power-electronics-ai999x — offline, verified power electronics 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

power-electronics-ai999x is an offline, deterministic Python solver that answers power electronics questions with verified results from 79 certified capabilities, gives 482 documented answers identically on Windows and Linux, and refuses rather than guesses when an answer cannot be verified.

power-electronics-ai999x is a compact, deterministic power electronics engine for Python. It reads a question in plain English (or a typed call), picks a certified power electronics 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. power-electronics-ai999x comes as the AI999X desktop app and as the command-line program power-electronics 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 Power Electronics desktop app 1.2.0	Windows, Linux — this model in its own app: search, ask in words, forms, live figures, explanations, offline	Windows (x86-64) · Linux (x86-64)
AI999X desktop app 1.2.0	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 (power-electronics)	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 power-electronics-ai999x

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

Rust versions

Power Electronics also comes as native programs compiled from Rust, version 1.2.0 of the AI999X desktop app and command line: they run offline on your computer and need no Python. Each file is in this repository's rust/ folder, on the GitHub release v1.0.0 and on the website.

Program	System	File	Size	Download	Use
Desktop app (Power Electronics)	Windows x86-64	ai999x-power-electronics-1.2.0-windows-x86_64.zip	7.1 MB	Hub · GitHub · website	unzip, then double-click ai999x-power-electronics.exe (no installer)
Desktop app (Power Electronics)	Linux x86-64	ai999x-power-electronics-1.2.0-linux-x86_64.tar.gz	8.0 MB	Hub · GitHub · website	extract, then run ./ai999x-power-electronics (X11 or Wayland desktop)
Command line (power-electronics)	Windows x86-64	ai999x-power-electronics-cli-1.2.0-windows-x86_64.zip	2.9 MB	Hub · GitHub · website (all models)	unzip, then power-electronics.exe ask "<question>" or power-electronics.exe solve <capability> name=value
Command line (power-electronics)	Linux x86-64	ai999x-power-electronics-cli-1.2.0-linux-x86_64.tar.gz	2.9 MB	Hub · GitHub · website (all models)	extract, then ./power-electronics ask "<question>" or ./power-electronics solve <capability> name=value

The command-line archive holds power-electronics with a README of its commands and examples, the licence and the notice. Checksums: rust/SHA256SUMS (sha256sum -c SHA256SUMS).

Quick start

Desktop app

Download AI999X Power Electronics for your system (above), unpack it and start it; 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. The AI999X desktop app has the same, with Power Electronics among all AI999X models.

Linux, from a terminal:

curl -LO https://www.ai999x.com/downloads/desktop/1.2.0/ai999x-power-electronics-1.2.0-linux-x86_64.tar.gz
curl -LO https://www.ai999x.com/downloads/desktop/1.2.0/SHA256SUMS
sha256sum -c --ignore-missing SHA256SUMS
tar xzf ai999x-power-electronics-1.2.0-linux-x86_64.tar.gz
./ai999x-power-electronics-1.2.0-linux-x86_64/ai999x-power-electronics

Command line

Linux:

curl -LO https://www.ai999x.com/downloads/desktop/1.2.0/ai999x-cli-1.2.0-linux-x86_64.tar.gz
curl -LO https://www.ai999x.com/downloads/desktop/1.2.0/SHA256SUMS
sha256sum -c --ignore-missing SHA256SUMS
mkdir -p ~/.local/share && tar xzf ai999x-cli-1.2.0-linux-x86_64.tar.gz -C ~/.local/share
echo 'export PATH="$HOME/.local/share/ai999x-cli-1.2.0-linux-x86_64:$PATH"' >> ~/.bashrc && source ~/.bashrc
power-electronics --version

Windows (PowerShell):

$F = "ai999x-cli-1.2.0-windows-x86_64.zip"
Invoke-WebRequest "https://www.ai999x.com/downloads/desktop/1.2.0/$F" -OutFile $F
Invoke-WebRequest "https://www.ai999x.com/downloads/desktop/1.2.0/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-1.2.0-windows-x86_64"; $env:Path += ";$D"
[Environment]::SetEnvironmentVariable("Path", [Environment]::GetEnvironmentVariable("Path", "User") + ";$D", "User")
power-electronics --version

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

Ask in words — Write the question as you would say it: the program reads the quantities and their units, picks the capability and answers.

$ power-electronics ask "Buck converter in CCM: Vin = 24 V, Vout = 5 V, L = 22 uH, fs = 200 kHz, Io = 3 A"
Read as: Buck converter in continuous conduction
D = 0.2083333333
dIL = 0.8996212121 A
IL_max = 3.449810606 A
IL_min = 2.550189394 A
ripple_ratio = 0.2998737374
Iin = 0.625 A
  by: Buck converter in continuous conduction  (D = Vout/Vin; dIL = (Vin - Vout) D/(L fs); IL = Io; IL_max,min = Io +- dIL/2; Iin = D Io)
  Io = 3 A
  L = 2.2e-05 H
  Vin = 24 V
  Vout = 5 V
  fs = 200000 Hz
  assumes: 0 < Vout < Vin; continuous conduction (IL_min > 0)

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.

$ power-electronics info pe_buck_ccm
Buck converter in continuous conduction
  D = Vout/Vin; dIL = (Vin - Vout) D/(L fs); IL = Io; IL_max,min = Io +- dIL/2; Iin = D Io
  in   Vin      input voltage [V]
  in   Vout     output voltage [V]
  in   L        inductance [H]
  in   fs       switching frequency [Hz]
  in   Io       load (output) current [A]
  out  D        duty cycle [1]
  out  dIL      peak-to-peak inductor ripple [A]
  out  IL_max   peak inductor current [A]
  out  IL_min   valley inductor current [A]
  out  ripple_ratio ripple ratio dIL/Io [1]
  out  Iin      average input current [A]

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.

$ power-electronics solve pe_buck_ccm "Vin=24 V" "Vout=5 V" "L=22 uH" "fs=200 kHz" "Io=3 A"
D = 0.2083333333
dIL = 0.8996212121 A
IL_max = 3.449810606 A
IL_min = 2.550189394 A
ripple_ratio = 0.2998737374
Iin = 0.625 A
  by: Buck converter in continuous conduction  (D = Vout/Vin; dIL = (Vin - Vout) D/(L fs); IL = Io; IL_max,min = Io +- dIL/2; Iin = D Io)
  Vin = 24 V
  Vout = 5 V
  L = 2.2e-05 H
  fs = 200000 Hz
  Io = 3 A
  assumes: 0 < Vout < Vin; continuous conduction (IL_min > 0)

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 | power-electronics batch

cases.jsonl:

{"key": "pe_buck_ccm", "inputs": {"Vin": "24 V", "Vout": "5 V", "L": "22 uH", "fs": "200 kHz", "Io": "3 A"}}
{"key": "pe_three_phase_controlled_rectifier", "inputs": {"VLL": "415 V", "alpha": "45 deg", "Id": "50 A"}}

$ power-electronics batch < cases.jsonl
{"outcome":"SOLVED","message":"","value":0.899621212121,"unit":"A","solved_for":"dIL","si":[["Vin",24.0],["Vout",5.0],["L",0.000022],["fs",200000.0],["Io",3.0]],"values":[["D",0.208333333333,""],["dIL",0.899621212121,"A"],["IL_max",3.44981060606,"A"],["IL_min",2.55018939394,"A"],["ripple_ratio",0.299873737374,""],["Iin",0.625,"A"]],"target":"dIL","phrase":"duty cycle","assumptions":["0 < Vout < Vin","continuous conduction (IL_min > 0)"],"name":"Buck converter in continuous conduction","equation":"D = Vout/Vin; dIL = (Vin - Vout) D/(L fs); IL = Io; IL_max,min = Io +- dIL/2; Iin = D Io"}
{"outcome":"SOLVED","message":"","value":396.295808299,"unit":"V","solved_for":"Vdc","si":[["VLL",415.0],["alpha",0.7853981633974483],["Id",50.0]],"values":[["Vdc",396.295808299,"V"],["Vrms",415.0,"V"],["P",19814.7904149,"W"],["I_thy_avg",16.6666666667,"A"],["I_thy_rms",28.8675134595,"A"],["Is_rms",40.8248290464,"A"],["mode","rectifying",""]],"target":"Vdc","phrase":"average output voltage","assumptions":["0 <= alpha < 180 degrees","continuous constant current, no source inductance"],"name":"Three-phase fully controlled thyristor bridge","equation":"Vdc = 3 sqrt(2) VLL cos(alpha)/pi; Vrms = Vm sqrt(1/2 + 3 sqrt(3) cos(2 alpha)/(4 pi)); I_thy = Id/3 (avg), Id/sqrt(3) (rms)"}

In your browser

Open Power Electronics 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: power_electronics_ai999x.wasm and the loader ax-wasm.js (https://www.ai999x.com/compute/rust/ax-wasm.js).

Python

pip install power-electronics-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.

Control-to-output transfer function of a DC-DC converter

The control-to-output (duty-to-voltage) transfer function features of an ideal buck, boost or buck-boost converter in continuous conduction: dc gain, double-pole frequency, quality factor and right-half-plane zero.

