Estimate Microstrip Substrate Relative Dielectric Constant

Download this article in .PDF format
Firas Mohammed Ali
Al-Raie
Department Head
The Polytechnic Higher Institute of Yefren, Libya,
Department of Electronic Engineering,
e-mail: [email protected]

Microstrip lines are commonly used in microwave circuits, although their performance is subject to the type of substrate material selected for a printed-circuit board (PCB). One of the more important material parameters for the substrate is the relative dielectric constant, which should be known before designing a high-frequency, microstrip-based circuit. What follows is a simple and practical method for estimating the relative dielectric constant of a microstrip substrate based on well-known microstrip line empirical equations.

where:

c = the speed of light and e_re = the effective dielectric constant of the microstrip line.

The value of e_re is significantly lower than that of e_r when the fields external to the substrate are taken into account. The guide wavelength, λ_g is related to the free-space wavelength, λ, as shown by Eq. 2:

Available numerical methods for the characterization of microstrip lines involve extensive computations. Closed-form expressions are necessary for the computer-aided design (CAD) and optimization of microstrip circuits. The closed-form expressions for Z_o and e_re apply to the following two sets of conditions2:

For W/h ≤ 1,

where:

And for W/h ≥ 1,

where:

These expressions provide accuracy of better than 2 percent.

Strip thickness

The results discussed above assume a negligible strip thickness. In reality, the thickness of the strip has some bearing on the electrical performance, and the strip thickness, t, affects the microstrip characteristics. However, for a thickness- to-height ratio (t/h) less than or equal to 0.005, the agreement between experimental and theoretical results obtained by assuming t/h = 0 is acceptable.2

The characteristic impedance of the open-circuit microstrip line is:

where X_oc is the measured input reactance of the microstrip line at the specified frequency (assuming it to be lossless).

Substituting Eq. 2 in Eq. 8 yields:

where:

λ = c/f

For a microstrip line with W/h ≥ 1, Z_o is given by Eq. 5 and can be rewritten as:

and e_re is given by Eq. 6 as:

where:

and

Substituting Eq. 11 into Eq. 10 and equating Eqs. 10 and 8 yields:

To find e_r, Eq. 14 can be solved using the Newton-Raphson iterative formula. Thus, Eq. 14 can be formulated as:

A computer program can be written to minimize f (e_r) according to the algorithm:

where:

and where e_re is substituted from Eq. 11. An initial value of e_r close to the actual value is necessary for the algorithm to converge as quickly as possible.

This estimation procedure was used to determine the relative dielectric constant for two types of substrate materials: Teflon-fiber glass (PTFE) and epoxy-glass boards. The relative dielectric constant of the PTFE substrate was found to be 2.68. This value is approximately constant over a wide range of frequencies. The microstrip line for this type of substrates can be considered lossless at frequencies to several gigahertz.