 3.10, 3.11, 3.12, 3.13, 3.14)
Runtime	CPU only, offline; no GPU, no network, no API key
License	Apache-2.0
Creator / engineer	Ibnul Jaif Farabi
Organization	AI999X

Python quick start

pip install electronics-engineering-ai999x
# or straight from this repository:
pip install https://huggingface.co/AI999X/electronics-engineering-ai999x/resolve/main/dist/electronics_engineering_ai999x-1.0.0-py3-none-any.whl

import electronics_engineering_ai999x as electronics_engineering

electronics_engineering.ask("BJT voltage-divider bias: VCC = 12 V, R1 = 47 kohm, R2 = 10 kohm, RC = 2.2 kohm, RE = 470 ohm, beta = 150, VBE = 0.7 V").show()
# -> SOLVED — VTH = 2.10526, RTH = 8245.61, IB = 1.77397e-05, IC = 0.00266096  (elx_bjt_voltage_divider_bias)
electronics_engineering.ask("Sallen-Key low-pass filter: R1 = 10 kohm, R2 = 10 kohm, C1 = 20 nF, C2 = 10 nF, K = 1").show()
# -> SOLVED — f0 = 1125.4, Q = 0.707107, H0 = 1, peak_gain = 0.707107  (elx_sallen_key_lowpass)
electronics_engineering.solve("elx_common_emitter_amplifier", IC=0.001, beta=100.0, VT=0.025, RC=5000.0, RL=10000.0, RE1=0.0, RB=20000.0, Rsig=1000.0).show()
# -> SOLVED — gm = 0.04, r_pi = 2500, Av = -133.333, Rin = 2222.22  (elx_common_emitter_amplifier)
electronics_engineering.ask("555 astable: RA = 1 kohm, RB = 10 kohm, C = 10 uF").show()
# -> SOLVED — t_high = 0.0762462, t_low = 0.0693147, T = 0.145561, f = 6.86998  (elx_555_astable)
electronics_engineering.ask("Diode operating point: VS = 5 V, R = 1 kohm, Is = 1e-14 A, n = 1, T = 300 K").show()
# -> SOLVED — ID = 0.00430746, VD = 0.692544, P_D = 0.0029831  (elx_diode_operating_point)
electronics_engineering.ask("Who created you?").message
# -> electronics-engineering-ai999x was created and engineered by Ibnul Jaif Farabi at AI999X.

The Python package also installs a Python command, electronics-engineering-ai999x "<question>"; for the native program use electronics-engineering (above). The outputs above were produced by running this exact code on electronics-engineering-ai999x 1.0.0.

How every answer is verified

1. Reading — the question is turned into typed quantities (value, unit, kind, role) and one asked quantity; every stated number must be used or the question is refused.
2. Planning — only certified capabilities may answer; a capability applies only under the conditions the wording states (for example a pendulum, a liquid at rest, boiling).
3. Computing — exact symbolic or high-precision numeric evaluation.
4. Checking — dimensions, substitution back into the defining equation, and agreement between independent readings; any disagreement becomes a refusal, never a number.

Capabilities are certified before they ship: each must reproduce hand-worked reference cases, reject a deliberately wrong variant (a control), and leave every benchmark answer unchanged.

Python worked examples, with their steps

Each question was put to electronics-engineering-ai999x; the answer is the one it returned, with the steps it took.

BJT voltage-divider bias: exact Q-point · BJT

BJT voltage-divider bias: VCC = 12 V, R1 = 47 kohm, R2 = 10 kohm, RC = 2.2 kohm, RE = 470 ohm, beta = 150, VBE = 0.7 V

→ VTH = 2.105263158 V, RTH = 8245.614035 ohm, IB = 1.773972436e-05 A, IC = 0.002660958654 A, IE = 0.002678698378 A, VB = 1.958988238 V, VE = 1.258988238 V, VC = 6.145890962 V, VCE = 4.886902724 V

Steps

• BJT voltage-divider bias: exact Q-point: VCC = 12 V, R1 = 47 kohm, R2 = 10 kohm, RC = 2.2 kohm, RE = 470 ohm, beta = 150, VBE = 0.7 V → VTH = 2.105263158 V, RTH = 8245.614035 ohm, IB = 1.773972436e-05 A, IC = 0.002660958654 A, IE = 0.002678698378 A, VB = 1.958988238 V, VE = 1.258988238 V, VC = 6.145890962 V, VCE = 4.886902724 V

Sallen-Key low-pass filter: f0, Q and gain · Active filters

Sallen-Key low-pass filter: R1 = 10 kohm, R2 = 10 kohm, C1 = 20 nF, C2 = 10 nF, K = 1

→ f0 = 1125.395395 Hz, Q = 0.7071067812, H0 = 1, peak_gain = 0.7071067812

Steps

• Sallen-Key low-pass filter: f0, Q and gain: R1 = 10 kohm, R2 = 10 kohm, C1 = 20 nF, C2 = 10 nF, K = 1 → f0 = 1125.395395 Hz, Q = 0.7071067812, H0 = 1, peak_gain = 0.7071067812

Common-emitter amplifier: gain, input and output resistance · BJT

m.solve("elx_common_emitter_amplifier", IC=0.001, beta=100.0, VT=0.025, RC=5000.0, RL=10000.0, RE1=0.0, RB=20000.0, Rsig=1000.0)

→ gm = 0.04 S, r_pi = 2500 ohm, Av = -133.3333333, Rin = 2222.222222 ohm, Rout = 5000 ohm, Gv = -91.95402299

Steps

• Common-emitter amplifier: gain, input and output resistance: IC = 0.001 A, beta = 100, VT = 0.025 V, RC = 5000 ohm, RL = 10000 ohm, RE1 = 0 ohm, RB = 20000 ohm, Rsig = 1000 ohm → gm = 0.04 S, r_pi = 2500 ohm, Av = -133.3333333, Rin = 2222.222222 ohm, Rout = 5000 ohm, Gv = -91.95402299

555 timer astable: frequency and duty cycle · Oscillators

555 astable: RA = 1 kohm, RB = 10 kohm, C = 10 uF

→ t_high = 0.07624618986 s, t_low = 0.06931471806 s, T = 0.1455609079 s, f = 6.869976385 Hz, duty = 0.5238095238