Command line:

$ power-electronics solve pe_converter_small_signal topology=boost "Vin=12 V" D=0.5 "L=50 uH" "C=100 uF" "R=10 ohm"
Gvd0 = 48 V
f0 = 1125.395395 Hz
Q = 7.071067812
f_rhpz = 7957.747155 Hz
rhp_zero = present
  by: Control-to-output transfer function of a DC-DC converter  (buck: Gvd0 = Vin, w0 = 1/sqrt(LC), Q = R sqrt(C/L); boost: Gvd0 = Vin/D'^2, w0 = D'/sqrt(LC), Q = D' R sqrt(C/L), wz = D'^2 R/L; buck-boost: wz = D'^2 R/(D L))
  Vin = 12 V
  D = 0.5
  L = 5e-05 H
  C = 0.0001 F
  R = 10 ohm
  topology = boost
  assumes: ideal converter in continuous conduction, no ESR

Python:

import power_electronics_ai999x as m

a = m.solve("pe_converter_small_signal", topology="boost", Vin="12 V", D="0.5", L="50 uH", C="100 uF", R="10 ohm")
print(a.values or (a.value, a.unit))

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

Peak current mode: subharmonic stability and slope compensation

The inductor current slopes, the period-to-period perturbation factor, the stability verdict and the smallest and the deadbeat compensation ramp slopes of a peak-current-mode buck converter.

Command line:

$ power-electronics solve pe_peak_current_mode_slope "Vin=12 V" "Vout=8 V" "L=10 uH" "fs=300 kHz" "ma=0.3 A/us"
D = 0.6666666667
m1 = 400000 A/s
m2 = 800000 A/s
alpha = -0.7142857143
stable = stable
ma_min = 200000 A/s
ma_deadbeat = 800000 A/s
  by: Peak current mode: subharmonic stability and slope compensation  (m1 = (Vin - Vo)/L, m2 = Vo/L; alpha = -(m2 - ma)/(m1 + ma); stable when |alpha| < 1; ma,min = (m2 - m1)/2)
  Vin = 12 V
  Vout = 8 V
  L = 1e-05 H
  fs = 300000 Hz
  ma = 300000 A/s
  assumes: 0 < Vout < Vin; continuous conduction

Python:

import power_electronics_ai999x as m

a = m.solve("pe_peak_current_mode_slope", Vin="12 V", Vout="8 V", L="10 uH", fs="300 kHz", ma="0.3 A/us")
print(a.values or (a.value, a.unit))

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

Boost converter in continuous conduction

The duty cycle, average inductor (input) current, inductor ripple, peak and valley currents and switch voltage stress of an ideal boost (step-up) converter in continuous conduction.

Command line:

$ power-electronics solve pe_boost_ccm "Vin=12 V" "Vout=48 V" "L=100 uH" "fs=100 kHz" "Io=1 A"
D = 0.75
IL = 4 A
dIL = 0.9 A
IL_max = 4.45 A
IL_min = 3.55 A
V_switch = 48 V
  by: Boost converter in continuous conduction  (D = 1 - Vin/Vout; IL = Io/(1 - D); dIL = Vin D/(L fs))
  Vin = 12 V
  Vout = 48 V
  L = 0.0001 H
  fs = 100000 Hz
  Io = 1 A
  assumes: Vout > Vin > 0; continuous conduction (IL_min > 0)

Python:

import power_electronics_ai999x as m

a = m.solve("pe_boost_ccm", Vin="12 V", Vout="48 V", L="100 uH", fs="100 kHz", Io="1 A")
print(a.values or (a.value, a.unit))

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

Boost converter in discontinuous conduction

The conversion ratio, output voltage, diode conduction interval D2, peak inductor current and load current of an ideal boost converter in discontinuous conduction.

Command line:

$ power-electronics solve pe_boost_dcm "Vin=12 V" "L=10 uH" "fs=100 kHz" "R=100 ohm" D=0.4
K = 0.02
M = 3.372281323
Vout = 40.46737588 V
D2 = 0.1686140662
I_pk = 4.8 A
Io = 0.4046737588 A
  by: Boost converter in discontinuous conduction  (M = (1 + sqrt(1 + 4 D^2/K))/2; D2 = D/(M - 1); I_pk = Vin D Ts/L)
  Vin = 12 V
  L = 1e-05 H
  fs = 100000 Hz
  R = 100 ohm
  D = 0.4
  assumes: K < D (1 - D)^2 (discontinuous conduction)

Python:

import power_electronics_ai999x as m

a = m.solve("pe_boost_dcm", Vin="12 V", L="10 uH", fs="100 kHz", R="100 ohm", D="0.4")
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/power-electronics-ai999x/

Highlights

• Verified, not generated. Every answer is computed by a certified power electronics 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 482 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 power-electronics 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: DC-DC converters, rectifiers, inverters and PWM, motor drives, device losses, thermal and magnetic design, resonant converters.

Model overview

Property	Value
Package	power-electronics-ai999x 1.0.0 (import power_electronics_ai999x)
Type	Deterministic neuro-symbolic power electronics engine (rule-based reading + certified symbolic capabilities)
Capabilities	79 certified procedures
Knowledge pack	version 1.0.0
Golden answers	482 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 power-electronics-ai999x
# or straight from this repository:
pip install https://huggingface.co/AI999X/power-electronics-ai999x/resolve/main/dist/power_electronics_ai999x-1.0.0-py3-none-any.whl

import power_electronics_ai999x as power_electronics

power_electronics.ask("Buck converter in CCM: Vin = 24 V, Vout = 5 V, L = 22 uH, fs = 200 kHz, Io = 3 A").show()
# -> SOLVED — D = 0.208333, dIL = 0.899621, IL_max = 3.44981, IL_min = 2.55019  (pe_buck_ccm)
power_electronics.ask("Three-phase controlled rectifier: VLL = 415 V, alpha = 45 deg, Id = 50 A").show()
# -> SOLVED — Vdc = 396.296, Vrms = 415, P = 19814.8, I_thy_avg = 16.6667  (pe_three_phase_controlled_rectifier)
power_electronics.ask("Space vector dwell times: Vref = 200 V, theta = 25 deg, Vdc = 600 V, fs = 10 kHz").show()
# -> SOLVED — sector = 1, T1 = 3.31155e-05, T2 = 2.43999e-05, T0 = 4.24847e-05  (pe_space_vector_times)
power_electronics.solve("pe_flyback_ccm", Vin=48.0, Vout=12.0, n=2.0, Lm=0.0002, fs=100000.0, Io=2.0).show()
# -> SOLVED — D = 0.333333, Im = 1.5, dIm = 0.8, Ipk_primary = 1.9  (pe_flyback_ccm)
power_electronics.ask("Steinmetz core loss: k = 3.2, alpha = 1.4, beta = 2.5, f = 100 kHz, Bpk = 100 mT, Vc = 5 cm^3").show()
# -> SOLVED — Pv = 101193, Pv_mW_cm3 = 101.193, P = 0.505964  (pe_steinmetz_core_loss)
power_electronics.ask("Who created you?").message
# -> power-electronics-ai999x was created and engineered by Ibnul Jaif Farabi at AI999X.

The Python package also installs a Python command, power-electronics-ai999x "<question>"; for the native program use power-electronics (above). The outputs above were produced by running this exact code on power-electronics-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 power-electronics-ai999x; the answer is the one it returned, with the steps it took.

Buck converter in continuous conduction · DC-DC converters

Buck converter in CCM: Vin = 24 V, Vout = 5 V, L = 22 uH, fs = 200 kHz, Io = 3 A

→ D = 0.2083333333, dIL = 0.8996212121 A, IL_max = 3.449810606 A, IL_min = 2.550189394 A, ripple_ratio = 0.2998737374, Iin = 0.625 A

Steps

• Buck converter in continuous conduction: Vin = 24 V, Vout = 5 V, L = 22 uH, fs = 200 kHz, Io = 3 A → D = 0.2083333333 A, dIL = 0.8996212121 A, IL_max = 3.449810606 A, IL_min = 2.550189394 A, ripple_ratio = 0.2998737374 A, Iin = 0.625 A

Three-phase fully controlled thyristor bridge · Rectifiers

Three-phase controlled rectifier: VLL = 415 V, alpha = 45 deg, Id = 50 A

→ Vdc = 396.2958083 V, Vrms = 415 V, P = 19814.79041 W, I_thy_avg = 16.66666667 A, I_thy_rms = 28.86751346 A, Is_rms = 40.82482905 A, mode = rectifying

Steps

• Three-phase fully controlled thyristor bridge: VLL = 415 V, alpha = 45 deg, Id = 50 A → Vdc = 396.2958083 V, Vrms = 415 V, P = 19814.79041 W, I_thy_avg = 16.66666667 A, I_thy_rms = 28.86751346 A, Is_rms = 40.82482905 A, mode = rectifying V

Space-vector modulation dwell times · Inverters

Space vector dwell times: Vref = 200 V, theta = 25 deg, Vdc = 600 V, fs = 10 kHz

→ sector = 1, T1 = 3.311545099e-05 s, T2 = 2.439987672e-05 s, T0 = 4.248467229e-05 s, m = 0.5773502692

