Ontological Layer: Local Linearization

Small-Signal Analysis

From Nonlinear Physics to Linear Mathematics

The Calculus of Approximations

Transistors are inherently nonlinear devices, governed by complex exponentials or quadratics. However, if we confine an AC signal to a very small region around a fixed DC operating point (the Q-point), we can approximate the curve with a straight line. This brilliant simplification replaces chaotic physics with elegant linear algebra, giving birth to the Equivalent Circuit Model.

NONLINEAR DEVICE → DC BIAS (Q-POINT) → SMALL PERTURBATION → LOCAL LINEARIZATION → HYBRID-π MODEL → AC EQUIVALENT CIRCUIT → IMPEDANCE & GAIN → AMPLIFIER DESIGN

The 91 Linear Nodes

Ontological Mapping: Realms 01 - 91

GROUP 01-08

Concept & Linearization

01 Small-Signal Concept
02 Large-Signal vs Small-Signal
04 Q-Point as Linearization Point
08 Differential Change (Calculus)

GROUP 09-17

Parameters & Gain

09 Small-Signal Parameters
12 Voltage & 13 Current Gain
15 Transconductance (gm)
10 Input / 11 Output Resistance

GROUP 18-27

BJT Models

18 BJT Small-Signal Model
19 Hybrid-π Model
20 T-Model & 21 Re Model
26 gm–IC Relationship & Early Effect

GROUP 28-35

FET Models

28 FET Small-Signal Model
31 MOSFET Transconductance (gm)
33 Channel-Length Modulation
34 Controlled-Source Representation

GROUP 36-40

Circuit Transformation

↓

36 Small-Signal Equivalent Circuit
37 AC Ground Concept
38 DC Source Suppression
39 Coupling / 40 Bypass Capacitors

GROUP 41-50

Frequency Domains

41 Midband Analysis
44 Miller Effect
47 Internal Device Capacitances
49 BJT/FET High-Frequency Models

GROUP 51-63

Two-Port & Loading

51 Hybrid (h, y, z) Parameters
52 Two-Port Small-Signal Model
61 Loading Effect (Source/Load)
62 Cascaded Stages & Gain Degradation

GROUP 64-69

Amplifier Topologies

64 Common-Emitter Analysis
65 Common-Base & 66 Common-Collector
67 Common-Source Analysis
68 Common-Gate & 69 Common-Drain

GROUP 70-75

Differential Concepts

70 Differential vs Common-Mode Input
73 Differential Gain (Ad)
74 Common-Mode Gain (Acm)
75 Common-Mode Rejection Ratio (CMRR)

GROUP 76-88

Noise & Stability

77 Thermal Noise & 78 Shot Noise
79 Equivalent Input Noise
81 Sensitivity to Parameter Variation
85 Model Selection & Accuracy

FINAL PROTOCOL

11 — The Small-Signal Laboratory

gm = ICQ / VT

rπ = β / gm

Av = -gm * (RC || RL)

[1] DC ANALYSIS (Q-POINT) [2] CALCULATE PARAMETERS [3] DRAW AC EQUIVALENT [4] EXTRACT GAIN/IMPEDANCE [5] BODE PLOT

Executing the algorithmic process of linearization. From calculating transconductance from DC bias currents, to replacing non-linear symbols with Hybrid-π models, to deriving the final voltage gain equations. The foundation for Realm 14: Amplifiers.

SIMULATION RUNNING