Steps

• 555 timer astable: frequency and duty cycle: RA = 1 kohm, RB = 10 kohm, C = 10 uF → t_high = 0.07624618986 s, t_low = 0.06931471806 s, T = 0.1455609079 s, f = 6.869976385 Hz, duty = 0.5238095238 Hz

Operating point of a diode in series with a resistor (exact) · Diodes

Diode operating point: VS = 5 V, R = 1 kohm, Is = 1e-14 A, n = 1, T = 300 K

→ ID = 0.004307456373 A, VD = 0.6925436275 V, P_D = 0.002983101461 W

Steps

• Operating point of a diode in series with a resistor (exact): VS = 5 V, R = 1 kohm, Is = 1e-14 A, n = 1, T = 300 K → ID = 0.004307456373 A, VD = 0.6925436275 V, P_D = 0.002983101461 W

CMOS gate propagation delay (switch-resistor model) · CMOS

CMOS propagation delay: Rn = 10000 ohm, Rp = 20000 ohm, CL = 100 fF

→ tpHL = 6.931471806e-10 s, tpLH = 1.386294361e-09 s, tp = 1.039720771e-09 s

Steps

• CMOS gate propagation delay (switch-resistor model): Rn = 10000 ohm, Rp = 20000 ohm, CL = 100 fF → tpHL = 6.931471806e-10 s, tpLH = 1.386294361e-09 s, tp = 1.039720771e-09 s

Figures

Every point was computed by electronics-engineering-ai999x.

Sallen-Key low-pass filter: f0, Q and gain

Q (quality factor) against R1 (first series resistor), with C1 = 2.00e-8 F, C2 = 1.00e-8 F, K = 1, R2 = 10000 ohm. 31 points computed by electronics-engineering-ai999x; marked: R1 = 10000, Q = 0.707107.

ADC output code, LSB and quantisation error

code (output code (decimal)) against Vin (analog input), with Vref = 5 V, n = 8. 31 points computed by electronics-engineering-ai999x; marked: Vin = 3.2, code = 163.

Operating point of a diode in series with a resistor (exact)

ID (diode current) against VS (source voltage), with Is = 1.00e-14 A, R = 1000 ohm, T = 300 K, n = 1. 31 points computed by electronics-engineering-ai999x; marked: VS = 5, ID = 0.00430746 A.

555 timer astable: frequency and duty cycle

f (frequency) against RA (resistor from Vcc to the discharge pin), with C = 1.00e-5 F, RB = 10000 ohm. 31 points computed by electronics-engineering-ai999x; marked: RA = 1000, f = 6.86998 Hz.

Capabilities

92 certified procedures. A few of them:

• Procedures: 555 timer astable: frequency and duty cycle; 555 timer monostable pulse width; ADC output code, LSB and quantisation error; ADC quantisation SNR with oversampling; Adjustable three-terminal regulator output voltage; BJT beta cutoff and transition frequency fT; BJT cascode: output resistance, transconductance and gain; BJT collector-feedback bias: Q-point; BJT current mirror with finite beta; BJT differential pair: differential and common-mode gain, CMRR; BJT dual-supply emitter bias: Q-point; BJT fixed (base-resistor) bias: Q-point; BJT small-signal (hybrid-pi) parameters; BJT terminal currents from beta; BJT voltage-divider bias: exact Q-point; Binary-weighted resistor DAC; Built-in potential of a pn junction; Butterworth filter: Q of each second-order section; Butterworth or Chebyshev low-pass filter order; CMOS dynamic (switching) power; CMOS gate propagation delay (switch-resistor model); CMOS inverter switching threshold VM; Capacitor-input filter: ripple voltage and DC output; Chebyshev type I filter: section Q and frequencies …

electronics_engineering_ai999x.templates() lists every question form ask() reads; the capability guide (PDF, every example executed) is included in this repository.

All 92 formulas

Show every formula, by topic

BJT / DC bias

BJT collector-feedback bias: Q-point

$$I_B=\frac{V_{CC}-V_{BE}}{R_F+(\beta +1)(R_C+R_E)}I\_B\; =\; \backslash frac\{V\_\{CC\}\; -\; V\_\{BE\}\}\{R\_F\; +\; (\backslash beta+1)(R\_C\; +\; R\_E)\}$$

BJT dual-supply emitter bias: Q-point

$$I_E=\frac{V_{EE}-V_{BE}}{R_E+R_B\mathrm{/}(\beta +1)}I\_E\; =\; \backslash frac\{V\_\{EE\}\; -\; V\_\{BE\}\}\{R\_E\; +\; R\_B/(\backslash beta+1)\}$$

BJT fixed (base-resistor) bias: Q-point

$$I_B=\frac{V_{CC}-V_{BE}}{R_B},V_{CE}=V_{CC}-\beta I_B{R}_{C}I\_B\; =\; \backslash frac\{V\_\{CC\}\; -\; V\_\{BE\}\}\{R\_B\},\backslash quad\; V\_\{CE\}\; =\; V\_\{CC\}\; -\; \backslash beta\; I\_BR\_C$$

BJT voltage-divider bias: exact Q-point

$$I_B=\frac{V_{TH}-V_{BE}}{R_{TH}+(\beta +1)R_E}I\_B\; =\; \backslash frac\{V\_\{TH\}\; -\; V\_\{BE\}\}\{R\_\{TH\}\; +\; (\backslash beta+1)R\_E\}$$

BJT / amplifiers

BJT cascode: output resistance, transconductance and gain

$$R_{out}=r_o+(1+g_m{r}_o)(r_o\parallel r_{\pi }),A_v=-G_m(R_{out}\parallel R_L)R\_\{out\}\; =\; r\_o\; +\; (1\; +\; g\_mr\_o)(r\_o\backslash parallel\; r\_\backslash pi),\backslash quad\; A\_v\; =\; -G\_m(R\_\{out\}\backslash parallel\; R\_L)$$

Common-base amplifier

$$A_v=g_m(R_C\parallel R_L),R_{in}=R_E\parallel r_{e}A\_v\; =\; g\_m(R\_C\backslash parallel\; R\_L),\backslash quad\; R\_\{in\}\; =\; R\_E\backslash parallel\; r\_e$$

