Why does a higher dielectric constant shrink the circuit?

Electromagnetic waves travel slower in a dielectric material by a factor of 1/sqrt(Er). A quarter-wave line at 2.4 GHz in free space is 31.25 mm long. On FR-4 (Er = 4.4), the effective Er for microstrip is about 3.3, so the wavelength shrinks by sqrt(3.3) = 1.82, making the quarter-wave line only 17.2 mm. On a high-Er ceramic substrate (Er = 10), the wavelength shrinks by sqrt(10) = 3.16, making it just 9.9 mm. This size reduction is valuable for compact designs like mobile phones and IoT modules. The trade-off is that higher-Er substrates have narrower trace widths for a given impedance, making fabrication tolerances tighter and losses higher.

What is loss tangent and why does it matter at mmWave?

Loss tangent (tan delta) is the ratio of the imaginary to real part of the complex permittivity. It quantifies how much electromagnetic energy the dielectric absorbs and converts to heat per cycle. Dielectric loss in dB per unit length is proportional to frequency times tan delta. At 1 GHz, FR-4 (tan delta = 0.02) loses about 0.1 dB/cm in a microstrip line. At 28 GHz, the same material loses 2.8 dB/cm, which is catastrophic for any trace longer than a few millimeters. This is why mmWave PCB designs use low-loss substrates like Rogers RO3003 (tan delta = 0.001) or liquid crystal polymer (tan delta = 0.002), where the loss at 28 GHz is only 0.14 dB/cm.

How does Er vary with frequency?

Most dielectric materials show a gradual decrease in Er with increasing frequency because the molecular dipoles cannot rotate fast enough to fully follow the electric field. FR-4 drops from Er = 4.7 at 100 MHz to about 4.2 at 10 GHz. This variation causes the impedance and electrical length of transmission lines to change with frequency, making broadband designs on FR-4 challenging above a few GHz. Low-loss RF substrates like Rogers RO4003C and PTFE-based materials have much more stable Er across frequency (variation less than 2% from DC to 40 GHz), which is critical for wideband filter and coupler designs where consistent electrical length across the bandwidth is essential.

Materials

Dielectric Constant

ε_r (relative permittivity)

A patch antenna on FR-4 (ε_r = 4.4) is 45% smaller than the same antenna in free space. On a Rogers TMM10i substrate (ε_r = 9.8), it shrinks by 69%. The dielectric constant controls wave speed: electromagnetic waves travel through a material at c/√ε_r, so every circuit element that depends on wavelength (patches, stubs, couplers, filters) scales inversely with √ε_r. But there is no free lunch. Higher ε_r means narrower traces for 50 Ω impedance, tighter fabrication tolerances, and usually higher loss tangent. The substrate under every RF trace is not just mechanical support; it is an active electromagnetic participant that shapes impedance, loss, and physical size.

Category: Materials

Key Relationship: v = c / √ε_r

Selection Driver: ε_r, tanδ, cost

Choosing a Substrate for Your Frequency

Substrate	ε_r	tanδ	Loss at 10 GHz (dB/cm)	Cost (rel.)	Max Freq.
FR-4 (standard)	4.2 to 4.7	0.020	0.5 to 0.8	1×	~3 GHz
Rogers RO4003C	3.38	0.0027	0.08	3×	~20 GHz
Rogers RO3003	3.00	0.0010	0.04	5×	~40 GHz
PTFE/Teflon	2.1 to 2.2	0.0009	0.03	6×	~77 GHz
Alumina (Al₂O₃)	9.8	0.0001	0.01	10×	~100 GHz
LCP (liquid crystal)	2.9 to 3.1	0.002	0.06	4×	~110 GHz

Wave velocity in dielectric:
v = c / √ε_r

Effective permittivity (microstrip):
ε_eff ≈ (ε_r + 1)/2 + (ε_r − 1)/2 × 1/√(1 + 12h/w)

Wavelength shrinkage factor:
λ_material = λ₀ / √ε_eff
Quarter-wave at 2.4 GHz: 31.25 mm (air) vs. 17.2 mm (FR-4, ε_eff ≈ 3.3)

Dielectric attenuation (dB per unit length):
α_d = 27.3 × (ε_r/√ε_eff) × (ε_eff − 1)/(ε_r − 1) × (tanδ/λ₀)
This shows why tanδ is the dominant loss mechanism at millimeter-wave frequencies where substrate selection drives system performance.

Common Questions

Frequently Asked Questions

Why does higher ε_r shrink circuits?

Waves travel at c/√ε_r. Wavelength shrinks by the same factor. On FR-4 (ε_r = 4.4): 1.82× smaller. On alumina (ε_r = 9.8): 3.13× smaller. Trade-off: narrower traces, tighter tolerances, sometimes higher loss.

Why does loss tangent matter at mmWave?

Dielectric loss scales linearly with frequency × tanδ. FR-4 at 1 GHz: 0.1 dB/cm. At 28 GHz: 2.8 dB/cm (unusable). Rogers RO3003 (tanδ = 0.001) at 28 GHz: 0.14 dB/cm. Substrate choice is survival at mmWave.

Does ε_r change with frequency?

Yes. FR-4 drops from 4.7 at 100 MHz to 4.2 at 10 GHz. Impedance and electrical length shift with frequency. Low-loss substrates (RO4003C, PTFE) vary <2% DC to 40 GHz, critical for wideband filters.

Related Terms

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Substrate Selection

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