Steps

• Space-vector modulation dwell times: Vref = 200 V, theta = 25 deg, Vdc = 600 V, fs = 10 kHz → sector = 1 s, T1 = 3.311545099e-05 s, T2 = 2.439987672e-05 s, T0 = 4.248467229e-05 s, m = 0.5773502692 s

Flyback converter in continuous conduction · Isolated converters

m.solve("pe_flyback_ccm", Vin=48.0, Vout=12.0, n=2.0, Lm=0.0002, fs=100000.0, Io=2.0)

→ D = 0.3333333333, Im = 1.5 A, dIm = 0.8 A, Ipk_primary = 1.9 A, Ipk_secondary = 3.8 A, V_switch = 72 V, V_diode = 36 V

Steps

• Flyback converter in continuous conduction: Vin = 48 V, Vout = 12 V, n = 2, Lm = 0.0002 H, fs = 100000 Hz, Io = 2 A → D = 0.3333333333, Im = 1.5 A, dIm = 0.8 A, Ipk_primary = 1.9 A, Ipk_secondary = 3.8 A, V_switch = 72 V, V_diode = 36 V

Core loss by the Steinmetz equation · Magnetics

Steinmetz core loss: k = 3.2, alpha = 1.4, beta = 2.5, f = 100 kHz, Bpk = 100 mT, Vc = 5 cm^3

→ Pv = 101192.8851 W/m^3, Pv_mW_cm3 = 101.1928851, P = 0.5059644256 W

Steps

• Core loss by the Steinmetz equation: k = 3.2, alpha = 1.4, beta = 2.5, f = 100 kHz, Bpk = 100 mT, Vc = 5 cm^3 → Pv = 101192.8851 W/m^3, Pv_mW_cm3 = 101.1928851 W/m^3, P = 0.5059644256 W

Control-to-output transfer function of a DC-DC converter · Control

Converter small-signal model: topology = boost, Vin = 12 V, D = 0.5, L = 50 uH, C = 100 uF, R = 10 ohm

→ Gvd0 = 48 V, f0 = 1125.395395 Hz, Q = 7.071067812, f_rhpz = 7957.747155 Hz, rhp_zero = present

Steps

• Control-to-output transfer function of a DC-DC converter: topology = boost, Vin = 12 V, D = 0.5, L = 50 uH, C = 100 uF, R = 10 ohm → Gvd0 = 48 V, f0 = 1125.395395 Hz, Q = 7.071067812 Hz, f_rhpz = 7957.747155 Hz, rhp_zero = present Hz

Figures

Every point was computed by power-electronics-ai999x.

Buck converter in continuous conduction

dIL (peak-to-peak inductor ripple) against Vin (input voltage), with Io = 3 A, L = 2.20e-5 H, Vout = 5 V, fs = 2.00e5 Hz. 31 points computed by power-electronics-ai999x; marked: Vin = 24, dIL = 0.899621 A.

Boost power-factor-correction stage

dI_max (largest inductor ripple over the line cycle) against Vac (RMS line voltage), with C = 3.30e-4 F, L = 5.00e-4 H, Po = 500 W, Vo = 400 V, eff = 0.95, f_line = 50 Hz, fs = 65000 Hz. 22 points computed by power-electronics-ai999x; marked: Vac = 230, dI_max = 3.07692 A.

Flyback converter in continuous conduction

D (duty cycle) against Vin (input voltage), with Io = 2 A, Lm = 2.00e-4 H, Vout = 12 V, fs = 1.00e5 Hz, n = 2. 31 points computed by power-electronics-ai999x; marked: Vin = 48, D = 0.333333.

Peak current mode: subharmonic stability and slope compensation

alpha (perturbation factor per switching period) against Vin (input voltage), with L = 1.00e-5 H, Vout = 8 V, fs = 3.00e5 Hz, ma = 3.00e5 A/s. 25 points computed by power-electronics-ai999x; marked: Vin = 12, alpha = -0.714286.

Capabilities

79 certified procedures. A few of them:

• Procedures: Armature current ripple of a chopper-fed DC motor; Average and RMS of a rectangular pulse train; BLDC motor with six-step (120 degree) drive; Boost converter in continuous conduction; Boost converter in discontinuous conduction; Boost converter output voltage ripple; Boost converter with inductor resistance: gain and efficiency; Boost power-factor-correction stage; Bootstrap capacitor for a high-side gate driver; Buck converter efficiency with conduction and switching losses; Buck converter in continuous conduction; Buck converter in discontinuous conduction; Buck converter inductor and capacitor design; Buck converter output voltage ripple; Buck converter switch, diode, inductor and input-capacitor currents; Buck-boost converter in continuous conduction; Buck-boost converter in discontinuous conduction; Capacitor-input rectifier filter: exact ripple and peak diode current; Chopper-fed separately excited DC motor: speed and torque; Commutation overlap in a single-phase thyristor bridge; Commutation overlap in a three-phase thyristor bridge; Control-to-output transfer function of a DC-DC converter; Core loss by the Steinmetz equation; Core loss under triangular flux (iGSE) …

power_electronics_ai999x.templates() lists every question form ask() reads; the capability guide (PDF, every example executed) is included in this repository.

All 79 formulas

Show every formula, by topic

DC-DC converters / Cuk

Cuk converter in continuous conduction

$$D=\frac{V_o}{V_{in}+V_o},V_{C1}=V_{in}+V_o,\mathrm{\Delta }{i}_{L1}=\frac{V_{in}D}{L_1{f}_s},\mathrm{\Delta }{v}_{C1}=\frac{I_{o}D}{C_1{f}_s}D\; =\; \backslash frac\{V\_o\}\{V\_\{in\}+V\_o\},\backslash \; V\_\{C1\}\; =\; V\_\{in\}+V\_o,\backslash \; \backslash Delta\; i\_\{L1\}\; =\; \backslash frac\{V\_\{in\}D\}\{L\_1\; f\_s\},\backslash \; \backslash Delta\; v\_\{C1\}\; =\; \backslash frac\{I\_o\; D\}\{C\_1\; f\_s\}$$

DC-DC converters / SEPIC

SEPIC converter in continuous conduction

$$D=\frac{V_o}{V_{in}+V_o},V_{C1}=V_{in},\mathrm{\Delta }{i}_{L1,2}=\frac{V_{in}D}{L_{1,2}{f}_s},\mathrm{\Delta }{v}_{C1}=\frac{I_{o}D}{C_1{f}_s}D\; =\; \backslash frac\{V\_o\}\{V\_\{in\}+V\_o\},\backslash \; V\_\{C1\}\; =\; V\_\{in\},\backslash \; \backslash Delta\; i\_\{L1,2\}\; =\; \backslash frac\{V\_\{in\}D\}\{L\_\{1,2\}\; f\_s\},\backslash \; \backslash Delta\; v\_\{C1\}\; =\; \backslash frac\{I\_o\; D\}\{C\_1\; f\_s\}$$

DC-DC converters / boost

Boost converter in continuous conduction

$$D=1-\frac{V_{in}}{V_o},I_L=\frac{I_o}{1-D},\mathrm{\Delta }{i}_L=\frac{V_{in}D}{L{f}_s}D\; =\; 1-\backslash frac\{V\_\{in\}\}\{V\_o\},\backslash quad\; I\_L\; =\; \backslash frac\{I\_o\}\{1-D\},\backslash quad\; \backslash Delta\; i\_L\; =\; \backslash frac\{V\_\{in\}D\}\{L\; f\_s\}$$

Boost converter output voltage ripple

$$\mathrm{\Delta }{v}_o=\frac{I_{o}D}{C{f}_s}\backslash Delta\; v\_o\; =\; \backslash frac\{I\_o\; D\}\{C\; f\_s\}$$

DC-DC converters / buck

Buck converter in continuous conduction

$$D=\frac{V_o}{V_{in}},\mathrm{\Delta }{i}_L=\frac{(V_{in}-V_o)D}{L{f}_s},I_{L,\max \mathrm{/}\min }=I_o\pm \frac{\mathrm{\Delta }{i}_L}{2}D\; =\; \backslash frac\{V\_o\}\{V\_\{in\}\},\backslash quad\; \backslash Delta\; i\_L\; =\; \backslash frac\{(V\_\{in\}-V\_o)D\}\{L\; f\_s\},\backslash quad\; I\_\{L,\backslash max/\backslash min\}\; =\; I\_o\; \backslash pm\; \backslash tfrac\{\backslash Delta\; i\_L\}\{2\}$$

Buck converter output voltage ripple

$$\mathrm{\Delta }{v}_o=\frac{\mathrm{\Delta }{i}_L}{8C{f}_s}=\frac{(1-D)V_o}{8LC{f}_s^2}\backslash Delta\; v\_o\; =\; \backslash frac\{\backslash Delta\; i\_L\}\{8\; C\; f\_s\}\; =\; \backslash frac\{(1-D)V\_o\}\{8\; L\; C\; f\_s^2\}$$