Common-collector (emitter-follower) amplifier

$$A_v=\frac{(\beta +1)R_E^{\mathrm{\prime }}}{r_{\pi }+(\beta +1)R_E^{\mathrm{\prime }}},R_{out}=R_E\parallel \frac{r_{\pi }+R_{sig}\parallel R_B}{\beta +1}A\_v\; =\; \backslash frac\{(\backslash beta+1)R\_E\text{'}\}\{r\_\backslash pi\; +\; (\backslash beta+1)R\_E\text{'}\},\backslash quad\; R\_\{out\}\; =\; R\_E\backslash parallel\backslash frac\{r\_\backslash pi\; +\; R\_\{sig\}\backslash parallel\; R\_B\}\{\backslash beta+1\}$$

Common-emitter amplifier: gain, input and output resistance

$$A_v=-\frac{\beta (R_C\parallel R_L)}{r_{\pi }+(\beta +1)R_{E1}}A\_v\; =\; -\backslash frac\{\backslash beta\backslash ,(R\_C\backslash parallel\; R\_L)\}\{r\_\backslash pi\; +\; (\backslash beta+1)R\_\{E1\}\}$$

BJT / current relations

BJT terminal currents from beta

$$I_C=\beta I_B,I_E=(\beta +1)I_B,\alpha =\frac{\beta }{\beta +1}I\_C\; =\; \backslash beta\; I\_B,\backslash quad\; I\_E\; =\; (\backslash beta+1)I\_B,\backslash quad\; \backslash alpha\; =\; \backslash frac\{\backslash beta\}\{\backslash beta+1\}$$

BJT / current sources

BJT current mirror with finite beta

$$I_{out}=\frac{\beta }{\beta +2}{I}_{ref}I\_\{out\}\; =\; \backslash frac\{\backslash beta\}\{\backslash beta+2\}I\_\{ref\}$$

Wilson current mirror output current

$$I_{out}=I_{ref}(1-\frac{2}{{\beta }^2+2\beta +2})I\_\{out\}\; =\; I\_\{ref\}\backslash left(1\; -\; \backslash frac\{2\}\{\backslash beta^2\; +\; 2\backslash beta\; +\; 2\}\backslash right)$$

BJT / differential amplifiers

BJT differential pair: differential and common-mode gain, CMRR

$$A_{cm}=-\frac{g_m{R}_C}{1+2R_{EE}(\beta +1)\mathrm{/}{r}_{\pi }},\text{CMRR}=\frac{1+2R_{EE}(\beta +1)\mathrm{/}{r}_{\pi }}{2}A\_\{cm\}\; =\; -\backslash frac\{g\_mR\_C\}\{1\; +\; 2R\_\{EE\}(\backslash beta+1)/r\_\backslash pi\},\backslash quad\; \backslash text\{CMRR\}\; =\; \backslash frac\{1\; +\; 2R\_\{EE\}(\backslash beta+1)/r\_\backslash pi\}\{2\}$$

BJT / frequency response

BJT beta cutoff and transition frequency fT

$$f_T=\frac{g_m}{2\pi (C_{\pi }+C_{\mu })},f_{\beta }=\frac{f_T}{\beta }f\_T\; =\; \backslash frac\{g\_m\}\{2\backslash pi(C\_\backslash pi\; +\; C\_\backslash mu)\},\backslash quad\; f\_\backslash beta\; =\; \backslash frac\{f\_T\}\{\backslash beta\}$$

Common-emitter high-frequency cutoff: Miller effect and open-circuit time constants

$$C_M=C_{\mu }(1+g_m{R}_L^{\mathrm{\prime }}),{\tau }_{oc}=C_{\pi }{R}_{\pi }^{\mathrm{\prime }}+C_{\mu }(R_{\pi }^{\mathrm{\prime }}+R_L^{\mathrm{\prime }}+g_m{R}_{\pi }^{\mathrm{\prime }}{R}_L^{\mathrm{\prime }})C\_M\; =\; C\_\backslash mu(1\; +\; g\_mR\_L\text{'}),\backslash quad\; \backslash tau\_\{oc\}\; =\; C\_\backslash pi\; R\_\backslash pi\text{'}\; +\; C\_\backslash mu\backslash left(R\_\backslash pi\text{'}\; +\; R\_L\text{'}\; +\; g\_mR\_\backslash pi\text{'}R\_L\text{'}\backslash right)$$

Common-emitter low-frequency cutoff by short-circuit time constants

$$f_L\approx \sum _i\frac{1}{2\pi R_i{C}_i}f\_L\; \backslash approx\; \backslash sum\_i\; \backslash frac\{1\}\{2\backslash pi\; R\_iC\_i\}$$

BJT / small-signal model

BJT small-signal (hybrid-pi) parameters

$$g_m=\frac{I_C}{V_T},r_{\pi }=\frac{\beta }{g_m},r_e=\frac{\alpha }{g_m},r_o=\frac{V_A}{I_C}g\_m\; =\; \backslash frac\{I\_C\}\{V\_T\},\backslash quad\; r\_\backslash pi\; =\; \backslash frac\{\backslash beta\}\{g\_m\},\backslash quad\; r\_e\; =\; \backslash frac\{\backslash alpha\}\{g\_m\},\backslash quad\; r\_o\; =\; \backslash frac\{V\_A\}\{I\_C\}$$

CMOS / inverter

CMOS inverter switching threshold VM

$$V_M=\frac{V_{tn}+r(V_{DD}-\mathrm{\mid }{V}_{tp}\mathrm{\mid })}{1+r},r=\sqrt{k_p\mathrm{/}{k}_n}V\_M\; =\; \backslash frac\{V\_\{tn\}\; +\; r\backslash ,(V\_\{DD\}\; -\; \backslash vert\; V\_\{tp\}\backslash vert\; )\}\{1\; +\; r\},\backslash quad\; r\; =\; \backslash sqrt\{k\_p/k\_n\}$$

CMOS / timing

CMOS gate propagation delay (switch-resistor model)

$$t_{pHL}=\ln 2R_n{C}_L,t_{pLH}=\ln 2R_p{C}_{L}t\_\{pHL\}\; =\; \backslash ln\; 2\backslash ,R\_nC\_L,\backslash quad\; t\_\{pLH\}\; =\; \backslash ln\; 2\backslash ,R\_pC\_L$$

