# 4.2: Substrates

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Planar line design involves choosing both the transmission line structure to use and the substrate. In this section the electrical and magnetic properties of substrate materials will be discussed.

## 4.2.1 Dielectric Effect

When the fields are in more than one medium (a **nonhomogeneous transmission line**), as for the microstrip line, the effective relative permittivity, \(\varepsilon_{r,e}\) (or usually just \(\varepsilon_{e} = \varepsilon_{r,e}\)), is used. The characteristics of the line are then more or less the same as for the same structure with a uniform dielectric of permittivity, \(\varepsilon_{\text{eff}} = \varepsilon_{e}\varepsilon_{0}\). The \(\varepsilon_{\text{eff}}\) changes with frequency as the proportion of energy stored in the different regions changes. This effect is called **dispersion **and causes a pulse to spread out as the different frequency components of a signal travel at different speeds.

## 4.2.2 Dielectric Loss Tangent, \(\tan\delta\)

Loss in a dielectric comes from (a) **dielectric damping** (also called **dielectric relaxation**), and (b) conduction losses in the dielectric. Dielectric damping originates from the movement of charge centers resulting in vibration of the lattice and thus energy is lost from the electric field. It is easy to see that this loss increases linearly with frequency and is zero at DC. In the frequency domain loss is incorporating in an imaginary term in the **permittivity**:

\[\label{eq:1}\varepsilon =\varepsilon_{r}\varepsilon_{0}=\varepsilon '-\jmath\varepsilon '' =\varepsilon_{0}\left(\varepsilon_{r} '-\jmath\varepsilon_{r} '' \right) \]

If there is no dielectric damping loss, \(\varepsilon '' = 0\). The other type of loss is due to the movement of charge carriers in the dielectric. The ability to move charges

Material |
\(10^{4}\tan\delta\) (at \(10\text{ GHz}\)) |
\(\varepsilon_{r}\) |
---|---|---|

Air (dry) | \(\approx 0\) | \(1\) |

Alumina, \(99.5\%\) | \(1-2\) | \(10.1\) |

Sapphire | \(0.4-0.7\) | \(9.4,\: 11.6\) |

Glass, typical | \(20\) | \(5\) |

Polyimide | \(50\) | \(3.2\) |

Quartz (fused) | \(1\) | \(3.8\) |

FR4 circuit board | \(100\) | \(4.3-4.5\) |

RT-duroid 5880 | \(5-15\) | \(2.16-2.24\) |

RT-duroid 6010 | \(10-60\) | \(10.2-10.7\) |

AT-1000 | \(20\) | \(10.0-13.0\) |

Si (high resistivity) | \(10-100\) | \(11.9\) |

GaAs | \(6\) | \(12.85\) |

InP | \(10\) | \(12.4\) |

SiO\(_{2}\) (on-chip) | \(—\) | \(4.0-4.2\) |

LTCC (typical, green tape(TM) 951) | \(15\) | \(7.8\) |

Table \(\PageIndex{1}\): Properties of common substrate materials. The dielectric loss tangent is scaled. For example, for glass, \(\tan\delta\) is typically \(0.002\).

is described by the conductivity, \(\sigma\), and this loss is independent of frequency. So the energy lost in the dielectric is proportional to \(\omega\varepsilon ''+\sigma\) and the energy stored in the electric field is proportional to \(\omega\varepsilon '\). Thus a loss tangent, \(\tan\delta\), is introduced:

\[\label{eq:2}\tan\delta =\frac{\omega\varepsilon ''+\sigma}{\omega\varepsilon '} \]

Also the relative permittivity can be redefined as

\[\label{eq:3}\varepsilon_{r}=\varepsilon_{r}'-\jmath\left(\varepsilon_{r}'' +\frac{\sigma}{\omega\varepsilon_{0}}\right) \]

With the exception of silicon, the loss tangent is very small for dielectrics that are useful at RF and microwave frequencies and so most of the time

\[\label{eq:4} |\varepsilon_{r} |\approx \varepsilon_{r}' \]

Thus

\[\label{eq:5}\varepsilon_{r} =\varepsilon_{r}'-\jmath\left(\varepsilon_{r}'' +\sigma /(\omega\varepsilon_{0} )\right)\approx \varepsilon_{r}' (1-\jmath\tan\delta ) \]

## 4.2.3 Magnetic Material Effect

Except for very special circumstances substrates used for microstrip lines are non magnetic so \(\mu =\mu_{0}\) and the relative permeability, \(\mu_{r}\), is defined so that

\[\label{eq:6} \mu =\mu_{r}\mu_{0} \]

## 4.2.4 Substrates for Planar Transmission Lines

The properties of common substrate materials are given in Table \(\PageIndex{1}\). Crystal substrates have very good dimensional tolerances and uniformity of electrical properties. Many other substrates have high surface roughness and electrical properties that can vary. For example, FR4 is the most common type of PCB substrate and is a weave of fiberglass embedded in resin. So the material is not uniform and there is an unpredictable localized variation in the proportion of resin and glass. High-performance microwave circuit boards have ceramic particles embedded in the resin.