ee

Core Circuit Laws

DC Analysis

V = IR                           Ohm's Law
P = VI = I²R = V²/R             Power
KVL: ΣV around loop = 0         Kirchhoff's Voltage Law
KCL: ΣI into node = 0           Kirchhoff's Current Law

Voltage Divider

Vout = Vin × R2 / (R1 + R2)
Rload effect: Vout_actual = Vin × (R2||Rload) / (R1 + R2||Rload)
For < 1% load error: Rload > 100 × R2

Thevenin / Norton

Vth = open-circuit voltage at terminals
Rth = resistance seen from terminals with all sources zeroed (V→short, I→open)
In  = Vth / Rth

Superposition

For linear circuits: activate one source at a time, zero others, sum results.

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Passive Components

Resistors

Parameter	Calculation	Notes
Power derate	P_rated × 0.5 at 70°C	Derate linearly to 0 at T_max
Noise (Johnson)	Vn = √(4kTRB)	k=1.38e-23, B=bandwidth
Tolerance effect	ΔVout/Vout = √(ΔR1²+ΔR2²) / (R1+R2)	Worst-case RSS

Standard E-series values: E12 (10% tol), E24 (5%), E48 (2%), E96 (1%), E192 (0.5%) SMD sizes: 0201, 0402 (¼W), 0603 (⅒W), 0805 (⅛W), 1206 (¼W), 2512 (1W)

Capacitors

Type	Voltage coeff	Temp coeff	Use case
C0G/NP0	None	±30 ppm/°C	Timing, RF, precision
X7R	Moderate (−80% at rated V)	±15% (-55→125°C)	Decoupling
X5R	Higher	±15% (-55→85°C)	Bulk, lower-cost decoupling
Y5V	Severe (−82% at rated V)	+22/−82%	Avoid for power
Electrolytic	Low	Varies	Bulk, low-freq only
Tantalum	Low	Stable	Bulk; high inrush risk

Derating rule: Use caps at ≤ 50% rated voltage for X7R (capacitance drops ~20% at 50%). Check datasheet derating curves.

Self-resonant frequency (SRF): Above SRF, cap is inductive. Rule of thumb: 0402 MLCC SRF ≈ 200–600 MHz, 0201 ≈ 1–3 GHz.

Decoupling placement: Place closest cap to IC power pin first. Cascade: bulk (10–100 µF) + mid (1–10 µF) + HF (100 nF) + ultra-HF (10 nF). Minimize loop area.

Inductors

V = L × dI/dt
Isat: current at which inductance drops 20–30%
Irms: continuous current at rated temperature rise
Q = ωL / R_dc                   Quality factor
SRF: above this, acts capacitive

DCR power loss: P = I² × DCR. Key spec for power inductors. Saturation: Never exceed Isat. Size to Ipeak × 1.3 minimum margin.

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RC / LC Circuits

RC Low-Pass Filter

fc = 1 / (2π × R × C)           Cutoff frequency (-3 dB)
Attenuation at f: A = 1 / √(1 + (f/fc)²)
Phase shift:      φ = -arctan(f/fc)

RC High-Pass Filter

fc = 1 / (2π × R × C)
A = (f/fc) / √(1 + (f/fc)²)

LC Resonant Circuit

f0 = 1 / (2π × √(L × C))        Resonant frequency
Q  = (1/R) × √(L/C)             Series resonance
BW = f0 / Q                     Bandwidth at -3 dB
Z  = √(L/C)                     Characteristic impedance

π / T filter (EMC)

π: cap–inductor–cap (low impedance source/load) T: inductor–cap–inductor (high impedance source/load)

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Op-Amps

Ideal Op-Amp Rules

1. V+ = V− (virtual short)
2. Input current = 0

Common Configurations

Config	Gain	Formula
Inverting	−Rf/Rin	Vout = -(Rf/Rin) × Vin
Non-inverting	1 + Rf/Rin	Vout = (1 + Rf/Rin) × Vin
Voltage follower	1	Vout = Vin
Differential	Rf/Rin	Vout = (Rf/Rin)(V+ − V−)
Integrator	−1/(RC×s)	Vout = −(1/RC)∫Vin dt
Differentiator	−RC×s	Vout = −RC × dVin/dt

Key Specs

• GBW (gain-bandwidth product): Gain × BW = constant. Av=10 → BW = GBW/10.
• Slew rate: Maximum dVout/dt. Limits large-signal bandwidth: fmax = SR / (2π × Vpeak).
• Input offset voltage (Vos): DC error. Total output offset = Vos × (1 + Rf/Rin).
• CMRR: Common-mode rejection. Target > 80 dB for precision.
• PSRR: Power supply rejection. Decouple op-amp supplies with 100 nF close.