MOSFET / DC bias

MOSFET voltage-divider bias with source resistor: Q-point

$$V_{OV}=\frac{2(V_G-V_t)}{1+\sqrt{1+2k_n{R}_S(V_G-V_t)}},I_D=\frac{1}{2}{k}_n{V}_{OV}^{2}V\_\{OV\}\; =\; \backslash frac\{2(V\_G\; -\; V\_t)\}\{1\; +\; \backslash sqrt\{1\; +\; 2k\_nR\_S(V\_G\; -\; V\_t)\}\},\backslash quad\; I\_D\; =\; \backslash tfrac\{1\}\{2\}k\_nV\_\{OV\}^2$$

MOSFET / amplifiers

Common-source amplifier with source degeneration and ro

$$A_v=-\frac{g_m{r}_o{R}_D^{\mathrm{\prime }}}{R_D^{\mathrm{\prime }}+r_o+R_S(1+g_m{r}_o)}A\_v\; =\; -\backslash frac\{g\_mr\_oR\_D\text{'}\}\{R\_D\text{'}\; +\; r\_o\; +\; R\_S(1\; +\; g\_mr\_o)\}$$

Source follower (common drain)

$$A_v=\frac{g_{m}R}{1+g_{m}R},R_{out}=R_S\parallel r_o\parallel \frac{1}{g_m}A\_v\; =\; \backslash frac\{g\_mR\}\{1\; +\; g\_mR\},\backslash quad\; R\_\{out\}\; =\; R\_S\backslash parallel\; r\_o\backslash parallel\backslash frac\{1\}\{g\_m\}$$

MOSFET / drain current

MOSFET drain current (cutoff, triode, saturation)

MOSFET / small-signal model

MOSFET small-signal gm, ro and intrinsic gain

$$g_m=\sqrt{2k_n{I}_D}=\frac{2I_D}{V_{OV}},r_o=\frac{1}{\lambda I_D}g\_m\; =\; \backslash sqrt\{2k\_nI\_D\}\; =\; \backslash frac\{2I\_D\}\{V\_\{OV\}\},\backslash quad\; r\_o\; =\; \backslash frac\{1\}\{\backslash lambda\; I\_D\}$$

MOSFET / threshold voltage

MOSFET threshold voltage with body effect

$$V_t=V_{t0}+\gamma (\sqrt{2{\varphi }_F+V_{SB}}-\sqrt{2{\varphi }_F})V\_t\; =\; V\_\{t0\}\; +\; \backslash gamma\backslash left(\backslash sqrt\{2\backslash phi\_F\; +\; V\_\{SB\}\}\; -\; \backslash sqrt\{2\backslash phi\_F\}\backslash right)$$

Active filters / Sallen-Key

Equal-component Sallen-Key low-pass design

$$R=\frac{1}{2\pi f_{c}C},K=3-\frac{1}{Q}R\; =\; \backslash frac\{1\}\{2\backslash pi\; f\_cC\},\backslash quad\; K\; =\; 3\; -\; \backslash frac\{1\}\{Q\}$$

Sallen-Key high-pass filter: f0, Q and gain

$$Q=\frac{\sqrt{R_1{R}_2{C}_1{C}_2}}{R_1(C_1+C_2)+R_2{C}_2(1-K)}Q\; =\; \backslash frac\{\backslash sqrt\{R\_1R\_2C\_1C\_2\}\}\{R\_1(C\_1+C\_2)\; +\; R\_2C\_2(1-K)\}$$

Sallen-Key low-pass filter: f0, Q and gain

$${\omega }_0=\frac{1}{\sqrt{R_1{R}_2{C}_1{C}_2}},Q=\frac{\sqrt{R_1{R}_2{C}_1{C}_2}}{R_1{C}_2+R_2{C}_2+R_1{C}_1(1-K)}\backslash omega\_0\; =\; \backslash frac\{1\}\{\backslash sqrt\{R\_1R\_2C\_1C\_2\}\},\backslash quad\; Q\; =\; \backslash frac\{\backslash sqrt\{R\_1R\_2C\_1C\_2\}\}\{R\_1C\_2\; +\; R\_2C\_2\; +\; R\_1C\_1(1-K)\}$$

Active filters / approximations

Butterworth filter: Q of each second-order section

$$Q_k=\frac{1}{2\sin \left(\frac{(2k-1)\pi }{2n}\right)}Q\_k\; =\; \backslash frac\{1\}\{2\backslash sin\backslash left(\backslash frac\{(2k-1)\backslash pi\}\{2n\}\backslash right)\}$$

Butterworth or Chebyshev low-pass filter order

$$n\ge \frac{{\log }_{10}D}{2{\log }_{10}(f_s\mathrm{/}{f}_p)},n\ge \frac{{\cosh }^{-1}\sqrt{D}}{{\cosh }^{-1}(f_s\mathrm{/}{f}_p)}n\; \backslash ge\; \backslash frac\{\backslash log\_\{10\}D\}\{2\backslash log\_\{10\}(f\_s/f\_p)\},\backslash quad\; n\; \backslash ge\; \backslash frac\{\backslash cosh^\{-1\}\backslash sqrt\{D\}\}\{\backslash cosh^\{-1\}(f\_s/f\_p)\}$$

Chebyshev type I filter: section Q and frequencies

$$p_k=-\sinh a\sin {\theta }_k+j\cosh a\cos {\theta }_k,a=\frac{1}{n}{\sinh }^{-1}\frac{1}{\epsilon },Q_k=\frac{\mathrm{\mid }{p}_k\mathrm{\mid }}{2\mathrm{\mid }\mathrm{\Re }{p}_k\mathrm{\mid }}p\_k\; =\; -\backslash sinh\; a\backslash sin\backslash theta\_k\; +\; j\backslash cosh\; a\backslash cos\backslash theta\_k,\backslash \; a\; =\; \backslash frac\{1\}\{n\}\backslash sinh^\{-1\}\backslash frac\{1\}\{\backslash varepsilon\},\backslash \; Q\_k\; =\; \backslash frac\{\backslash vert\; p\_k\backslash vert\; \}\{2\backslash vert\; \backslash Re\; p\_k\backslash vert\; \}$$

Active filters / first order

First-order active low-pass filter (inverting)

$$H(j\omega )=-\frac{R_f\mathrm{/}{R}_1}{1+j\omega R_{f}C},f_c=\frac{1}{2\pi R_{f}C}H(j\backslash omega)\; =\; -\backslash frac\{R\_f/R\_1\}\{1\; +\; j\backslash omega\; R\_fC\},\backslash quad\; f\_c\; =\; \backslash frac\{1\}\{2\backslash pi\; R\_fC\}$$

