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Foundations & Laws

1. Basic Quantities​

Definitions

  • Voltage (V): Difference in potential. The "force" or "pressure" that moves charges.

    • Sign: Positive Vab means point a is higher potential than b.

  • Current (I): Rate of flow of charge (I = dQ / dt).

    • Direction: Conventional Flow (Positive to Negative). Electron flow is opposite, but we ignore that in design.

  • Conductance (G): Inverse of resistance (G = 1 / R).

    • Math: Measured in Siemens (S).
Conservation Laws

  • KCL (Kirchhoff's Current Law): The sum of currents entering and leaving a node is Zero.

    • Why: Charge conservation (matter isn't created/destroyed).

  • KVL (Kirchhoff's Voltage Law): The sum of voltage drops around any closed loop is Zero.

    • Why: Energy conservation (you can't gain energy by walking in a circle).

2. AC & Signals​

Why Sinusoids? We use sine waves (V = Asin(2πft)) because:

  1. It's what comes out of the wall (generators rotate).
  2. Linearity: A linear circuit with a sinusoidal input will always output a sinusoid (just scaled/shifted). This is unique.
AC Power Quantities

  • RMS (Root Mean Square): The "DC Equivalent" for power heating.

    • Sine: Vrms = Vpeak / √2 ≈ 0.707 Vpeak.

  • Power Types:

    • Average Power (P): Real work done. P = Vrms · Irms · cos(φ).
    • Apparent Power (S): Total energy moving. S = Vrms · Irms.
    • Power Factor (PF): Efficiency ratio. PF = P / S = cos(φ).
Decibels (dB)

A logarithmic scale to compare huge differences in magnitude.

Formulas

  • Power: dB = 10 · log10(P2 / P1)
  • Amplitude (Volts): dB = 20 · log10(A2 / A1)

Cheatsheet

  • +3 dB: 2x Power
  • +6 dB: 2x Voltage
  • -3 dB: Half Power
  • +20 dB: 10x Voltage

Phase: A time delay between signals.

  • "90° out of phase" means when one signal peaks, the other is crossing zero.

Phase: I leads V by 90°

VITime →

Current (Red) is already at its peak when Voltage (Yellow) is just starting at zero.

3. Circuit Theorems (Analysis Tools)​

These theorems allow us to simplify complex "black box" circuits into two components.

TheoremEquivalent ModelHow to Calculate
TheveninVoltage Source + Series ResistorVth: Open circuit voltage at the port.
Rth: Resistance looking into the port.
NortonCurrent Source + Parallel ResistorIn: Short circuit current at the port.
Rn: Same as Rth.

4. Impedance & Loading (Design Philosophy)​

This is the practical application of Thevenin. Every real-world source (Sensor, Battery, Output Pin) has an internal Source Resistance.

Simulation: Source Loading Effect

4.95 V
Actual Output Voltage
99.0% of voltage reaches load

The "Ideal" vs "Real"​

  1. Source Resistance (Rs): You want this LOW.

    • High Rs causes internal voltage drop when current is drawn.
    • Low Rs means the source is "stiff" (voltage doesn't sag under load).

  2. Load Resistance (RL): You usually want this HIGH.

    • "Bridging": High RL draws minimal current, preventing voltage sag on the source.
    • Example: Multimeters have 10MΩ input impedance so they don't disturb the circuit they measure.

Max Power Transfer​

If you care about efficiency (battery life), keep RL High. If you care about raw power (RF antenna, Audio amp), you match them:

  • DC: Rload = Rsource
  • AC: Zload = Zsource* (Complex Conjugate)

5. Frequency Domain​

Frequency Response How a circuit's gain/phase changes as frequency changes.

Transfer Functions (H(s))

  • Definition: The ratio of Output to Input in the frequency domain.
  • H(s) = Output(s) / Input(s)
  • Used to map Poles (instability) and Zeros (blocking).

6. Real World Noise & Parasitics​

Noise​

Random electrical fluctuations.