DC-DC converters / buck-boost

Buck-boost converter in continuous conduction

$$D=\frac{\mathrm{\mid }{V}_o\mathrm{\mid }}{V_{in}+\mathrm{\mid }{V}_o\mathrm{\mid }},I_L=\frac{I_o}{1-D},\mathrm{\Delta }{i}_L=\frac{V_{in}D}{L{f}_s}D\; =\; \backslash frac\{\backslash vert\; V\_o\backslash vert\; \}\{V\_\{in\}+\backslash vert\; V\_o\backslash vert\; \},\backslash quad\; I\_L\; =\; \backslash frac\{I\_o\}\{1-D\},\backslash quad\; \backslash Delta\; i\_L\; =\; \backslash frac\{V\_\{in\}D\}\{L\; f\_s\}$$

DC-DC converters / component stresses

Buck converter switch, diode, inductor and input-capacitor currents

$$I_{sw,rms}=\sqrt{D(I_o^2+\frac{\mathrm{\Delta }{i}_L^2}{12})},I_{C_{in},rms}=\sqrt{I_{sw,rms}^2-(D{I}_o{)}^2}I\_\{sw,rms\}\; =\; \backslash sqrt\{D\backslash left(I\_o^2\; +\; \backslash tfrac\{\backslash Delta\; i\_L^2\}\{12\}\backslash right)\},\backslash quad\; I\_\{C\_\{in\},rms\}\; =\; \backslash sqrt\{I\_\{sw,rms\}^2\; -\; (D\; I\_o)^2\}$$

DC-DC converters / conduction boundary

Critical inductance at the CCM/DCM boundary

$$L_{c,buck}=\frac{(1-D)R}{2f_s},L_{c,boost}=\frac{D(1-D{)}^{2}R}{2f_s},L_{c,buck\text{-}boost}=\frac{(1-D{)}^{2}R}{2f_s}L\_\{c,buck\}\; =\; \backslash frac\{(1-D)R\}\{2f\_s\},\backslash \; L\_\{c,boost\}\; =\; \backslash frac\{D(1-D)^2R\}\{2f\_s\},\backslash \; L\_\{c,buck\backslash text\{-\}boost\}\; =\; \backslash frac\{(1-D)^2R\}\{2f\_s\}$$

DC-DC converters / conversion ratio

Ideal DC-DC conversion ratio from the duty cycle

$$M_{buck}=D,M_{boost}=\frac{1}{1-D},M_{buck\text{-}boost,Cuk,SEPIC}=\frac{D}{1-D}M\_\{buck\}\; =\; D,\backslash quad\; M\_\{boost\}\; =\; \backslash frac\{1\}\{1-D\},\backslash quad\; M\_\{buck\backslash text\{-\}boost,\backslash ,Cuk,\backslash ,SEPIC\}\; =\; \backslash frac\{D\}\{1-D\}$$

DC-DC converters / discontinuous conduction

Boost converter in discontinuous conduction

$$M=\frac{1+\sqrt{1+4D^2\mathrm{/}K}}{2},D_2=\frac{D}{M-1}M\; =\; \backslash frac\{1+\backslash sqrt\{1+4D^2/K\}\}\{2\},\backslash quad\; D\_2\; =\; \backslash frac\{D\}\{M-1\}$$

Buck converter in discontinuous conduction

$$M=\frac{2}{1+\sqrt{1+4K\mathrm{/}{D}^2}},K=\frac{2L}{R{T}_s}M\; =\; \backslash frac\{2\}\{1+\backslash sqrt\{1+4K/D^2\}\},\backslash quad\; K\; =\; \backslash frac\{2L\}\{R\; T\_s\}$$

Buck-boost converter in discontinuous conduction

$$M=\frac{D}{\sqrt{K}},D_2=\sqrt{K}M\; =\; \backslash frac\{D\}\{\backslash sqrt\{K\}\},\backslash quad\; D\_2\; =\; \backslash sqrt\{K\}$$

DC-DC converters / filter design

Buck converter inductor and capacitor design

$$L=\frac{(V_{in}-V_o)D}{r{I}_o{f}_s},C=\frac{r{I}_o}{8f_s\mathrm{\Delta }{v}_o}L\; =\; \backslash frac\{(V\_\{in\}-V\_o)D\}\{r\; I\_o\; f\_s\},\backslash quad\; C\; =\; \backslash frac\{r\; I\_o\}\{8\; f\_s\; \backslash Delta\; v\_o\}$$

DC-DC converters / interleaving

Interleaved multiphase buck: output current ripple cancellation

$$\frac{\mathrm{\Delta }{i}_{tot}}{\mathrm{\Delta }{i}_{ph}}=\frac{N(D-\frac{m}{N})(\frac{m+1}{N}-D)}{D(1-D)},m=\lfloor ND\rfloor \backslash frac\{\backslash Delta\; i\_\{tot\}\}\{\backslash Delta\; i\_\{ph\}\}\; =\; \backslash frac\{N\backslash left(D-\backslash frac\{m\}\{N\}\backslash right)\backslash left(\backslash frac\{m+1\}\{N\}-D\backslash right)\}\{D(1-D)\},\backslash \; m\; =\; \backslash lfloor\; ND\backslash rfloor$$

Control / averaged small-signal model

Control-to-output transfer function of a DC-DC converter

$$G_{vd}(s)=G_{d0}\frac{1-s\mathrm{/}{\omega }_z}{1+s\mathrm{/}(Q{\omega }_0)+(s\mathrm{/}{\omega }_0{)}^2},{\omega }_{z,boost}=\frac{D^{\mathrm{\prime }2}R}{L}G\_\{vd\}(s)\; =\; G\_\{d0\}\backslash frac\{1\; -\; s/\backslash omega\_z\}\{1\; +\; s/(Q\backslash omega\_0)\; +\; (s/\backslash omega\_0)^2\},\backslash quad\; \backslash omega\_\{z,boost\}\; =\; \backslash frac\{D\text{'}^2R\}\{L\}$$

Control / current mode

Peak current mode: subharmonic stability and slope compensation

Devices / conduction loss

MOSFET conduction loss at junction temperature

$$R_{ds}(T_j)=R_{ds,25}(1+\alpha (T_j-{25}^{\circ }\mathrm{C})),P=I_{rms}^2{R}_{ds}(T_j)R\_\{ds\}(T\_j)\; =\; R\_\{ds,25\}\backslash left(1+\backslash alpha(T\_j-25^\backslash circ\backslash mathrm\{C\})\backslash right),\backslash quad\; P\; =\; I\_\{rms\}^2R\_\{ds\}(T\_j)$$

Devices / gate drive

Bootstrap capacitor for a high-side gate driver

$$C_{boot}\ge \frac{Q_g+I_{lk}{D}_{max}\mathrm{/}{f}_s}{\mathrm{\Delta }V}C\_\{boot\}\; \backslash ge\; \backslash frac\{Q\_g\; +\; I\_\{lk\}D\_\{max\}/f\_s\}\{\backslash Delta\; V\}$$

Gate-drive power and currents

$$P_G=Q_g{V}_{drv}{f}_s,I_{G,pk}=\frac{V_{drv}}{R_g}P\_\{G\}\; =\; Q\_gV\_\{drv\}f\_s,\backslash quad\; I\_\{G,pk\}\; =\; \backslash frac\{V\_\{drv\}\}\{R\_g\}$$

Devices / paralleling

Current sharing between paralleled MOSFETs

$$I_k=I\frac{1\mathrm{/}{R}_k}{\sum _{j}1\mathrm{/}{R}_j},P_k=I_k^2{R}_{k}I\_k\; =\; I\backslash frac\{1/R\_k\}\{\backslash sum\_j\; 1/R\_j\},\backslash quad\; P\_k\; =\; I\_k^2R\_k$$

Devices / reverse recovery

Diode reverse-recovery charge and losses

$$Q_{rr}=\frac{1}{2}{I}_{rm}{t}_{rr},E_D=\frac{V_r{I}_{rm}{t}_b}{6},E_{sw}=V_r{I}_{rm}(\frac{t_a}{2}+\frac{t_b}{3})Q\_\{rr\}\; =\; \backslash tfrac12I\_\{rm\}t\_\{rr\},\backslash quad\; E\_D\; =\; \backslash frac\{V\_rI\_\{rm\}t\_b\}\{6\},\backslash quad\; E\_\{sw\}\; =\; V\_rI\_\{rm\}\backslash left(\backslash frac\{t\_a\}\{2\}+\backslash frac\{t\_b\}\{3\}\backslash right)$$

Devices / snubbers

RC snubber from measured ringing frequencies

$$C_p=\frac{C_{add}}{(f_0\mathrm{/}{f}_1{)}^2-1},L_p=\frac{1}{(2\pi f_0{)}^2{C}_p},R_s=\sqrt{L_p\mathrm{/}{C}_p}C\_p\; =\; \backslash frac\{C\_\{add\}\}\{(f\_0/f\_1)^2-1\},\backslash quad\; L\_p\; =\; \backslash frac\{1\}\{(2\backslash pi\; f\_0)^2C\_p\},\backslash quad\; R\_s\; =\; \backslash sqrt\{L\_p/C\_p\}$$