Active filters / multiple feedback

Multiple-feedback (MFB) band-pass filter

$${\omega }_0=\sqrt{\frac{R_1+R_2}{R_1{R}_2{R}_3{C}_1{C}_2}},Q=\frac{{\omega }_0{C}_1{C}_2{R}_3}{C_1+C_2}\backslash omega\_0\; =\; \backslash sqrt\{\backslash frac\{R\_1+R\_2\}\{R\_1R\_2R\_3C\_1C\_2\}\},\backslash quad\; Q\; =\; \backslash frac\{\backslash omega\_0C\_1C\_2R\_3\}\{C\_1+C\_2\}$$

Multiple-feedback (MFB) low-pass filter

$$Q=\frac{\sqrt{R_2{R}_3{C}_1{C}_2}}{C_2(R_2+R_3+R_2{R}_3\mathrm{/}{R}_1)},H_0=-\frac{R_2}{R_1}Q\; =\; \backslash frac\{\backslash sqrt\{R\_2R\_3C\_1C\_2\}\}\{C\_2\backslash left(R\_2\; +\; R\_3\; +\; R\_2R\_3/R\_1\backslash right)\},\backslash quad\; H\_0\; =\; -\backslash frac\{R\_2\}\{R\_1\}$$

Data converters / ADC

ADC output code, LSB and quantisation error

$$D=\lfloor \frac{V_{in}}{V_{ref}}{2}^n\rfloor ,\text{LSB}=\frac{V_{ref}}{2^n}D\; =\; \backslash left\backslash lfloor\; \backslash frac\{V\_\{in\}\}\{V\_\{ref\}\}2^n\backslash right\backslash rfloor,\backslash quad\; \backslash text\{LSB\}\; =\; \backslash frac\{V\_\{ref\}\}\{2^n\}$$

ADC quantisation SNR with oversampling

$$\text{SNR}=6.02n+1.76+10{\log }_{10}\text{OSR dB}\backslash text\{SNR\}\; =\; 6.02\backslash ,n\; +\; 1.76\; +\; 10\backslash log\_\{10\}\backslash text\{OSR\}\backslash \; \backslash text\{dB\}$$

Dual-slope integrating ADC

$$T_2=T_1\frac{V_{in}}{V_{ref}},N_2=N_1\frac{V_{in}}{V_{ref}}T\_2\; =\; T\_1\backslash frac\{V\_\{in\}\}\{V\_\{ref\}\},\backslash quad\; N\_2\; =\; N\_1\backslash frac\{V\_\{in\}\}\{V\_\{ref\}\}$$

Effective number of bits (ENOB) from SINAD

$$\text{ENOB}=\frac{\text{SINAD}-1.76}{6.02}\backslash text\{ENOB\}\; =\; \backslash frac\{\backslash text\{SINAD\}\; -\; 1.76\}\{6.02\}$$

Successive-approximation (SAR) ADC conversion

$$b_k=[V_{in}\ge V_{DAC}(D_k+2^k)]b\_k\; =\; \backslash left[V\_\{in\}\; \backslash ge\; V\_\{DAC\}(D\_\{k\}\; +\; 2^k)\backslash right]$$

Data converters / DAC

Binary-weighted resistor DAC

$$V_{out}=-V_{ref}\frac{R_f}{R}\sum _{k=0}^{n-1}\frac{b_k}{2^k}=-V_{ref}\frac{R_f}{R}\frac{D}{2^{n-1}}V\_\{out\}\; =\; -V\_\{ref\}\backslash frac\{R\_f\}\{R\}\backslash sum\_\{k=0\}^\{n-1\}\backslash frac\{b\_k\}\{2^k\}\; =\; -V\_\{ref\}\backslash frac\{R\_f\}\{R\}\backslash frac\{D\}\{2^\{n-1\}\}$$

DAC output voltage (R-2R or any binary DAC)

$$V_{out}=V_{ref}\frac{D}{2^n}V\_\{out\}\; =\; V\_\{ref\}\backslash frac\{D\}\{2^n\}$$

Data converters / sampling

SNR limit from sampling-clock (aperture) jitter

$${\text{SNR}}_j=-20{\log }_{10}(2\pi f{t}_j)\backslash text\{SNR\}\_j\; =\; -20\backslash log\_\{10\}(2\backslash pi\; f\; t\_j)$$

Sample-and-hold droop

$$\frac{dV}{dt}=\frac{I_{leak}}{C}\backslash frac\{dV\}\{dt\}\; =\; \backslash frac\{I\_\{leak\}\}\{C\}$$

Diodes / LEDs

LED series (current-limiting) resistor

$$R=\frac{V_S-n{V}_F}{I_F}R\; =\; \backslash frac\{V\_S\; -\; nV\_F\}\{I\_F\}$$

Diodes / Shockley equation

Diode current from the Shockley equation

$$I_D=I_S(e^{V_D\mathrm{/}n{V}_T}-1)I\_D\; =\; I\_S\backslash left(e^\{V\_D/nV\_T\}\; -\; 1\backslash right)$$

Diode voltage for a given current

$$V_D=n{V}_T\ln (1+\frac{I_D}{I_S})V\_D\; =\; nV\_T\backslash ln\backslash left(1\; +\; \backslash frac\{I\_D\}\{I\_S\}\backslash right)$$

Diodes / Zener regulators

Zener regulator design: series resistor and Zener rating

$$R_S=\frac{V_{in,min}-V_Z}{I_{Z,min}+I_{L,max}}R\_S\; =\; \backslash frac\{V\_\{in,min\}\; -\; V\_Z\}\{I\_\{Z,min\}\; +\; I\_\{L,max\}\}$$

Zener shunt regulator: load voltage and currents

$$I_Z=\frac{V_{in}-V_Z}{R_S}-\frac{V_Z}{R_L}I\_Z\; =\; \backslash frac\{V\_\{in\}\; -\; V\_Z\}\{R\_S\}\; -\; \backslash frac\{V\_Z\}\{R\_L\}$$

Diodes / operating point

Operating point of a diode in series with a resistor (exact)

$$V_S=I_{D}R+n{V}_T\ln (1+\frac{I_D}{I_S})V\_S\; =\; I\_DR\; +\; nV\_T\backslash ln\backslash left(1\; +\; \backslash frac\{I\_D\}\{I\_S\}\backslash right)$$

