Review of Essential Material Parameters

Fabrication Processes for RF and Microwave Circuits

3.2 Review of Essential Material Parameters

k k

83

3

k k

G C

𝜔C

G

(a) (b) (c)

1/Z 𝛿

Figure 3.1 Parallel-plate capacitor (a), with its equivalent circuit (b) and vector admittance diagram (c).

Also shown in Figure 3.1c is the admittance vector diagram for theGCcombination, from which it can be seen that tan𝛿= 1

𝜔CR. (3.1)

tan𝛿is referred to as the loss tangent of the dielectric. From simple circuit theory Eq. (3.1) can also be written as tan𝛿= 1

Q, (3.2)

whereQis theQ-factor¹of the dielectric.

An alternative to using a conductanceGto represent the dielectric loss is simply to write the relative permittivity of the dielectric in a complex format as

𝜀r=𝜀^′−j𝜀^′′, (3.3)

where𝜀^′is the value of permittivity used at DC and low frequencies, and 𝜀^′′ is a quantity representing the loss in the dielectric. It can be shown [1] that the loss tangent and the components of the complex permittivity are related by

tan𝛿=𝜀^′′

𝜀^′. (3.4)

The values of𝜀^′ and tan𝛿remain reasonably constant with frequency over the lower part of the RF spectrum, but at microwave (and particularly millimetre-wave) frequencies, significant changes can occur. Consequently, dielectric characterization techniques are of particular importance when selecting or developing dielectric materials for microwave applications.

Propagation of electromagnetic waves through dielectric material may be described using the concepts introduced in Chapter 1, namely that the propagation constant,𝛾, is of the form

𝛾=𝛼+j𝛽, (3.5)

where𝛼is the attenuation constant in the units of Np/m, and𝛽is the phase propagation constant in the units of rad/m.

In the particular case of microstrip the dielectric loss in dB per line wavelength (𝜆s) is given by the widely referenced formula [2]

𝛼d=27.3𝜀r(𝜀^MSTRIP_r,eff −1)tan𝛿

𝜀^MSTRIP_r,eff (𝜀r−1) dB∕𝜆s, (3.6)

where the symbols have their usual meanings. Note that the effective relative permittivity is used to take account of fringing, since some of the energy travels in the air above the substrate.

Example 3.1 A low-loss dielectric has a complex relative permittivity given by 𝜀r=2.6−j0.018.

Determine:

(i) The loss tangent of the dielectric (ii) TheQ-factor of the dielectric.

1Q-factor is defined later in the chapter in Eq. (3.16)

k k

3.2 Review of Essential Material Parameters 85

Solution

(i) Using Eq. (3.4):

tan𝛿= 0.018

2.6 =6.92×10⁻³. (ii) Using Eq. (3.2):

Q= 1

tan𝛿 = 1

6.92×10⁻³ =144.51.

Example 3.2 A dielectric has a real value of relative permittivity of 2.6, aQof 210, and a propagation constant given by

𝛾=2𝜋 𝜆d

(0.5 tan𝛿+j), where the symbols have their usual meanings.

Determine (at a frequency of 10 GHz):

(i) The wavelength in the material

(ii) The ratio of the velocity of propagation in the material to that in air (iii) The phase change through a 10 mm length of the material

(iv) The attenuation constant in dB/m

(v) The loss in dB through 30 mm of the material.

Solution (i) 𝜆_d= 𝜆o

√2.6

= 30

√2.6

mm=18.61 mm.

(ii) v_p(material) v_p(air) = 1

√2.6

=0.62.

(iii) 𝜙= 2𝜋 𝜆d

×l= 2𝜋

18.61×10 rad=1.07𝜋rad≡192.60∘. (iv) Q=210 ⇒ tan𝛿= 1

210 =4.46×10⁻³, 𝛼 = 2𝜋

𝜆d

×0.5 tan𝛿= 2𝜋

18.61×0.5×4.76×10⁻³Np∕mm

=8.04×10⁻⁴Np∕mm=8.04×10⁻¹Np∕m

=8.04×8.686×10⁻¹dB∕m=6.98 dB∕m.

(v) Loss=6.98×0.030 dB=0.21 dB.

k k Example 3.3 Determine the dielectric loss in dB of a 100 mm length of 50Ωmicrostrip line at 15 GHz, given that

the line is fabricated on a substrate that has aQof 195, a relative permittivity of 9.8, and a thickness of 0.4 mm.

Solution

Q=195 ⇒ tan𝛿= 1

195 =5.13×10⁻³. Using microstrip design graphs (or CAD):Z_o=50Ω ⇒ 𝜀^MSTRIP_r,eff =6.6.

15 GHz ⇒ 𝜆o=20 mm, 𝜆s= 20

√6.6

mm=7.78 mm.