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Power Supply Design

LDO Linear Regulator

Vout = Vref × (1 + R1/R2)       Adjustable output
Pdiss = (Vin - Vout) × Iout     Power dissipation (heat!)
η = Vout / Vin                  Efficiency (poor for large dropout)

When to use LDO: Low noise, small dropout (< 0.5V), < 500 mA, noise-sensitive analog/RF. Min dropout voltage: Vin ≥ Vout + Vdropout (typically 100–300 mV for modern LDOs). Thermal check: θJA × Pdiss < Tj_max − Tambient. Use exposed pad or heatsink if > 1W.

Buck Converter (Step-Down)

D = Vout / Vin                  Duty cycle (ideal, continuous mode)
ΔIL = (Vin - Vout) × D / (L × fsw)    Inductor ripple current
ΔVout = ΔIL / (8 × C × fsw)    Output voltage ripple
Lmin = (Vin - Vout) × D / (2 × Iout × fsw)   Min L for CCM

Component selection:

• L: Isat > Iout + ΔIL/2. L value for 20–40% ripple ratio.
• Cin: rated for Vin, low ESR. Irms_cin = Iout × √(D(1-D)).
• Cout: C > ΔIL / (8 × fsw × ΔVout_spec). ESR < ΔVout / ΔIL.

Layout rules: Short, fat traces on switching node. Input cap right at Vin pin. GND plane under switcher. Keep Lx node away from feedback resistors.

Boost Converter (Step-Up)

D = 1 - Vin/Vout               Duty cycle
ΔIL = Vin × D / (L × fsw)      Inductor ripple
Isat_req = Iout/(1-D) + ΔIL/2  Peak inductor current

Power Budget Template

Rail	Voltage	Current	Power
+3.3V_IO	3.3V	xxx mA	xxx mW
+1.8V_DDR	1.8V	xxx mA	xxx mW
+1.0V_CORE	1.0V	xxx mA	xxx mW
Total			xxx mW

Add 20% margin for thermal and headroom.

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Transistors

BJT

IC = β × IB                    Collector current
VCE_sat ≈ 0.2V (ON), VBE ≈ 0.7V
IB_req = IC / (β × 0.1)       Force saturation: overdrive 10×
Pdiss = VCE × IC (linear) or VCEsat × IC (switch)

Check: IC < IC_max, VCE < VCEO, Pdiss < Pd_max.

MOSFET

ID = (k/2)(VGS - Vth)²         Saturation
VGS > Vth + safety margin       Fully enhanced
Rds(on) varies with VGS and Tj: derate 2× from datasheet at 125°C vs 25°C
Pdiss (switch) ≈ ID² × Rds(on) + Qg × VGS × fsw

Gate drive: Sufficient VGS for low Rds(on). Drive impedance limits switching speed → EMI trade-off. Body diode: Always present; check reverse recovery for high-side switches.

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Signal Integrity

Transmission Lines

Z0 = √(L/C)                    Characteristic impedance
v  = 1/√(LC) = c/√(εr_eff)     Propagation velocity
λ  = v/f                       Wavelength

Rule of thumb: Treat trace as transmission line when length > λ/10 at the signal's knee frequency (≈ 0.35/tr for digital).

Microstrip (PCB, trace over ground plane):

Z0 ≈ (87/√(εr+1.41)) × ln(5.98H / (0.8W + T))
εr_eff ≈ (εr+1)/2 + (εr-1)/2 × (1+12H/W)^(-0.5)

• H = height to ground plane, W = trace width, T = trace thickness
• FR4: εr ≈ 4.0–4.5 (use 4.2 at 1 GHz), εr_eff ≈ 3.0

Stripline (buried trace between planes): Fully enclosed, εr_eff = εr, no dispersion. Use for tight impedance control.

Termination:

• Series: R = Z0, at source. Eliminates reflections at load (point-to-point).
• Parallel: R = Z0 to GND, at load. Eliminates reflections at source (multi-drop).
• AC: cap in series with R. DC-blocking parallel termination.

Return Paths

Signal current returns via lowest impedance path — not the shortest ground path. At high frequency, this is directly beneath the signal trace (the image current in the reference plane).

Rules:

• Never split ground plane under a high-speed signal. Splits force current around the gap → loop antenna.
• Cross splits only through bypass caps bridging the split.
• Via stitching closes return path at layer transitions.

Crosstalk

NEXT (near-end) ≈ (Cm/C0 + Lm/L0) / 4
FEXT (far-end)  ≈ (Cm/C0 - Lm/L0) / 4 × TD

Reduce crosstalk: Increase trace spacing (3W rule: spacing ≥ 3× trace width), reduce parallel run length, use ground guard traces, use differential pairs.