  • Thermal (Johnson): Noise generated by electrons bouncing around in a resistor. Depends on Temp and Resistance.
  • Shot Noise: Noise from electrons crossing a barrier (PN junction).
  • 1/f (Flicker): Low-frequency noise, dominant in MOSFETs/active devices.

Parasitics​

Every component is actually an R, L, and C combined.

  1. Everything is an Inductor: All wires/traces have length, therefore they have inductance.
    • Result: High frequency signals face higher impedance than expected.
  2. Everything is a Capacitor: Any two conductors near each other form a capacitor.
    • Result: PCB traces can cross-talk; Inductors have self-capacitance.
  3. Skin Effect: At high AC frequencies, current crowds to the outside edge of a wire, effectively reducing the cross-section and increasing Resistance.

7. Noise in Electronics​

Noise is random in nature and typically follows a Gaussian distribution. It is a fundamental limit to system resolution.

1. Noise Types​

Thermal Noise (Johnson Noise)​

Thermal noise is associated with the random motion of thermally excited electrons in a conductor. It is present in all resistive elements regardless of current flow.

  • Characteristics: It has a flat power spectral density (White Noise).
  • Dependency: Directly proportional to absolute temperature (Kelvin) and resistance.
  • Magnitude: At room temperature, a 1kΩ resistor has a noise voltage density of approximately 4 nV/√Hz.

Shot Noise​

Shot noise is associated with DC current flow across potential barriers, such as PN junctions in diodes and BJTs.

  • Mechanism: Caused by the discrete nature of charge carriers (electrons/holes) crossing the barrier.
  • Dependency: Proportional to the square root of the DC current.

1/f Noise (Flicker Noise)​

1/f noise is associated with DC current flow and is related to carrier traps and crystal imperfections in semiconductor devices.

  • Behavior: The noise energy is inversely proportional to frequency. As frequency decreases, the noise amplitude increases.
  • 1/f Corner: The frequency at which 1/f noise amplitude equals the broadband thermal noise. Below this frequency, 1/f noise dominates.

Popcorn Noise (Burst Noise)​

A low-frequency noise typically associated with heavy metal ion contamination or defects in silicon lattice. It appears as step-function shifts in voltage (random telegraph signals).

2. Noise Spectral Density (NSD)​

Noise is described statistically using Spectral Density, representing noise amplitude per unit of bandwidth.

  • Voltage NSD Units: nV/√Hz
  • Current NSD Units: fA/√Hz or pA/√Hz

Calculating Total RMS Noise​

To determine the total RMS noise voltage in a circuit, you must integrate the Noise Power Spectral Density curve over the frequency range of interest.

  • Concept: Total noise corresponds to the area under the density curve.
  • Bandwidth: A wider system bandwidth results in a larger area under the curve, increasing total noise.
  • Calculation: 1/f noise and broadband noise are estimated separately and combined using the square root of the sum of the squares (Root Sum Square):

    Total Noise = √( (Noise1/f)2 + (Noisebroadband)2 )

3. Amplifier Noise Model​

Amplifiers (constructed from transistors, resistors, and capacitors) have internal noise sources. These are modeled as equivalent Voltage Noise (en) and Current Noise (in) generators placed at the amplifier inputs.

Specifications​

Datasheets typically specify noise in two formats:

  1. Low Frequency (0.1 Hz to 10 Hz): Expressed as a Peak-to-Peak voltage (μVpp). This characterizes the 1/f noise contribution.
  2. Broadband (1 kHz or 10 kHz): Expressed as Spectral Density (nV/√Hz).

Current Noise Variation​

Current noise varies significantly by transistor technology:

  • JFET/CMOS Op-Amps: Very low current noise (fA/√Hz range).
  • Bipolar Op-Amps: Higher current noise (pA/√Hz range).

4. Power vs. Noise Relationship​

There is generally an inverse relationship between noise performance and power consumption.

  • Mechanism: In bipolar transistors, voltage noise is inversely proportional to the square root of the collector current (IC).
  • Trade-off: Reducing voltage noise requires increasing the bias current of the input transistors. Consequently, low-noise amplifiers typically consume more power.