RCD turn-off snubber design

$$C_s=\frac{I{t}_f}{2V},E_{sw}=\frac{I^2{t}_f^2}{24C_s}=\frac{VI{t}_f}{12},E_R=\frac{1}{2}{C}_s{V}^{2}C\_s\; =\; \backslash frac\{I\; t\_f\}\{2V\},\backslash quad\; E\_\{sw\}\; =\; \backslash frac\{I^2t\_f^2\}\{24C\_s\}\; =\; \backslash frac\{VIt\_f\}\{12\},\backslash quad\; E\_R\; =\; \backslash tfrac12C\_sV^2$$

Devices / soft switching

Zero-voltage switching: resonant transition of a switch node

$$v(t)=I_{0}Z\sin \omega t,Z=\sqrt{L\mathrm{/}C},t_{zvs}=\frac{1}{\omega }{\sin }^{-1}\frac{V_{in}}{I_{0}Z}v(t)\; =\; I\_0Z\backslash sin\backslash omega\; t,\backslash quad\; Z\; =\; \backslash sqrt\{L/C\},\backslash quad\; t\_\{zvs\}\; =\; \backslash frac\{1\}\{\backslash omega\}\backslash sin^\{-1\}\backslash frac\{V\_\{in\}\}\{I\_0Z\}$$

Devices / switching loss

Hard-switching loss of a power transistor

$$P_{sw}=\frac{1}{2}VI(t_r+t_f)f_s,P_{oss}=\frac{1}{2}{C}_{oss}{V}^2{f}_{s}P\_\{sw\}\; =\; \backslash tfrac12\; V\; I\backslash ,(t\_r\; +\; t\_f)\backslash ,f\_s,\backslash quad\; P\_\{oss\}\; =\; \backslash tfrac12C\_\{oss\}V^2f\_s$$

Front ends / energy storage

Hold-up (bulk) capacitor

$$C=\frac{2P_o{t}_{hold}}{V_1^2-V_2^2}C\; =\; \backslash frac\{2P\_ot\_\{hold\}\}\{V\_1^2\; -\; V\_2^2\}$$

Front ends / power factor correction

Boost power-factor-correction stage

$$\mathrm{\Delta }{i}_L(v)=\frac{v(1-v\mathrm{/}{V}_o)}{L{f}_s},\mathrm{\Delta }{i}_{L,max}=\frac{V_o}{4L{f}_s},\mathrm{\Delta }{V}_o=\frac{P_o}{2\pi f_{line}C{V}_o}\backslash Delta\; i\_L(v)\; =\; \backslash frac\{v\backslash ,(1-v/V\_o)\}\{Lf\_s\},\backslash quad\; \backslash Delta\; i\_\{L,max\}\; =\; \backslash frac\{V\_o\}\{4Lf\_s\},\backslash quad\; \backslash Delta\; V\_o\; =\; \backslash frac\{P\_o\}\{2\backslash pi\; f\_\{line\}CV\_o\}$$

Inverters / dead time

Inverter dead-time voltage error

$$\mathrm{\Delta }V=t_d{f}_s{V}_{dc},\mathrm{\Delta }{\widehat{V}}_1=\frac{4}{\pi }{t}_d{f}_s{V}_{dc}\backslash Delta\; V\; =\; t\_d\; f\_s\; V\_\{dc\},\backslash quad\; \backslash Delta\backslash hat\; V\_1\; =\; \backslash frac\{4\}\{\backslash pi\}t\_d\; f\_s\; V\_\{dc\}$$

Inverters / load current

Square-wave inverter with an RL load: current

$$I_{max}=\frac{V_{dc}}{R}\frac{1-e^{-T\mathrm{/}2\tau }}{1+e^{-T\mathrm{/}2\tau }},\tau =\frac{L}{R}I\_\{max\}\; =\; \backslash frac\{V\_\{dc\}\}\{R\}\backslash ,\backslash frac\{1-e^\{-T/2\backslash tau\}\}\{1+e^\{-T/2\backslash tau\}\},\backslash quad\; \backslash tau\; =\; \backslash frac\{L\}\{R\}$$

Inverters / quasi-square-wave

Quasi-square-wave (phase-shifted) full-bridge inverter

$$V_{n,rms}=\frac{4V_{dc}}{n\pi \sqrt{2}}\cos (n\alpha ),V_{rms}=V_{dc}\sqrt{1-\frac{2\alpha }{\pi }}V\_\{n,rms\}\; =\; \backslash frac\{4V\_\{dc\}\}\{n\backslash pi\backslash sqrt2\}\backslash cos(n\backslash alpha),\backslash quad\; V\_\{rms\}\; =\; V\_\{dc\}\backslash sqrt\{1-\backslash frac\{2\backslash alpha\}\{\backslash pi\}\}$$

Inverters / sinusoidal PWM

Sinusoidal PWM: fundamental output voltage

$${\widehat{V}}_{1,leg}=m_a\frac{V_{dc}}{2},V_{LL1,rms}=\frac{\sqrt{3}}{2\sqrt{2}}{m}_a{V}_{dc},f_c=m_f{f}_1\backslash hat\; V\_\{1,leg\}\; =\; m\_a\backslash frac\{V\_\{dc\}\}\{2\},\backslash quad\; V\_\{LL1,rms\}\; =\; \backslash frac\{\backslash sqrt3\}\{2\backslash sqrt2\}m\_aV\_\{dc\},\backslash quad\; f\_c\; =\; m\_f\; f\_1$$

Inverters / space-vector PWM

Space-vector modulation dwell times

$$T_1=\frac{\sqrt{3}{T}_s{V}_{ref}}{V_{dc}}\sin \text{\hspace{0.17em}⁣}(\frac{\pi }{3}-{\theta }^{\mathrm{\prime }}),T_2=\frac{\sqrt{3}{T}_s{V}_{ref}}{V_{dc}}\sin {\theta }^{\mathrm{\prime }},T_0=T_s-T_1-T_{2}T\_1\; =\; \backslash frac\{\backslash sqrt3\; T\_s\; V\_\{ref\}\}\{V\_\{dc\}\}\backslash sin\backslash !\backslash left(\backslash tfrac\{\backslash pi\}\{3\}-\backslash theta\text{'}\backslash right),\backslash quad\; T\_2\; =\; \backslash frac\{\backslash sqrt3\; T\_s\; V\_\{ref\}\}\{V\_\{dc\}\}\backslash sin\backslash theta\text{'},\backslash quad\; T\_0\; =\; T\_s\; -\; T\_1\; -\; T\_2$$

Inverters / square-wave

Square-wave inverter output and harmonics

$$V_{n,rms}=\frac{4A}{n\pi \sqrt{2}},THD=\sqrt{\frac{{\pi }^2}{8}-1}V\_\{n,rms\}\; =\; \backslash frac\{4A\}\{n\backslash pi\backslash sqrt2\},\backslash quad\; THD\; =\; \backslash sqrt\{\backslash frac\{\backslash pi^2\}\{8\}-1\}$$

Inverters / three-phase

Three-phase six-step (square-wave) inverter

$$V_{LL1}=\frac{\sqrt{6}}{\pi }{V}_{dc},V_{LL}=\sqrt{\frac{2}{3}}{V}_{dc},V_{LL,h}=\frac{V_{LL1}}{h}V\_\{LL1\}\; =\; \backslash frac\{\backslash sqrt6\}\{\backslash pi\}V\_\{dc\},\backslash quad\; V\_\{LL\}\; =\; \backslash sqrt\{\backslash tfrac23\}V\_\{dc\},\backslash quad\; V\_\{LL,h\}\; =\; \backslash frac\{V\_\{LL1\}\}\{h\}$$

Isolated converters / bridge

Push-pull, half-bridge and full-bridge converters

$$V_o=\frac{2D{V}_{in}}{n}(\text{push-pull, full-bridge}),V_o=\frac{D{V}_{in}}{n}(\text{half-bridge})V\_o\; =\; \backslash frac\{2\; D\; V\_\{in\}\}\{n\}\backslash \; (\backslash text\{push-pull,\; full-bridge\}),\backslash quad\; V\_o\; =\; \backslash frac\{D\; V\_\{in\}\}\{n\}\backslash \; (\backslash text\{half-bridge\})$$

Isolated converters / flyback

Flyback converter in continuous conduction

$$\frac{V_o}{V_{in}}=\frac{1}{n}\frac{D}{1-D},I_m=\frac{I_o}{n(1-D)},\mathrm{\Delta }{i}_m=\frac{V_{in}D}{L_m{f}_s}\backslash frac\{V\_o\}\{V\_\{in\}\}\; =\; \backslash frac\{1\}\{n\}\backslash frac\{D\}\{1-D\},\backslash quad\; I\_m\; =\; \backslash frac\{I\_o\}\{n(1-D)\},\backslash quad\; \backslash Delta\; i\_m\; =\; \backslash frac\{V\_\{in\}D\}\{L\_m\; f\_s\}$$

Isolated converters / flyback design

Flyback magnetizing inductance for discontinuous conduction

$$L_m=\frac{(V_{in}D{)}^2}{2P_{in}{f}_s},I_{pk}=\frac{V_{in}D}{L_m{f}_s},I_{rms}=I_{pk}\sqrt{D\mathrm{/}3}L\_m\; =\; \backslash frac\{(V\_\{in\}D)^2\}\{2\; P\_\{in\}\; f\_s\},\backslash quad\; I\_\{pk\}\; =\; \backslash frac\{V\_\{in\}D\}\{L\_m\; f\_s\},\backslash quad\; I\_\{rms\}\; =\; I\_\{pk\}\backslash sqrt\{D/3\}$$

