Ohm’s Law

• Ohm’s Law is the most fundamental relationship in electronics. It tells us how voltage, current, and resistance interact. In embedded systems, this governs everything from GPIO pin safety to sensor interfacing. If you know two values, you can calculate the third.
  Base Equation – V=I * R
  where R: Resistance (Ohms), V: Voltage across a resistor (Volts), I: Current through resistor (Amps).
  Use Cases :
  -> LED Current Limiting
  MCU GPIO pins can only source a safe current (e.g., 20 mA).
  Example: GPIO = 3.3V, LED drop = 2V, desired I=10mA.
  R = V / I = 3.3 – 2 / 0.01 = 130 Ohms
  Choose 150 Ohms resistor for safety.
  -> voltage Divider for ADC
  Safe 12v Battery to 3.3v ADC
  Vout = Vin * R2 / R1 + R2
  With R_1=27k, R_2=10k
  Vout = 12 * 10 / 37 = 3.24v
  Safe for ADC input.

Kirchhoff’s Current Law (KCL)

• Kirchhoff’s Current Law states that the total current entering a node equals the total current leaving it. This ensures charge conservation in circuits and is critical for PCB design and embedded power distribution.
  Base Equation – ΣI_in = ΣI_ou
  Where: I_in = currents entering a node, I_out = currents leaving a node.
  Use Cases:
  -> Power Distribution on PCB
  If a regulator supplies 500 mA to a node, and three sensors draw 100 mA, 150 mA, and 200 mA, then:
  ΣI_in = 500 mA, ΣI_out = 100 + 150 + 200 = 450 mA.
  Remaining 50 mA goes to MCU and decoupling capacitors.
  -> Ground Bounce in ADC
  When multiple digital pins switch simultaneously, return currents flow through ground. KCL ensures the sum of currents equals supply current, helping design proper ground planes.

Kirchhoff’s Voltage Law (KVL)

• Kirchhoff’s Voltage Law states that the sum of voltage drops around a closed loop equals the applied voltage. This is essential for analyzing embedded power rails and battery systems.
  Base Equation – ΣV_loop = 0
  Where: V_loop = sum of voltages around a closed path.
  Use Cases:
  -> Battery Internal Resistance
  Battery = 12V, internal resistance = 0.5Ω, load current = 2A.
  Voltage drop = I × R = 2 × 0.5 = 1V.
  Loop equation: 12V – 1V – 11V(load) = 0.
  -> Decoupling Capacitors
  Ensures voltage drops across traces + capacitors balance supply voltage, preventing MCU brown-outs.

Thevenin’s Theorem

• Thevenin’s Theorem reduces any linear circuit to a single voltage source and series resistance. This simplifies sensor and ADC interface design.
  Base Equation – V_th = V_open, R_th = V_open / I_short
  Where: V_th = Thevenin voltage, R_th = Thevenin resistance.
  Use Cases:
  -> Sensor Modeling
  Sensor open-circuit voltage = 2.5V, short-circuit current = 5 mA.
  R_th = 2.5 / 0.005 = 500Ω.
  ADC input must have impedance >> 500Ω to avoid loading error.
  -> Button Networks
  Pull-up resistor + switch modeled as Thevenin source to predict voltage levels at MCU pin.

Superposition Principle

• Superposition states that in linear circuits, the total response equals the sum of responses from each source independently.
  Base Equation – V_total = V1 + V2 + … (linear systems)
  Where: V_total = combined voltage response, V1, V2 = individual contributions from each source.
  Use Cases:
  -> Noise Budgeting
  Regulator noise = 10 mV RMS, MCU clock noise = 15 mV RMS.
  Total noise = √(10² + 15²) = 18 mV RMS.
  -> Mixed-Signal ADC Input
  If DAC injects 1V and interference adds 0.2V, total = 1.2V.

Faraday’s Law of Induction

• Faraday’s Law states that a changing magnetic flux induces voltage. This governs motors, relays, and wireless charging in embedded systems.
  Base Equation – V = –N × dΦ/dt
  Where: N = coil turns, Φ = magnetic flux.
  Use Cases:
  -> Relay Kickback
  Coil inductance = 10 mH, current change = 0.5A in 1 ms.
  V = –N × dΦ/dt ≈ –L × di/dt = –0.01 × 0.5/0.001 = –5V spike.
  Flyback diode clamps this safely.
  -> Motor Back-EMF
  Motor generates voltage proportional to speed, sensed by MCU for control.

Coulomb’s Law / Capacitance

• Coulomb’s Law describes charge interaction; capacitance defines charge storage.
  Base Equation – F = k × q1 × q2 / r², C = Q / V
  Where: F = force, q = charge, r = distance, C = capacitance.
  Use Cases:
  -> Capacitive Touch
  Finger changes capacitance from 20 pF to 40 pF. MCU detects ΔC via charge-transfer method.
  -> RC Timing
  R = 10kΩ, C = 100nF → τ = R × C = 1 ms.
  GPIO toggling + capacitor creates debounce delay.

Fourier Theorem

• Fourier’s Theorem states any signal can be decomposed into sinusoidal components.
  Base Equation – x(t) = Σ X_n e^(j2πnf₀t)
  Where: x(t) = time‑domain signal, X_n = Fourier coefficients (amplitude of each frequency component), f_0 = fundamental frequency
  Use Cases:
  -> PWM Spectrum
  PWM at 1 kHz produces harmonics at multiples of 1 kHz. LC filter removes higher harmonics.
  -> DSP FFT
  MCU samples audio at 8 kHz, FFT shows peaks at 1 kHz and 2 kHz tones.

Shannon’s Channel Capacity

• Shannon’s theorem defines maximum data rate over a noisy channel.
  Base Equation – C = B × log₂(1 + S/N)
  Where: B = bandwidth, S/N = signal-to-noise ratio.
  Use Cases:
  -> UART Throughput
  Cable bandwidth = 1 MHz, S/N = 20 dB (≈100).
  C = 1e6 × log₂(1+100) ≈ 6.6 Mbps.
  Safe UART rate = 1 Mbps.
  -> Wireless Links
  RSSI and noise floor determine usable data rate for BLE/Wi-Fi.

Maximum Power Transfer Theorem

• Maximum power is delivered when load resistance equals source resistance.
  Base Equation – R_L = R_S
  Where: R_L = Load resistance, R_S = Source resistance.
  Use Cases:
  -> Sensor Excitation
  Source resistance = 1kΩ, load matched at 1kΩ → maximum signal power to ADC.
  -> RF Antenna Matching
  Transceiver output impedance = 50Ω, antenna tuned to 50Ω → maximum RF power transfer.