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RF Design

dB Reference Table

Power ratio	dB
2×	+3 dB
10×	+10 dB
0.5×	−3 dB
0.1×	−10 dB

dBm: Power relative to 1 mW. 0 dBm = 1 mW, +30 dBm = 1 W. dBW: Relative to 1 W. 0 dBW = +30 dBm.

RF Chain Budget

Pout = Pin + Gain − Losses
NF_total = NF1 + (NF2-1)/G1 + (NF3-1)/(G1×G2) + ...   (Friis formula)
IP3_total: 1/IP3_in = 1/IP3_1 + G1/IP3_2 + G1G2/IP3_3 ...

Sensitivity: Sens = kTB + NF + SNRmin = −174 + 10log(BW) + NF + SNRmin [dBm]

S-Parameters

Parameter	Meaning
S11	Input reflection (return loss). Good: < −10 dB
S21	Forward gain (or insertion loss if passive)
S22	Output reflection
S12	Reverse isolation

Return loss: RL = −20 log|Γ|. VSWR = (1+|Γ|)/(1−|Γ|). Insertion loss: IL = −20 log|S21| for a 2-port.

Impedance Matching (L-network)

Given Rsource → Rload (both real, Rsource > Rload):

Q = √(Rsource/Rload - 1)
Xs (series element) = Q × Rload
Xp (shunt element) = Rsource / Q

BW ≈ f0/Q. Use π or T networks for narrower BW.

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Thermal Design

Heat Flow

Tj = Ta + Pdiss × (θJC + θCS + θSA)
θJA = θJC + θCS + θSA          Junction-to-ambient total

• θJC: Junction-to-case (datasheet)
• θCS: Case-to-sink (thermal interface material — TIM)
• θSA: Sink-to-ambient (heatsink spec, depends on airflow)
• Ta: Ambient temperature

Copper area as heatsink: 1 in² of 1 oz copper ≈ 50–70°C/W (still air). Doubles with 2 oz copper.

Thermal via: Each via ≈ 3–10°C/W. Use arrays under exposed pads (QFN, BGA). Guideline: 1 via per 100 mW for QFN.

Derate components: At T > 25°C, many parameters degrade. Check derating curves: Rds(on) of MOSFETs typically doubles 25→125°C.

Junction Temp Check

Tj_max (datasheet) — Tj_operating ≥ 10°C margin
Tj = Ta + Pdiss × θJA

If Tj > limit: reduce Pdiss, increase copper area, add heatsink, improve airflow, choose lower Rds(on) part.

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EMC

Emission Reduction

Common-mode filter: Series CM choke + shunt caps (π filter) on cable exits. Differential-mode filter: LC filter on power lines. Shielding: Enclosure or shielded connector. Ground the shield at one point (low-freq) or both (high-freq > 1 MHz).

Layout Rules for EMC

1. Minimize loop areas — current loops are antennas. Keep signal and return traces close.
2. Solid ground plane — no splits under switching circuits or clock lines.
3. Separate grounds — AGND and DGND joined at single star point (or solid plane with careful routing).
4. Decoupling every IC — 100 nF + bulk cap, right at VCC pins, shortest possible trace.
5. Clock/oscillator — keep under metal (internal layer or add copper pour), surround with GND vias.
6. High-current loops first — SMPS switching loop, gate drive loop. Minimize physically.

Common Failure Modes

Symptom	Likely cause
Oscillation in amplifier	Parasitic feedback, missing decoupling
SMPS noise on analog rail	Insufficient filtering, layout ground loop
Erratic digital behavior	Ground bounce, inadequate bulk caps
ESD latchup	Missing ESD diodes on I/O, wrong ground return
EMC emission at clock frequency	Clock harmonics, inadequate shielding

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Protection Circuits

ESD Protection

• TVS diode: Clamp voltage, bidirectional or unidirectional. Select Vclamp < IC's abs max.
• Rail-to-rail TVS: One device per supply rail.
• Line protection: Series R (33–100 Ω) + TVS to GND. Limits ESD current into IC.

Overcurrent Protection

Ifuse = Imax_load × 1.5        Fuse rating (with 50% margin)
Rsense = Vsense / Ilimit       Current sense resistor (Vsense typically 50–100 mV)

Polyfuse (PPTC): Self-resetting. Trips when Joule heating exceeds threshold. Slow — not for fast faults. Ideal diode / load switch: MOSFET-based, fast, no voltage drop.

Reverse Polarity

• Series diode (Schottky): Simple, 0.3–0.5V drop.
• P-channel MOSFET: Near-zero drop, controlled by gate. Source to input+, drain to load+, gate through R to GND, TVS gate-source.

Overvoltage

• Clamp: TVS or Zener in parallel with load.
• Crowbar (SCR): Fires on OV event, blows fuse. Latching — requires power cycle.
• Ideal OVP: Comparator + MOSFET series switch. Non-latching.