Isolated converters / forward

Forward converter with reset winding

$$V_o=\frac{V_{in}D}{n},D_{max}=\frac{1}{1+N_r\mathrm{/}{N}_p},V_{sw}=V_{in}(1+\frac{N_p}{N_r})V\_o\; =\; \backslash frac\{V\_\{in\}\; D\}\{n\},\backslash quad\; D\_\{max\}\; =\; \backslash frac\{1\}\{1+N\_r/N\_p\},\backslash quad\; V\_\{sw\}\; =\; V\_\{in\}\backslash left(1+\backslash frac\{N\_p\}\{N\_r\}\backslash right)$$

Losses / efficiency

Buck converter efficiency with conduction and switching losses

$$D=\frac{V_o+V_F+I_o{R}_L}{V_{in}-I_o{R}_{ds}+V_F},P_{sw}=\frac{1}{2}{V}_{in}{I}_o{t}_{sw}{f}_s,\eta =\frac{P_o}{P_o+\sum P}D\; =\; \backslash frac\{V\_o+V\_F+I\_oR\_L\}\{V\_\{in\}-I\_oR\_\{ds\}+V\_F\},\backslash quad\; P\_\{sw\}\; =\; \backslash tfrac12\; V\_\{in\}I\_o\; t\_\{sw\}\; f\_s,\backslash quad\; \backslash eta\; =\; \backslash frac\{P\_o\}\{P\_o+\backslash sum\; P\}$$

Losses / inverter devices

IGBT and diode conduction losses in a sinusoidal PWM inverter

$$P_T=V_{ce0}\widehat{I}(\frac{1}{2\pi }+\frac{M\cos \varphi }{8})+r_{ce}{\widehat{I}}^2(\frac{1}{8}+\frac{M\cos \varphi }{3\pi })P\_T\; =\; V\_\{ce0\}\backslash hat\; I\backslash left(\backslash frac\{1\}\{2\backslash pi\}+\backslash frac\{M\backslash cos\backslash phi\}\{8\}\backslash right)\; +\; r\_\{ce\}\backslash hat\; I^2\backslash left(\backslash frac18+\backslash frac\{M\backslash cos\backslash phi\}\{3\backslash pi\}\backslash right)$$

Losses / non-ideal converters

Boost converter with inductor resistance: gain and efficiency

$$M=\frac{1}{1-D}\frac{1}{1+\frac{R_L}{(1-D{)}^{2}R}},\eta =\frac{1}{1+\frac{R_L}{(1-D{)}^{2}R}}M\; =\; \backslash frac\{1\}\{1-D\}\backslash ,\backslash frac\{1\}\{1+\backslash frac\{R\_L\}\{(1-D)^2R\}\},\backslash quad\; \backslash eta\; =\; \backslash frac\{1\}\{1+\backslash frac\{R\_L\}\{(1-D)^2R\}\}$$

Magnetics / core loss

Core loss by the Steinmetz equation

$$P_v=k{f}^{\alpha }{\widehat{B}}^{\beta }P\_v\; =\; k\; f^\{\backslash alpha\}\; \backslash hat\; B^\{\backslash beta\}$$

Core loss under triangular flux (iGSE)

$$P_v=k_i(\mathrm{\Delta }B{)}^{\beta }{f}^{\alpha }(D^{1-\alpha }+(1-D{)}^{1-\alpha }),k_i=\frac{k}{(2\pi {)}^{\alpha -1}{\int }_0^{2\pi }\mathrm{\mid }\cos \theta {\mathrm{\mid }}^{\alpha }{2}^{\beta -\alpha }d\theta }P\_v\; =\; k\_i(\backslash Delta\; B)^\{\backslash beta\}f^\{\backslash alpha\}\backslash left(D^\{1-\backslash alpha\}+(1-D)^\{1-\backslash alpha\}\backslash right),\backslash \; k\_i\; =\; \backslash frac\{k\}\{(2\backslash pi)^\{\backslash alpha-1\}\backslash int\_0^\{2\backslash pi\}\backslash vert\; \backslash cos\backslash theta\backslash vert\; ^\{\backslash alpha\}2^\{\backslash beta-\backslash alpha\}d\backslash theta\}$$

Magnetics / core selection

Inductor core area product

$$A_p=A_c{W}_a=\frac{L{I}_{pk}{I}_{rms}}{B_{max}J{K}_u}A\_p\; =\; A\_cW\_a\; =\; \backslash frac\{L\backslash ,I\_\{pk\}I\_\{rms\}\}\{B\_\{max\}JK\_u\}$$

Transformer core area product

$$A_p=\frac{P_t}{K_f{K}_u{B}_{m}fJ},P_t=P_o(1+\frac{1}{\eta })A\_p\; =\; \backslash frac\{P\_t\}\{K\_fK\_uB\_mfJ\},\backslash quad\; P\_t\; =\; P\_o\backslash left(1+\backslash frac\{1\}\{\backslash eta\}\backslash right)$$

Magnetics / inductors

Gapped-core inductor: turns and air gap

$$N=\frac{L{I}_{pk}}{B_{max}{A}_c},l_g=\frac{{\mu }_0{N}^2{A}_c}{L}N\; =\; \backslash frac\{LI\_\{pk\}\}\{B\_\{max\}A\_c\},\backslash quad\; l\_g\; =\; \backslash frac\{\backslash mu\_0N^2A\_c\}\{L\}$$

Magnetics / transformers

Transformer turns from Faraday's law

$$N_p=\frac{V}{K_{f}f{B}_{max}{A}_c},K_f=\sqrt{2}\pi (\text{sine}),4(\text{square})N\_p\; =\; \backslash frac\{V\}\{K\_f\; f\; B\_\{max\}A\_c\},\backslash quad\; K\_f\; =\; \backslash sqrt2\backslash pi\backslash \; (\backslash text\{sine\}),\backslash \; 4\backslash \; (\backslash text\{square\})$$

Magnetics / winding losses

Skin effect in a round wire

$$\delta =\sqrt{\frac{\rho }{\pi f{\mu }_0}},\frac{R_{ac}}{R_{dc}}=\mathrm{R}\mathrm{e}\left[\frac{kr}{2}\frac{J_0(kr)}{J_1(kr)}\right],k=\frac{1-j}{\delta }\backslash delta\; =\; \backslash sqrt\{\backslash frac\{\backslash rho\}\{\backslash pi\; f\backslash mu\_0\}\},\backslash quad\; \backslash frac\{R\_\{ac\}\}\{R\_\{dc\}\}\; =\; \backslash mathrm\{Re\}\backslash left[\backslash frac\{kr\}\{2\}\backslash frac\{J\_0(kr)\}\{J\_1(kr)\}\backslash right],\backslash \; k\; =\; \backslash frac\{1-j\}\{\backslash delta\}$$

Winding AC resistance factor (Dowell)

$$F_R=\mathrm{\Delta }[\frac{\sinh 2\mathrm{\Delta }+\sin 2\mathrm{\Delta }}{\cosh 2\mathrm{\Delta }-\cos 2\mathrm{\Delta }}+\frac{2(m^2-1)}{3}\frac{\sinh \mathrm{\Delta }-\sin \mathrm{\Delta }}{\cosh \mathrm{\Delta }+\cos \mathrm{\Delta }}]F\_R\; =\; \backslash Delta\backslash left[\backslash frac\{\backslash sinh2\backslash Delta+\backslash sin2\backslash Delta\}\{\backslash cosh2\backslash Delta-\backslash cos2\backslash Delta\}\; +\; \backslash frac\{2(m^2-1)\}\{3\}\backslash frac\{\backslash sinh\backslash Delta-\backslash sin\backslash Delta\}\{\backslash cosh\backslash Delta+\backslash cos\backslash Delta\}\backslash right]$$

Motor drives / BLDC

BLDC motor with six-step (120 degree) drive

$$D{V}_{dc}=2R_{ph}I+2K_{e,ph}\omega ,T=2K_{e,ph}IDV\_\{dc\}\; =\; 2R\_\{ph\}I\; +\; 2K\_\{e,ph\}\backslash omega,\backslash quad\; T\; =\; 2K\_\{e,ph\}I$$

Motor drives / DC motor

Chopper-fed separately excited DC motor: speed and torque

$$V_a=D{V}_s,I_a=\frac{T_L}{K},\omega =\frac{V_a-I_a{R}_a}{K}V\_a\; =\; DV\_s,\backslash quad\; I\_a\; =\; \backslash frac\{T\_L\}\{K\},\backslash quad\; \backslash omega\; =\; \backslash frac\{V\_a\; -\; I\_aR\_a\}\{K\}$$