Diodes / rectifier filters

Capacitor-input filter: ripple voltage and DC output

$$V_r=\frac{V_p}{f_r{R}_{L}C},V_{dc}=V_p-\frac{V_r}{2}V\_r\; =\; \backslash frac\{V\_p\}\{f\_rR\_LC\},\backslash quad\; V\_\{dc\}\; =\; V\_p\; -\; \backslash frac\{V\_r\}\{2\}$$

Diodes / rectifiers

Rectifier output: average, RMS, ripple factor and PIV

$$V_{dc}=\frac{k}{2\pi }[2V_m\cos {\theta }_1-V(\pi -2{\theta }_1)],{\theta }_1=\arcsin \frac{V}{V_m}V\_\{dc\}\; =\; \backslash frac\{k\}\{2\backslash pi\}\backslash left[2V\_m\backslash cos\backslash theta\_1\; -\; V(\backslash pi\; -\; 2\backslash theta\_1)\backslash right],\backslash \; \backslash theta\_1\; =\; \backslash arcsin\backslash frac\{V\}\{V\_m\}$$

Op-amps / basic amplifiers

Inverting op-amp amplifier

$$V_{out}=-\frac{R_f}{R_{in}}{V}_{in}V\_\{out\}\; =\; -\backslash frac\{R\_f\}\{R\_\{in\}\}V\_\{in\}$$

Inverting summing amplifier

$$V_{out}=-R_f\sum _i\frac{V_i}{R_i}V\_\{out\}\; =\; -R\_f\backslash sum\_i\backslash frac\{V\_i\}\{R\_i\}$$

Non-inverting op-amp amplifier

$$V_{out}=(1+\frac{R_f}{R_g})V_{in}V\_\{out\}\; =\; \backslash left(1\; +\; \backslash frac\{R\_f\}\{R\_g\}\backslash right)V\_\{in\}$$

Op-amps / comparators

Inverting Schmitt trigger thresholds

$$V_{TH,TL}=\frac{V_{sat}{R}_1+V_{ref}{R}_f}{R_1+R_f}V\_\{TH,TL\}\; =\; \backslash frac\{V\_\{sat\}R\_1\; +\; V\_\{ref\}R\_f\}\{R\_1\; +\; R\_f\}$$

Op-amps / difference amplifiers

Op-amp difference amplifier (with resistor mismatch)

$$V_{out}=\frac{R_4}{R_3+R_4}(1+\frac{R_2}{R_1})V_2-\frac{R_2}{R_1}{V}_{1}V\_\{out\}\; =\; \backslash frac\{R\_4\}\{R\_3+R\_4\}\backslash left(1+\backslash frac\{R\_2\}\{R\_1\}\backslash right)V\_2\; -\; \backslash frac\{R\_2\}\{R\_1\}V\_1$$

Three-op-amp instrumentation amplifier

$$V_{out}=(1+\frac{2R_1}{R_G})\frac{R_3}{R_2}(V_2-V_1)V\_\{out\}\; =\; \backslash left(1\; +\; \backslash frac\{2R\_1\}\{R\_G\}\backslash right)\backslash frac\{R\_3\}\{R\_2\}(V\_2\; -\; V\_1)$$

Op-amps / feedback

Negative feedback: closed-loop gain and bandwidth extension

$$A_f=\frac{A}{1+A\beta },f_{H,f}=f_H(1+A\beta )A\_f\; =\; \backslash frac\{A\}\{1\; +\; A\backslash beta\},\backslash quad\; f\_\{H,f\}\; =\; f\_H(1\; +\; A\backslash beta)$$

Op-amps / frequency limits

Closed-loop bandwidth from the gain-bandwidth product

$$f_{3dB}=\frac{\text{GBW}}{1+R_f\mathrm{/}{R}_1}f\_\{3dB\}\; =\; \backslash frac\{\backslash text\{GBW\}\}\{1\; +\; R\_f/R\_1\}$$

Slew-rate limit: full-power bandwidth

$$f_{max}=\frac{SR}{2\pi V_p}f\_\{max\}\; =\; \backslash frac\{SR\}\{2\backslash pi\; V\_p\}$$

Op-amps / integrator and differentiator

Op-amp differentiator: gain and output amplitude

$$\mathrm{\mid }H(j\omega )\mathrm{\mid }=\omega RC\backslash vert\; H(j\backslash omega)\backslash vert\; =\; \backslash omega\; RC$$

Op-amp integrator: output ramp and unity-gain frequency

$$V_{out}(t)=V_0-\frac{1}{RC}{\int }_0^t{V}_{in}dtV\_\{out\}(t)\; =\; V\_0\; -\; \backslash frac\{1\}\{RC\}\backslash int\_0^t\; V\_\{in\}\backslash ,dt$$

Op-amps / non-ideal

Inverting amplifier with finite open-loop gain

$$G=\frac{-R_f\mathrm{/}{R}_1}{1+{\displaystyle \frac{1+R_f\mathrm{/}{R}_1}{A_0}}}G\; =\; \backslash frac\{-R\_f/R\_1\}\{1\; +\; \backslash dfrac\{1\; +\; R\_f/R\_1\}\{A\_0\}\}$$

Op-amp output offset from input offset voltage and bias currents

$$V_{out}=(V_{os}-I_{B+}{R}_p)(1+\frac{R_f}{R_1})+I_{B-}{R}_{f}V\_\{out\}\; =\; \backslash left(V\_\{os\}\; -\; I\_\{B+\}R\_p\backslash right)\backslash left(1\; +\; \backslash frac\{R\_f\}\{R\_1\}\backslash right)\; +\; I\_\{B-\}R\_f$$

Op-amps / stability

Phase margin of a two-pole amplifier with feedback

$$PM={180}^{\circ }-\arctan \frac{f_c}{f_1}-\arctan \frac{f_c}{f_2}PM\; =\; 180^\backslash circ\; -\; \backslash arctan\backslash frac\{f\_c\}\{f\_1\}\; -\; \backslash arctan\backslash frac\{f\_c\}\{f\_2\}$$

