Docs on Dark Current Labs

MOSFET Gate Threshold Voltage Temperature Dependence

Tue, 21 Jul 2026 00:00:00 +0000

$$ AN-806 $$$$ Tyler Mesko $$

MOSFET Threshold Voltage: Physical Derivation and Temperature Dependence
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Author: Tyler Mesko

1. Setup
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Derive $V_{TH}$’s dependence on temperature $T$, asserting $V_{SB}=0$ initially. Assuming fixed $V_{GS}$ and $V_{SB}$, the gate structure amounts to a single reverse-biased PN junction under an applied gate voltage $V_{GS}$, in which the negative charge in the p-type substrate is facilitated by the depletion region.

Subthreshold MOSFET gm/Id Transconductance Efficiency

Sun, 19 Jul 2026 00:00:00 +0000

$$ AN-810 $$$$ Tyler Mesko $$

1. Setup and Key Relations
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For an NMOS device, the basic (strong-inversion) drain current relation and overdrive voltage are:

$$I_D = K\left(\frac{W}{L}\right)\left(V_{OV}\right)^2, \qquad K = \mu_n C_{ox}, \qquad V_{OV} = V_{GS}-V_{TH}$$

$V_{TH}$ itself depends on bandgap energy, the body-effect coefficient, and the Fermi potential — its full physical derivation (including temperature dependence) is treated separately. For this document, the relevant threshold-voltage parameters are:

Control Loop Basics

Thu, 03 Apr 2025 00:00:00 +0000

$$ AN-401 $$$$ Tyler Mesko $$

1. The Basic Feedback Loop
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The simplest control loop can be conceptualized as a desired output value relating to some input value (setpoint) through both a feedforward gain path and a negative feedback gain path:

Where:

$V_i$ = input signal
$V_e$ = error signal
$A$ = feedforward (open-loop) gain
$\beta$ = feedback gain
$V_o$ = output signal
$V_f$ = feedback signal

Deriving the Closed-Loop Gain
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We will now derive the expression for the closed loop gain

Johnson Noise

Mon, 17 Mar 2025 00:00:00 +0000

$$ AN-701 $$

$$ Tyler Mesko $$

Introduction
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Johnson noise was first discovered by John Bertrand Johnson at Bell Labs in 1927 when he was investigating the causes of background noise in vacuum tube amplifiers. He set up several experiments to try to isolate the source of the noise, and discovered that the noise was due to random voltage fluctuations on conductors. He noticed that it was independent of frequency (today known as white noise), and it was directly correlated with temperature. In short order, Harry Nyquist provided a theoretical solution to the problem in 1928 in his paper titled “Thermal Agitation of Electric Charges in Conductors”. It is the result of this paper that we are going to set up from first principles to show how absolutely insane of a feat this accomplishment really was.

Miller's Theorem

Sat, 08 Mar 2025 00:00:00 +0000

$$ AN-805 $$$$ Tyler Mesko $$

Introduction
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Say you have two voltage nodes, $X$ and $Y$, connected by some floating impedance $Z_{xy}$ (with other branches/circuitry hanging off each node) as shown below:

The current through impedance $Z_{xy}$ is

$$ i_{xy} = \frac{V_x - V_y}{Z_{xy}} $$

This floating impedance makes solving the circuit difficult — but Miller’s theorem gives us a way out. $Z_{xy}$ can be replaced by two impedances, each shunted from its respective node ($X$ or $Y$) illustrated below:

Crystal Lattices and Solid-State Fundamentals

Mon, 12 Aug 2024 00:00:00 +0000

$$ AN-801 $$$$ Tyler Mesko $$

Introduction
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Solid-state physics is the branch of condensed matter physics that investigates how macroscopic properties of solid materials like conductivity, magnetism, optical, thermal, ect. emerge from the microscopic quantum-mechanical interactions of insanely large numbers of atoms. Because analyzing each individual atom is impossible to do classically, it is more convenient to represent these groups of atoms as systems where these atoms pack densely together, typically organizing into regular, repeating periodic structures that we can analyze more easily. Semiconductors are a subset of the broader Solid-state physics pie (groups II through VI on the periodic table) that all can be combined in novel ways to create specific crystal lattices that have favorable properties, allowing us to construct useful electronic devices (e.g. diodes and transistors).

BJT Basics

Sat, 03 Feb 2024 00:00:00 +0000

$$ AN-803 $$$$ Tyler Mesko $$

Four Fundamental Single BJT Configurations
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1. Common-Emitter Amplifier
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Basic characteristics
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Incomplete - Need to clean up and add diagrams / re-add all derivation steps

DC bias: $V_{DD} \to R_C \to$ collector; base driven through source resistance $R_S$; emitter grounded.

Docs on Dark Current Labs

MOSFET Gate Threshold Voltage Temperature Dependence

MOSFET Threshold Voltage: Physical Derivation and Temperature Dependence #

1. Setup #

Subthreshold MOSFET gm/Id Transconductance Efficiency

1. Setup and Key Relations #

Control Loop Basics

1. The Basic Feedback Loop #

Deriving the Closed-Loop Gain #

Johnson Noise

Introduction #

Miller's Theorem

Introduction #

Crystal Lattices and Solid-State Fundamentals

Introduction #

BJT Basics

Four Fundamental Single BJT Configurations #

1. Common-Emitter Amplifier #

Basic characteristics #

MOSFET Threshold Voltage: Physical Derivation and Temperature Dependence
#

1. Setup
#

1. Setup and Key Relations
#

1. The Basic Feedback Loop
#

Deriving the Closed-Loop Gain
#

Introduction
#

Introduction
#

Introduction
#

Four Fundamental Single BJT Configurations
#

1. Common-Emitter Amplifier
#

Basic characteristics
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