Phase-controlled rectifier DC motor drive

$$V_a=\frac{2\sqrt{2}{V}_s}{\pi }\cos \alpha (1\varphi ),\frac{3\sqrt{2}{V}_{LL}}{\pi }\cos \alpha (3\varphi ),\omega =\frac{V_a-I_a{R}_a}{K}V\_a\; =\; \backslash frac\{2\backslash sqrt2V\_s\}\{\backslash pi\}\backslash cos\backslash alpha\backslash \; (1\backslash phi),\backslash \; \backslash frac\{3\backslash sqrt2V\_\{LL\}\}\{\backslash pi\}\backslash cos\backslash alpha\backslash \; (3\backslash phi),\backslash quad\; \backslash omega\; =\; \backslash frac\{V\_a\; -\; I\_aR\_a\}\{K\}$$

Motor drives / PMSM

PMSM steady state in the dq frame: voltages and torque

$$v_d=R_s{i}_d-{\omega }_e{L}_q{i}_q,v_q=R_s{i}_q+{\omega }_e(L_d{i}_d+{\psi }_f),T_e=\frac{3}{2}p({\psi }_f{i}_q+(L_d-L_q)i_d{i}_q)v\_d\; =\; R\_si\_d\; -\; \backslash omega\_eL\_qi\_q,\backslash \; v\_q\; =\; R\_si\_q\; +\; \backslash omega\_e(L\_di\_d+\backslash psi\_f),\backslash \; T\_e\; =\; \backslash tfrac32p\backslash left(\backslash psi\_fi\_q\; +\; (L\_d-L\_q)i\_di\_q\backslash right)$$

Motor drives / PMSM control

Maximum torque per ampere (MTPA) of an interior PMSM

$$i_d=\frac{{\psi }_f-\sqrt{{\psi }_f^2+8(L_q-L_d{)}^2{I}_s^2}}{4(L_q-L_d)},i_q=\sqrt{I_s^2-i_d^2}i\_d\; =\; \backslash frac\{\backslash psi\_f\; -\; \backslash sqrt\{\backslash psi\_f^2\; +\; 8(L\_q-L\_d)^2\; I\_s^2\}\}\{4(L\_q-L\_d)\},\backslash quad\; i\_q\; =\; \backslash sqrt\{I\_s^2\; -\; i\_d^2\}$$

Motor drives / current ripple

Armature current ripple of a chopper-fed DC motor

$$I_{max}=\frac{V_s}{R_a}\frac{1-e^{-DT\mathrm{/}\tau }}{1-e^{-T\mathrm{/}\tau }}-\frac{E_a}{R_a},I_{min}=\frac{V_s}{R_a}\frac{e^{DT\mathrm{/}\tau }-1}{e^{T\mathrm{/}\tau }-1}-\frac{E_a}{R_a}I\_\{max\}\; =\; \backslash frac\{V\_s\}\{R\_a\}\backslash frac\{1-e^\{-DT/\backslash tau\}\}\{1-e^\{-T/\backslash tau\}\}\; -\; \backslash frac\{E\_a\}\{R\_a\},\backslash \; I\_\{min\}\; =\; \backslash frac\{V\_s\}\{R\_a\}\backslash frac\{e^\{DT/\backslash tau\}-1\}\{e^\{T/\backslash tau\}-1\}\; -\; \backslash frac\{E\_a\}\{R\_a\}$$

Motor drives / field weakening

Surface PMSM field weakening: base speed, maximum speed and torque

$${\omega }_b=\frac{V_{max}}{p\sqrt{{\psi }_f^2+(L_s{I}_{max}{)}^2}},{\omega }_{max}=\frac{V_{max}}{p({\psi }_f-L_s{I}_{max})},i_d=\frac{(V_{max}\mathrm{/}p\omega {)}^2-{\psi }_f^2-L_s^2{I}_{max}^2}{2{\psi }_f{L}_s}\backslash omega\_b\; =\; \backslash frac\{V\_\{max\}\}\{p\backslash sqrt\{\backslash psi\_f^2\; +\; (L\_sI\_\{max\})^2\}\},\backslash \; \backslash omega\_\{max\}\; =\; \backslash frac\{V\_\{max\}\}\{p(\backslash psi\_f\; -\; L\_sI\_\{max\})\},\backslash \; i\_d\; =\; \backslash frac\{(V\_\{max\}/p\backslash omega)^2\; -\; \backslash psi\_f^2\; -\; L\_s^2I\_\{max\}^2\}\{2\backslash psi\_fL\_s\}$$

Motor drives / induction motor

Induction motor constant V/f speed control

$$\frac{V}{f}=\text{const},n=\frac{120f}{P}-(n_{s,r}-n_r)\frac{T}{T_r}\backslash frac\{V\}\{f\}\; =\; \backslash text\{const\},\backslash quad\; n\; =\; \backslash frac\{120f\}\{P\}\; -\; \backslash left(n\_\{s,r\}\; -\; n\_r\backslash right)\backslash frac\{T\}\{T\_r\}$$

Motor drives / regenerative braking

Regenerative braking of a DC motor with a chopper

$$I_a=\frac{K\omega -(1-D)V_s}{R_a},P_{regen}=(1-D)V_s{I}_{a}I\_a\; =\; \backslash frac\{K\backslash omega\; -\; (1-D)V\_s\}\{R\_a\},\backslash quad\; P\_\{regen\}\; =\; (1-D)V\_sI\_a$$

Power quality / THD

Total harmonic distortion from harmonic values

$$THD=\frac{\sqrt{\sum _{h\ge 2}{V}_h^2}}{V_1},V_{rms}=V_1\sqrt{1+TH{D}^2}THD\; =\; \backslash frac\{\backslash sqrt\{\backslash sum\_\{h\backslash ge2\}\; V\_h^2\}\}\{V\_1\},\backslash quad\; V\_\{rms\}\; =\; V\_1\backslash sqrt\{1+THD^2\}$$

Power quality / power factor

Power factor with harmonic distortion

$$PF=\frac{\cos {\varphi }_1}{\sqrt{1+TH{D}_i^2}}PF\; =\; \backslash frac\{\backslash cos\backslash phi\_1\}\{\backslash sqrt\{1+THD\_i^2\}\}$$

Power quality / waveform values

Average and RMS of a rectangular pulse train

$$V_{avg}=DA,V_{rms}=\sqrt{D}A,RF=\sqrt{\frac{1}{D}-1}V\_\{avg\}\; =\; DA,\backslash quad\; V\_\{rms\}\; =\; \backslash sqrt\{D\}\backslash ,A,\backslash quad\; RF\; =\; \backslash sqrt\{\backslash tfrac1D\; -\; 1\}$$

Rectifiers / AC voltage control

Single-phase AC voltage controller (resistive load)

$$V_o=V_s\sqrt{1-\frac{\alpha }{\pi }+\frac{\sin 2\alpha }{2\pi }},PF=\frac{V_o}{V_s}V\_\{o\}\; =\; V\_s\backslash sqrt\{1-\backslash frac\{\backslash alpha\}\{\backslash pi\}+\backslash frac\{\backslash sin2\backslash alpha\}\{2\backslash pi\}\},\backslash quad\; PF\; =\; \backslash frac\{V\_o\}\{V\_s\}$$

Rectifiers / RL load

Half-wave diode rectifier with an RL load

$$\sin (\beta -\theta )+\sin \theta e^{-\beta \mathrm{/}\omega \tau }=0,V_{avg}=\frac{V_m(1-\cos \beta )}{2\pi }\backslash sin(\backslash beta-\backslash theta)+\backslash sin\backslash theta\backslash ,e^\{-\backslash beta/\backslash omega\backslash tau\}\; =\; 0,\backslash quad\; V\_\{avg\}\; =\; \backslash frac\{V\_m(1-\backslash cos\backslash beta)\}\{2\backslash pi\}$$

Rectifiers / capacitor filter

Capacitor-input rectifier filter: exact ripple and peak diode current

$$\theta =\pi -{\tan }^{-1}(\omega RC),\sin \alpha =\sin \theta e^{-(P+\alpha -\theta )\mathrm{/}\omega RC},I_{D,pk}=V_p(\omega C\cos \alpha +\frac{\sin \alpha }{R})\backslash theta\; =\; \backslash pi\; -\; \backslash tan^\{-1\}(\backslash omega\; RC),\backslash \; \backslash sin\backslash alpha\; =\; \backslash sin\backslash theta\backslash ,e^\{-(P+\backslash alpha-\backslash theta)/\backslash omega\; RC\},\backslash \; I\_\{D,pk\}\; =\; V\_p\backslash left(\backslash omega\; C\backslash cos\backslash alpha+\backslash frac\{\backslash sin\backslash alpha\}\{R\}\backslash right)$$

Rectifiers / commutation

Commutation overlap in a single-phase thyristor bridge

$$\cos (\alpha +u)=\cos \alpha -\frac{2\omega L_s{I}_d}{V_m},V_{dc}=\frac{2V_m}{\pi }\cos \alpha -\frac{2\omega L_s{I}_d}{\pi }\backslash cos(\backslash alpha+u)\; =\; \backslash cos\backslash alpha\; -\; \backslash frac\{2\backslash omega\; L\_s\; I\_d\}\{V\_m\},\backslash quad\; V\_\{dc\}\; =\; \backslash frac\{2V\_m\}\{\backslash pi\}\backslash cos\backslash alpha\; -\; \backslash frac\{2\backslash omega\; L\_sI\_d\}\{\backslash pi\}$$