Oscillators / 555 timer

555 timer astable: frequency and duty cycle

$$t_H=\ln 2(R_A+R_B)C,t_L=\ln 2R_{B}C,f=\frac{1}{\ln 2(R_A+2R_B)C}t\_H\; =\; \backslash ln\; 2\backslash ,(R\_A\; +\; R\_B)C,\backslash quad\; t\_L\; =\; \backslash ln\; 2\backslash ,R\_BC,\backslash quad\; f\; =\; \backslash frac\{1\}\{\backslash ln\; 2\backslash ,(R\_A\; +\; 2R\_B)C\}$$

555 timer monostable pulse width

$$T=\ln 3RC\approx 1.1RCT\; =\; \backslash ln\; 3\backslash ,RC\; \backslash approx\; 1.1\backslash ,RC$$

Oscillators / LC

Colpitts oscillator frequency and feedback ratio

$$f_0=\frac{1}{2\pi \sqrt{L\frac{C_1{C}_2}{C_1+C_2}}},A_{min}=\frac{C_2}{C_1}f\_0\; =\; \backslash frac\{1\}\{2\backslash pi\backslash sqrt\{L\backslash ,\backslash frac\{C\_1C\_2\}\{C\_1+C\_2\}\}\},\backslash quad\; A\_\{min\}\; =\; \backslash frac\{C\_2\}\{C\_1\}$$

Hartley oscillator frequency

$$f_0=\frac{1}{2\pi \sqrt{(L_1+L_2+2M)C}}f\_0\; =\; \backslash frac\{1\}\{2\backslash pi\backslash sqrt\{(L\_1\; +\; L\_2\; +\; 2M)C\}\}$$

Oscillators / RC

RC phase-shift oscillator frequency and gain

$$f_0=\frac{1}{2\pi \sqrt{6}RC},\mathrm{\mid }A\mathrm{\mid }\ge 29f\_0\; =\; \backslash frac\{1\}\{2\backslash pi\backslash sqrt\{6\}\backslash ,RC\},\backslash quad\; \backslash vert\; A\backslash vert\; \backslash ge\; 29$$

Wien-bridge oscillator frequency and required gain

$$f_0=\frac{1}{2\pi \sqrt{R_1{R}_2{C}_1{C}_2}},A=1+\frac{R_1}{R_2}+\frac{C_2}{C_1}f\_0\; =\; \backslash frac\{1\}\{2\backslash pi\backslash sqrt\{R\_1R\_2C\_1C\_2\}\},\backslash quad\; A\; =\; 1\; +\; \backslash frac\{R\_1\}\{R\_2\}\; +\; \backslash frac\{C\_2\}\{C\_1\}$$

Oscillators / crystal

Quartz crystal: series and parallel resonance, Q

$$f_s=\frac{1}{2\pi \sqrt{L_m{C}_m}},f_p=f_s\sqrt{1+\frac{C_m}{C_0}}f\_s\; =\; \backslash frac\{1\}\{2\backslash pi\backslash sqrt\{L\_mC\_m\}\},\backslash quad\; f\_p\; =\; f\_s\backslash sqrt\{1\; +\; \backslash frac\{C\_m\}\{C\_0\}\}$$

Oscillators / phase-locked loops

Type-2 PLL with active PI loop filter: natural frequency and damping

$${\omega }_n=\sqrt{\frac{K_d{K}_o}{{\tau }_1}},\zeta =\frac{{\tau }_2{\omega }_n}{2}\backslash omega\_n\; =\; \backslash sqrt\{\backslash frac\{K\_dK\_o\}\{\backslash tau\_1\}\},\backslash quad\; \backslash zeta\; =\; \backslash frac\{\backslash tau\_2\backslash omega\_n\}\{2\}$$

Oscillators / relaxation

Op-amp relaxation (astable) oscillator

$$T=2RC\ln \frac{1+\beta }{1-\beta },\beta =\frac{R_1}{R_1+R_2}T\; =\; 2RC\backslash ln\backslash frac\{1+\backslash beta\}\{1-\backslash beta\},\backslash quad\; \backslash beta\; =\; \backslash frac\{R\_1\}\{R\_1+R\_2\}$$

Power supplies / linear regulators

Adjustable three-terminal regulator output voltage

$$V_{out}=V_{ref}(1+\frac{R_2}{R_1})+I_{adj}{R}_{2}V\_\{out\}\; =\; V\_\{ref\}\backslash left(1\; +\; \backslash frac\{R\_2\}\{R\_1\}\backslash right)\; +\; I\_\{adj\}R\_2$$

Linear regulator dissipation and efficiency

$$P_D=(V_{in}-V_{out})I_L+V_{in}{I}_q,\eta =\frac{V_{out}{I}_L}{V_{in}(I_L+I_q)}P\_D\; =\; (V\_\{in\}\; -\; V\_\{out\})I\_L\; +\; V\_\{in\}I\_q,\backslash quad\; \backslash eta\; =\; \backslash frac\{V\_\{out\}I\_L\}\{V\_\{in\}(I\_L\; +\; I\_q)\}$$

Zener-referenced series-pass (emitter-follower) regulator

$$V_{out}=V_Z-V_{BE},I_Z=\frac{V_{in}-V_Z}{R_S}-\frac{I_L}{\beta +1}V\_\{out\}\; =\; V\_Z\; -\; V\_\{BE\},\backslash quad\; I\_Z\; =\; \backslash frac\{V\_\{in\}\; -\; V\_Z\}\{R\_S\}\; -\; \backslash frac\{I\_L\}\{\backslash beta+1\}$$

Power supplies / thermal design

Heat-sink thermal path: junction, case and sink temperatures

$$T_j=T_a+P({\theta }_{jc}+{\theta }_{cs}+{\theta }_{sa})T\_j\; =\; T\_a\; +\; P\backslash ,(\backslash theta\_\{jc\}\; +\; \backslash theta\_\{cs\}\; +\; \backslash theta\_\{sa\})$$

Required heat-sink thermal resistance

$${\theta }_{sa}\le \frac{T_{j,max}-T_a}{P}-{\theta }_{jc}-{\theta }_{cs}\backslash theta\_\{sa\}\; \backslash le\; \backslash frac\{T\_\{j,max\}\; -\; T\_a\}\{P\}\; -\; \backslash theta\_\{jc\}\; -\; \backslash theta\_\{cs\}$$