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Test & Measurement

Oscilloscope Setup

Setting	Rule of Thumb
Bandwidth	≥ 5× signal bandwidth (≥ 3.5× for digital: 0.35/tr)
Sample rate	≥ 5× signal bandwidth
Probe compensation	Square-wave comp at 1 kHz before measuring
Ground clip	Shortest possible — loop is antenna
Probe loading	10 MΩ ‖ 10 pF at 1× → use 10× probe (10 MΩ ‖ 1 pF) for fast signals

Measure power supply noise: AC-couple, 20 MHz BW limit, 100 mV/div. Short probe ground.

DMM Tips

• Resistance: Power off, discharge caps, avoid measuring in-circuit (parallel paths).
• Diode test: 0.3–0.5V = Schottky/Ge, 0.6–0.7V = Si, OL = open, ~0 = short/zener-in-circuit.
• Continuity: Not reliable for detecting shared return paths (other paths sink current).

Spectrum Analyzer / Tinker SA

• Resolution bandwidth (RBW): narrower → slower sweep, better sensitivity.
• Reference level: Set 10 dB above expected signal.
• Span: Start wide, then zoom in.
• Input protection: Know your max input power. +10 dBm (10 mW) is common; check before connecting.

Calibration / Null Measurements

• Use 4-wire (Kelvin) sensing for resistance < 10 Ω to eliminate lead resistance.
• Thermal EMF (Seebeck effect) corrupts µV-level DC measurements. Use DC reversal method.
• Lock-in amplifier: Detect signals buried in noise; phase-lock to known reference.

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Component Selection Checklist

For every component in a new design:

• MPN specified (no generic "10k 0402")
• Package confirmed against footprint (SOT-23-3 vs SOT-23-5 etc.)
• Voltage rated at ≥ 2× nominal (caps), or Vds/Vce > max circuit voltage
• Current rated at ≥ 1.5× max operating current
• Temperature range covers operating range (-40→+85°C industrial, -40→+125°C automotive)
• Lifecycle — not obsolete/NRND; check DigiKey/Mouser lifecycle status
• Lead time and stock verified at target quantity
• Datasheet read — verify application circuit, decoupling, Abs Max ratings
• Datasheet pinout confirmed against KiCad symbol (especially SOT-23 BJTs/MOSFETs)

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Quick Reference — Standard Values

Resistor Values (E24 common subset)

1.0, 1.1, 1.2, 1.3, 1.5, 1.6, 1.8, 2.0, 2.2, 2.4, 2.7, 3.0, 3.3, 3.6, 3.9, 4.3, 4.7, 5.1, 5.6, 6.2, 6.8, 7.5, 8.2, 9.1 (× 10^n)

Capacitor Common Values

1, 1.5, 2.2, 3.3, 4.7, 10, 22, 47, 100 nF; 1, 2.2, 4.7, 10, 22, 47, 100 µF

Typical I²C Pull-Up Values

• 3.3V, 400 kHz (fast-mode): 2.2 kΩ – 4.7 kΩ
• 3.3V, 100 kHz (standard): 4.7 kΩ – 10 kΩ
• 1.8V, 400 kHz: 1 kΩ – 2.2 kΩ

Crystal Load Capacitors

CL_ext = 2 × CL_spec − Cstray     (Cstray ≈ 3–5 pF)
Typical: 12 pF spec → 18–22 pF external caps

USB Signal Integrity

• USB 2.0 FS/HS differential impedance: 90 Ω ± 15%
• USB 3.x differential impedance: 85 Ω ± 15%
• USB 3.x max length: 1m (channel loss < 8 dB at Nyquist)

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Integration with KiCad Skills

This skill feeds the rest of the EDA workflow:

Calculation	→ Use in
Voltage divider for VREF	circuit-weaver validate design.yaml --enhanced: verify feedback resistors
LDO dropout check	circuit-weaver validate design.yaml --enhanced: verify rail headroom
Inductor current ripple	bom: confirm Isat rating from DigiKey
Crystal load caps	circuit-weaver validate design.yaml --enhanced: verify Cload in schematic
I²C pull-up values	analyze_schematic.py bus detection output
Signal trace impedance	analyze_pcb.py trace width + stackup
Thermal check	circuit-weaver validate design.yaml --enhanced: flag missing thermal evidence
EMC filter values	circuit-weaver simulate design.yaml where supported; external RF analysis otherwise

Platform Guidance

• Claude Code: Present calculation results inline with units. Use AskUserQuestion when multiple approaches exist.
• Codex/OpenCode: Present formulas and results as text. Ask user for missing parameters.
• CLI: Reference calculations available via the ee skill. No dedicated CLI subcommand.