Commutation overlap in a three-phase thyristor bridge

$$\cos (\alpha +u)=\cos \alpha -\frac{2\omega L_s{I}_d}{\sqrt{2}{V}_{LL}},V_{dc}=\frac{3\sqrt{2}{V}_{LL}}{\pi }\cos \alpha -\frac{3\omega L_s{I}_d}{\pi }\backslash cos(\backslash alpha+u)\; =\; \backslash cos\backslash alpha\; -\; \backslash frac\{2\backslash omega\; L\_sI\_d\}\{\backslash sqrt2\; V\_\{LL\}\},\backslash quad\; V\_\{dc\}\; =\; \backslash frac\{3\backslash sqrt2V\_\{LL\}\}\{\backslash pi\}\backslash cos\backslash alpha\; -\; \backslash frac\{3\backslash omega\; L\_sI\_d\}\{\backslash pi\}$$

Rectifiers / phase-controlled

Single-phase phase-controlled thyristor rectifier

$$V_{dc,HW}=\frac{V_m(1+\cos \alpha )}{2\pi },V_{dc,FB}=\frac{2V_m\cos \alpha }{\pi },V_{dc,semi}=\frac{V_m(1+\cos \alpha )}{\pi }V\_\{dc,HW\}\; =\; \backslash frac\{V\_m(1+\backslash cos\backslash alpha)\}\{2\backslash pi\},\backslash \; V\_\{dc,FB\}\; =\; \backslash frac\{2V\_m\backslash cos\backslash alpha\}\{\backslash pi\},\backslash \; V\_\{dc,semi\}\; =\; \backslash frac\{V\_m(1+\backslash cos\backslash alpha)\}\{\backslash pi\}$$

Three-phase fully controlled thyristor bridge

$$V_{dc}=\frac{3\sqrt{2}{V}_{LL}}{\pi }\cos \alpha ,I_{s,rms}=\sqrt{\frac{2}{3}}{I}_{d}V\_\{dc\}\; =\; \backslash frac\{3\backslash sqrt2\; V\_\{LL\}\}\{\backslash pi\}\backslash cos\backslash alpha,\backslash quad\; I\_\{s,rms\}\; =\; \backslash sqrt\{\backslash tfrac23\}\backslash ,I\_d$$

Rectifiers / power quality

Rectifier line current: THD, displacement and power factor

$$PF=\frac{I_{s1}}{I_s}\cos \alpha ,\frac{I_{s1}}{I_s}=\frac{2\sqrt{2}}{\pi }(1\varphi ),\frac{3}{\pi }(3\varphi )PF\; =\; \backslash frac\{I\_\{s1\}\}\{I\_s\}\backslash cos\backslash alpha,\backslash quad\; \backslash frac\{I\_\{s1\}\}\{I\_s\}\; =\; \backslash frac\{2\backslash sqrt2\}\{\backslash pi\}\backslash \; (1\backslash phi),\backslash \; \backslash frac\{3\}\{\backslash pi\}\backslash \; (3\backslash phi)$$

Rectifier line-current harmonic spectrum

$$I_h=\frac{I_1}{h},h=6k\pm 1(3\varphi ),h\text{ odd }(1\varphi )I\_h\; =\; \backslash frac\{I\_1\}\{h\},\backslash quad\; h\; =\; 6k\backslash pm1\backslash \; (3\backslash phi),\backslash \; h\backslash \; \backslash text\{odd\}\backslash \; (1\backslash phi)$$

Rectifiers / three-phase uncontrolled

Three-phase diode bridge rectifier with resistive load

$$V_{dc}=\frac{3\sqrt{2}{V}_{LL}}{\pi },V_{rms}=\sqrt{2}{V}_{LL}\sqrt{\frac{1}{2}+\frac{3\sqrt{3}}{4\pi }}V\_\{dc\}\; =\; \backslash frac\{3\backslash sqrt2\backslash ,V\_\{LL\}\}\{\backslash pi\},\backslash quad\; V\_\{rms\}\; =\; \backslash sqrt2\; V\_\{LL\}\backslash sqrt\{\backslash tfrac12+\backslash tfrac\{3\backslash sqrt3\}\{4\backslash pi\}\}$$

Three-phase half-wave (three-pulse) diode rectifier

$$V_{dc}=\frac{3\sqrt{3}{V}_m}{2\pi },V_{rms}=V_m\sqrt{\frac{1}{2}+\frac{3\sqrt{3}}{8\pi }},PIV=\sqrt{3}{V}_{m}V\_\{dc\}\; =\; \backslash frac\{3\backslash sqrt3\backslash ,V\_m\}\{2\backslash pi\},\backslash quad\; V\_\{rms\}\; =\; V\_m\backslash sqrt\{\backslash tfrac12+\backslash tfrac\{3\backslash sqrt3\}\{8\backslash pi\}\},\backslash quad\; PIV\; =\; \backslash sqrt3\; V\_m$$

Resonant converters / LLC

LLC resonant converter: tank gain (first harmonic)

$$M=\mid \frac{j\omega L_m\parallel R_{ac}}{j\omega L_r+\frac{1}{j\omega C_r}+j\omega L_m\parallel R_{ac}}\mid ,R_{ac}=\frac{8n^{2}R}{{\pi }^2}M\; =\; \backslash left\backslash vert\; \backslash frac\{j\backslash omega\; L\_m\backslash parallel\; R\_\{ac\}\}\{j\backslash omega\; L\_r\; +\; \backslash frac\{1\}\{j\backslash omega\; C\_r\}\; +\; j\backslash omega\; L\_m\backslash parallel\; R\_\{ac\}\}\backslash right\backslash vert\; ,\backslash quad\; R\_\{ac\}\; =\; \backslash frac\{8n^2R\}\{\backslash pi^2\}$$

Resonant converters / dual active bridge

Dual active bridge: power transfer and inductor current

$$P=\frac{n{V}_1{V}_2\varphi (\pi -\varphi )}{2{\pi }^2{f}_{s}L},P_{max}=\frac{n{V}_1{V}_2}{8f_{s}L}P\; =\; \backslash frac\{nV\_1V\_2\backslash ,\backslash phi(\backslash pi-\backslash phi)\}\{2\backslash pi^2f\_sL\},\backslash quad\; P\_\{max\}\; =\; \backslash frac\{nV\_1V\_2\}\{8f\_sL\}$$

Resonant converters / series resonant

Series resonant converter: tank gain (first harmonic)

$$M=\frac{1}{\sqrt{1+Q^2{(f_n-\frac{1}{f_n})}^2}},Q=\frac{\sqrt{L_r\mathrm{/}{C}_r}}{R}M\; =\; \backslash frac\{1\}\{\backslash sqrt\{1+Q^2\backslash left(f\_n-\backslash frac\{1\}\{f\_n\}\backslash right)^2\}\},\backslash quad\; Q\; =\; \backslash frac\{\backslash sqrt\{L\_r/C\_r\}\}\{R\}$$

Thermal / steady state

Several power devices on one heat sink

$$T_s=T_a+R_{sa}\sum _k{P}_k,T_{j,k}=T_s+P_k(R_{jc}+R_{cs})T\_s\; =\; T\_a\; +\; R\_\{sa\}\backslash sum\_kP\_k,\backslash quad\; T\_\{j,k\}\; =\; T\_s\; +\; P\_k(R\_\{jc\}+R\_\{cs\})$$

Thermal / transient

Junction temperature under periodic power pulses

$$\mathrm{\Delta }{T}_{pk}=\sum _{i}P{R}_i\frac{1-e^{-t_p\mathrm{/}{\tau }_i}}{1-e^{-T\mathrm{/}{\tau }_i}}\backslash Delta\; T\_\{pk\}\; =\; \backslash sum\_iPR\_i\backslash frac\{1-e^\{-t\_p/\backslash tau\_i\}\}\{1-e^\{-T/\backslash tau\_i\}\}$$

Transient thermal impedance of a Foster network

$$Z_{th}(t)=\sum _i{R}_i(1-e^{-t\mathrm{/}{\tau }_i}),\mathrm{\Delta }{T}_j=P{Z}_{th}(t_p)Z\_\{th\}(t)\; =\; \backslash sum\_i\; R\_i\backslash left(1-e^\{-t/\backslash tau\_i\}\backslash right),\backslash quad\; \backslash Delta\; T\_j\; =\; P\backslash ,Z\_\{th\}(t\_p)$$

Out of scope

• Open-ended explanation, essay writing or tutoring dialogue: it is a solver, not a chatbot.
• Questions outside its certified power electronics 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 (DC-DC converters, rectifiers, inverters and PWM, motor drives, device losses, thermal and magnetic design, resonant converters); 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{power_electronics_ai999x_1_0_0,
  title  = {power-electronics-ai999x: an offline, verified power electronics solver},
  author = {Farabi, Ibnul Jaif}, organization = {AI999X},
  year   = {2026}, version = {1.0.0},
  url    = {https://huggingface.co/AI999X/power-electronics-ai999x}
}

Contact

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

Support the project

power-electronics-ai999x is free and open source. If it helps you, you can support further development on Buy Me a Coffee.

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