Semiconductors / carrier statistics

Electron and hole concentrations of doped (compensated) material

$$n=\frac{N_D-N_A}{2}+\sqrt{{\left(\frac{N_D-N_A}{2}\right)}^2+n_i^2},p=\frac{n_i^2}{n}n\; =\; \backslash frac\{N\_D\; -\; N\_A\}\{2\}\; +\; \backslash sqrt\{\backslash left(\backslash frac\{N\_D\; -\; N\_A\}\{2\}\backslash right)^2\; +\; n\_i^2\},\backslash quad\; p\; =\; \backslash frac\{n\_i^2\}\{n\}$$

Intrinsic carrier concentration

$$n_i=\sqrt{N_c{N}_v}{e}^{-E_g\mathrm{/}2kT},N_{c,v}(T)=N_{c,v}(300){\left(\frac{T}{300}\right)}^{3\mathrm{/}2}n\_i\; =\; \backslash sqrt\{N\_cN\_v\}\backslash ,e^\{-E\_g/2kT\},\backslash quad\; N\_\{c,v\}(T)\; =\; N\_\{c,v\}(300)\backslash left(\backslash tfrac\{T\}\{300\}\backslash right)^\{3/2\}$$

Semiconductors / conduction

Conductivity and resistivity of a semiconductor

$$\sigma =q(n{\mu }_n+p{\mu }_p),\rho =1\mathrm{/}\sigma \backslash sigma\; =\; q\backslash ,(n\backslash mu\_n\; +\; p\backslash mu\_p),\backslash quad\; \backslash rho\; =\; 1/\backslash sigma$$

Semiconductors / diffusion

Einstein relation and diffusion length

$$D=\mu \frac{kT}{q},L=\sqrt{D\tau }D\; =\; \backslash mu\backslash frac\{kT\}\{q\},\backslash quad\; L\; =\; \backslash sqrt\{D\backslash tau\}$$

Semiconductors / drift

Drift velocity and drift current in a semiconductor bar

$$J=qn\mu E,E=V\mathrm{/}LJ\; =\; q\backslash ,n\backslash ,\backslash mu\backslash ,E,\backslash quad\; E\; =\; V/L$$

Semiconductors / pn junction

Built-in potential of a pn junction

$$V_{bi}=\frac{kT}{q}\ln \frac{N_A{N}_D}{n_i^2}V\_\{bi\}\; =\; \backslash frac\{kT\}\{q\}\backslash ln\backslash frac\{N\_AN\_D\}\{n\_i^2\}$$

Depletion width, field and capacitance of an abrupt pn junction

$$W=\sqrt{\frac{2\epsilon (V_{bi}+V_R)}{q}(\frac{1}{N_A}+\frac{1}{N_D})}W\; =\; \backslash sqrt\{\backslash frac\{2\backslash varepsilon(V\_\{bi\}+V\_R)\}\{q\}\backslash left(\backslash frac\{1\}\{N\_A\}+\backslash frac\{1\}\{N\_D\}\backslash right)\}$$

Junction (depletion) capacitance versus bias

$$C_j=\frac{C_{j0}}{{(1+V_R\mathrm{/}{V}_{bi})}^m}C\_j\; =\; \backslash frac\{C\_\{j0\}\}\{\backslash left(1\; +\; V\_R/V\_\{bi\}\backslash right)^\{m\}\}$$

Semiconductors / thermal voltage

Thermal voltage kT/q

$$V_T=\frac{kT}{q}V\_T\; =\; \backslash frac\{kT\}\{q\}$$

Timing / RC

RC delay, 10-90 % rise time and bandwidth

$$t_r=\ln 9RC\approx 2.2RC,t_r{f}_{3dB}\approx 0.35t\_r\; =\; \backslash ln\; 9\backslash ,RC\; \backslash approx\; 2.2RC,\backslash quad\; t\_rf\_\{3dB\}\; \backslash approx\; 0.35$$

Timing / interconnect

Elmore delay of an RC ladder (interconnect)

$${\tau }_D=\sum _i{C}_i\sum _{j\le i}{R}_j\backslash tau\_D\; =\; \backslash sum\_i\; C\_i\backslash sum\_\{j\backslash le\; i\}R\_j$$

Timing / power

CMOS dynamic (switching) power

$$P_{dyn}=\alpha C{V}_{DD}^{2}fP\_\{dyn\}\; =\; \backslash alpha\backslash ,C\backslash ,V\_\{DD\}^2f$$

Timing / synchronous logic

Flip-flop setup and hold slack, maximum clock frequency

$$T\ge t_{cq}+t_{pd}+t_{su}-t_{skew},t_{ccq}+t_{cd}\ge t_h+t_{skew}T\; \backslash ge\; t\_\{cq\}\; +\; t\_\{pd\}\; +\; t\_\{su\}\; -\; t\_\{skew\},\backslash quad\; t\_\{ccq\}\; +\; t\_\{cd\}\; \backslash ge\; t\_h\; +\; t\_\{skew\}$$

Synchronizer metastability MTBF

$$\text{MTBF}=\frac{e^{t_r\mathrm{/}\tau }}{T_0{f}_{clk}{f}_{data}}\backslash text\{MTBF\}\; =\; \backslash frac\{e^\{t\_r/\backslash tau\}\}\{T\_0f\_\{clk\}f\_\{data\}\}$$

Out of scope

• Open-ended explanation, essay writing or tutoring dialogue: it is a solver, not a chatbot.
• Questions outside its certified electronics engineering capabilities, and any question whose stated numbers it cannot all use — it refuses with the reason instead.
• High-stakes engineering, medical or legal decisions without independent review.

Limitations

• It answers what its certified capabilities cover (semiconductors, diode and transistor circuits, amplifiers, active filters, oscillators, data converters, timing and power supplies); anything else is refused with a reason.
• ask() reads documented question forms and common phrasings; unusual wording may be refused — use solve() for guaranteed results.
• Results are as correct as the certified relations and their stated conditions; a certified relation is a checked model, not a claim about every physical situation.

Citation

@software{electronics_engineering_ai999x_1_0_0,
  title  = {electronics-engineering-ai999x: an offline, verified electronics engineering solver},
  author = {Farabi, Ibnul Jaif}, organization = {AI999X},
  year   = {2026}, version = {1.0.0},
  url    = {https://huggingface.co/AI999X/electronics-engineering-ai999x}
}

Contact

Questions, bug reports and requests: open a discussion on https://huggingface.co/AI999X/electronics-engineering-ai999x/discussions or visit https://ai999x.com.

Support the project

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Created and engineered by Ibnul Jaif Farabi · AI999X · ai999x.com