Substrate Wavelength Calculator

Wavelength in Substrate Formula:

\[ \lambda = \frac{c}{f \sqrt{\varepsilon_r}} \]

Wavelength in Substrate (λ):

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1. What is Substrate Wavelength?

Substrate wavelength refers to the wavelength of an electromagnetic wave as it propagates through a dielectric material. It differs from free-space wavelength due to the material's relative permittivity (dielectric constant) which affects the wave's propagation speed.

2. How Does the Calculator Work?

The calculator uses the substrate wavelength formula:

\[ \lambda = \frac{c}{f \sqrt{\varepsilon_r}} \]

Where:

\( \lambda \) — Wavelength in substrate (m)
\( c \) — Speed of light in vacuum (3×10⁸ m/s)
\( f \) — Frequency of the wave (Hz)
\( \varepsilon_r \) — Relative permittivity of the substrate (unitless)

Explanation: The formula calculates how the wavelength shortens when electromagnetic waves propagate through a dielectric material compared to free space.

3. Importance of Wavelength Calculation

Details: Accurate wavelength calculation in substrates is crucial for designing microwave circuits, antennas, transmission lines, and other RF components where the physical dimensions are related to the wavelength.

4. Using the Calculator

Tips: Enter frequency in Hz and relative permittivity (a positive value greater than 1 for dielectric materials). Both values must be valid positive numbers.

5. Frequently Asked Questions (FAQ)

Q1: Why does wavelength change in a substrate?
A: The wavelength decreases in dielectric materials because the wave propagates slower than in vacuum, proportional to 1/√ε_r.

Q2: What are typical values for relative permittivity?
A: Common values range from 2-10 for PCB substrates (FR4: ~4.5, Rogers materials: 2.2-10.2), air: 1, water: ~80 at RF frequencies.

Q3: How does this relate to antenna design?
A: Antenna elements are typically designed at fractions of the wavelength (λ/2, λ/4), so the substrate wavelength determines the physical size of antenna elements.

Q4: Does this formula work for all frequencies?
A: Yes, the formula is valid across the electromagnetic spectrum as long as the material properties don't change significantly with frequency.

Q5: What if my substrate has loss?
A: This calculator assumes lossless dielectric. For lossy materials, the calculation becomes more complex involving the complex permittivity.