Linear Wave Shaping in Pulse and Digital Circuits

Introduction

• Subject: Pulse and Digital Circuits
• Target Audience: B-Tech ECE/Triple branches (2nd year)
• Focus: Operation of pulse and digital circuits with various inputs (sinusoidal, ramp, pulse, exponential, step).

Key Concepts

Linear and Non-Linear Networks

• Linear Network

  • Consists of linear components: Resistance (R), Capacitance (C), Inductance (L)
  • Requires an active source (voltage/current source) to energize the circuit.
  • Examples: RC network, RL network, RLC network.

• Non-Linear Network

  • Contains non-linear components in addition to R, C, L (e.g., diodes, transistors).
  • Active sources are also present.

Linear Wave Shaping

• Definition: The process of changing the shape of a non-sinusoidal signal when passing through a linear network.
• Characteristics:
  • Sinusoidal Input: The output remains sinusoidal; no shape change.
  • Non-Sinusoidal Input: The output waveform is distorted/changed based on the input.

Linear Network Example

Circuit Description

• Components: Resistor (R) at input, Capacitor (C) at output.
• Input applied between resistor and ground; output taken across the capacitor.
• Sinusoidal Input: Output is sinusoidal, indicating it's a linear network.

Frequency Response of Low Pass RC Network

• Purpose: Pass low frequencies while attenuating higher frequencies.
• Frequency Response Characteristics:
  • Passband: Lower frequencies where signals are passed.
  • Stopband: Higher frequencies that are eliminated.
  • Cutoff Frequencies: F1 (low) and F2 (high).
  • Gain Response: Maximum response at 1, with 3dB down at the cutoff frequency.

Gain Calculation

Gain of RC Network

• Formula: Gain (A) = Output Voltage (Vout) / Input Voltage (Vin)
• Expression for Vout across the capacitor: 1/(Cs) * I(s)
• Expression for Vin: R + 1/(Cs) * I(s)

LaPlace Transform Network

• Convert circuit components into their LaPlace Transform equivalents:
  • Vout = Vnaught(s) = 1/(Cs)
  • Vin = Vinput(s) = R + 1/(Cs)

Gain Expression

• Gain formula: A = 1 / (1 + J2πfRC)
• At cutoff frequency (fh): Gain = 1 / √2
• Cutoff frequency (fh) derived as: fh = 1 / (2πRC)

Final Gain Equation

• Magnitude and phase angle relationships expressed through:
  • Mod A = 1 / √(1 + (F/Fh)²)
  • Angle θ = tan⁻¹(B/A) where B = F/Fh

Conclusion

• Understanding linear wave shaping is crucial for analyzing pulse and digital circuits, especially in the context of signal processing and filtering.