Calculate RL Differentiator

Calculator and formulas for calculating an RL differentiator circuit

RL Differentiator Calculator

RL Differentiator Circuit

With this function, the properties of a differentiator circuit made from a resistor and an inductor can be calculated. The function calculates the inductor, resistance, or period/frequency.

Results
Resistance:
Inductance:
Frequency:
Period:
Time constant:

Circuit Diagram & Signals

Input signal

T = Period, t1 = Pulse

RL Differentiator
Operation

The differentiator functions as a pulse shaper stage. The RL circuit generates pulse-like AC voltage at the output from a square wave voltage at the input.

Time Constant
\[τ = \frac{L}{R}\]

τ (Tau) determines the time behavior of the differentiator.

Example Calculations

Practical Calculation Examples

Example 1: Audio Pulse Generator

Given: L = 10 mH, R = 100 Ω, Pulse width t1 = 1 ms

\[τ = \frac{L}{R} = \frac{10 \times 10^{-3}}{100} = 100 \text{ µs}\]
\[\frac{t1}{τ} = \frac{1 \text{ ms}}{100 \text{ µs}} = 10\]
Optimal ratio for differentiation
Sharp pulses at output
Example 2: Digital Edge Detection

Given: L = 1 µH, Pulse width t1 = 100 ns, τ = 20 ns

\[R = \frac{L}{τ} = \frac{1 \times 10^{-6}}{20 \times 10^{-9}} = 50 \text{ Ω}\]
\[\frac{t1}{τ} = \frac{100 \text{ ns}}{20 \text{ ns}} = 5\]
Good edge detection
For TTL/CMOS logic
Example 3: RF Pulse Generator

Given: R = 50 Ω, t1 = 10 ns, τ = 2 ns

\[L = τ \cdot R = 2 \times 10^{-9} \cdot 50 = 100 \text{ nH}\]
\[\frac{t1}{τ} = \frac{10 \text{ ns}}{2 \text{ ns}} = 5\]
Short RF pulses
Radar/Radio applications
Optimal Ratios
Pulse length to time constant:
t1 = 5τ: Good differentiation
t1 = 10τ: Optimal differentiation
t1 > 10τ: Short pulses
t1 < 5τ: Distortion
Typical Applications:
Audio: 10 µH - 10 mH
Digital: 100 nH - 10 µH
RF: 10 nH - 1 µH
Pulse radar: 1 - 100 nH

RL Differentiator - Theory and Formulas

What is an RL Differentiator?

An RL differentiator is a pulse shaping circuit that generates short, needle-shaped pulses from square wave pulses. The output signal approximately corresponds to the mathematical derivative of the input signal. The time constant τ = L/R determines the properties of the pulse shaping.

Calculation Formulas

Time Constant and Basic Formulas
Time Constant
\[τ = \frac{L}{R}\]

Determines the time behavior of the circuit

Calculate Resistance
\[R = \frac{L}{τ}\]

Resistance for desired time constant

Calculate Inductance
\[L = τ \cdot R\]

Inductance for desired time constant

Transfer Function
\[H(s) = \frac{sL}{R + sL}\]

Laplace transfer function

Pulse Ratios for Optimal Differentiation
5τ Rule
\[t1 = 5τ = \frac{5L}{R}\]

Basic rule for good differentiation

10τ Rule
\[t1 = 10τ = \frac{10L}{R}\]

Optimal differentiation with short pulses

Universal Formula
\[R = \frac{n \cdot L}{t1} \text{ with } n = 5...10\]

n = 5 for basic, n = 10 for optimal differentiation

Signal Behavior and Output Characteristics
Pulse Width (Output)
\[t_{out} ≈ 2.2τ = \frac{2.2L}{R}\]

Width of output needle pulses

Amplitude (relative)
\[U_{out} ≈ U_{in} \cdot \frac{τ}{t1}\]

Height of output needle pulses

Pulse Behavior
  • Positive edge: Positive output needle pulse
  • Negative edge: Negative output needle pulse
  • Constant voltage: No output signal
  • Pulse duration: About 2-3 time constants
Design Rules and Optimization
Inductor Selection
  • Low frequencies: mH range
  • Audio/Digital: µH range
  • RF/Pulse: nH range
  • Quality: Use highest possible Q
Resistor Selection
  • Impedance matching: Match to source/load
  • Bandwidth: Smaller R = larger bandwidth
  • Noise: Compromise between R and bandwidth
  • Power rating: Sufficient power handling

Practical Applications

Signal Processing:
• Edge detection
• Pulse shaping
• Clock conditioning
• Trigger generation
RF Technology:
• Radar pulses
• Time domain reflectometry
• Impulse UWB
• Fast rise time generators
Digital Technology:
• Clock edge detection
• Reset pulses
• Watchdog timer
• Interrupt generation

Design Guidelines

Important Design Aspects
  • Time constant: τ should be 5-10 times smaller than pulse width
  • Bandwidth: f₃dB ≈ 1/(2πτ) - limits maximum operating frequency
  • Impedance matching: R should match system impedance
  • Parasitic effects: Consider inductor self-capacitance at high frequencies
  • Saturation: Inductor core must not saturate at high currents
  • Temperature: Consider temperature drift of L and R

Signal Behavior

Characteristic Properties
  • Input signal: Square wave pulses with defined edges
  • Output signal: Short needle pulses at each edge
  • Positive edge: Positive output needle pulse
  • Negative edge: Negative output needle pulse
  • Pulse duration: About 2.2 time constants (10%-90% criterion)
  • Amplitude: Depends on τ/t₁ ratio

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