Using Eq. (3.6):

𝛼d=27.3×9.8× (6.6−1) ×5.13×10⁻³

6.6× (9.8−1) dB per 7.78 mm

=0.13 dB per 7.78 mm.

Therefore, the loss in a 100 mm length of line is:

Loss=0.13× 100

7.78 dB=1.67 dB.

3.2.2 Conductors

The attenuation due to the finite conductivity of conductors is one of the main contributing factors to loss in RF and microwave circuits. Conductor losses may be divided into those due to the bulk resistivity of the material and those due to surface roughness. As the frequency increases surface losses tend to dominate due to the skin effect, which was discussed in Chapter 2.

The conductor loss in a microstrip line is normally evaluated using an expression developed by Hammerstad and Bekkadal [2], namely

𝛼c=0.072

√f wZ_o𝜆s

( 1+ 2

𝜋tan⁻¹ [1.4Δ²

𝛿²s

])

dB∕𝜆s, (3.7)

where

f=frequency of operation in GHz w=width of microstrip line

Z_o=characteristic impedance of microstrip line Δ =RMS surface roughness

𝛿s=skin depth.

Example 3.4 Determine the dielectric and conductor losses (in dB/m) at 5 GHz for a 70Ωmicrostrip line given the following material parameters:

Dielectric: Relative permittivity=9.8 Thickness=0.6 mm Loss tangent=0.0004 Conductor: Copper (𝜎=56×10⁶S/m)

RMS surface roughness=0.63μm

k k

3.2 Review of Essential Material Parameters 87

Solution

Using the microstrip design curves (or CAD):

70Ω ⇒ w

h =0.35 ⇒ 𝜀^MSTRIP_r,eff =6.25.

𝜆s= 𝜆o

√6.25

= 60

√6.25

mm=24 mm.

w=0.35×h=0.35×0.6 mm=0.21 mm.

Using Eq. (3.6):

𝛼d=27.3×9.8× (6.25−1) ×0.0004

6.25× (9.8−1) dB∕𝜆s=0.01 dB∕𝜆s. 𝛼d=0.01×1000

24 dB∕m=0.42 dB∕m.

The skin depth,𝛿s, is found using Eq. (2.8) in Chapter 2:

𝛿s=

√ 2 𝜔𝜇o𝜎 =

√ 2

2𝜋×5×10⁹×4𝜋×10⁻⁷×56×10⁶ m

=0.951μm,

where we have assumed the relative permeability of the conductor to be unity.

Using Eq. (3.7):

𝛼c=0.072×

√5

0.21×70×24× (

1+2 𝜋tan⁻¹

[1.4× (0.63)² (0.951)²

]) dB∕𝜆s

=0.355 dB∕𝜆s

=0.355×1000

24 dB∕m=14.79 dB∕m.

Example 3.5 In Example 3.4, what percentage of the conductor loss is due to surface roughness?

Solution

PuttingΔ =0 in Eq. (3.7) gives the bulk conductor loss, i.e.

𝛼c(bulk) =0.072×

√5

0.21×70×24 dB∕𝜆s

=0.263 dB∕𝜆s. Then

𝛼c(surface) = (0.355−0.263)dB∕𝜆s

=0.092 dB∕𝜆s. Percentage of loss due to surface roughness is

0.092

0.355×100% =25.92%.

The results from Example 3.4 show that the conductor loss is significantly greater than the dielectric loss, and this is normally the case for modern low-loss substrate materials. However, the expression given in Eq. (3.7) becomes less accurate as the frequency increases, and the skin depth becomes comparable with the RMS surface roughness. Figure 3.2 shows how the skin depth varies with frequency for three common materials used in RF and microwave circuits. It can be seen that

k k

0 0.2 0.4 0.6 0.8

0 20 40 60 80 100

Frequency (GHz)

Skin depth (μm)

Ag

Au Cu

Figure 3.2 Variation of skin depth with frequency for gold (Au), copper (Cu), and silver (Ag).

the skin depth decreases dramatically above 10 GHz, and consequently a number of authors suggest that this should be the upper frequency limit for the use of Eq. (3.7).

Although the expression for conductor loss given in Eq. (3.7) is still widely used, it was developed some time ago.

More recent research using full 3D simulation, and more complex models to represent the surface of conductors, has led to improved techniques to determine surface losses at higher microwave frequencies. Using a high-frequency struc- ture simulator, Sain and Melde [3] have shown good agreement between simulated and measured conductor loss on a conductor-backed coplanar line at frequencies up to 40 GHz. In a similar work, Iwai and Mizatani [4] proposed a modifica- tion to Eq. (3.7) to better represent surface losses at millimetre-wave frequencies. Their work led to a modified expression for conductor loss

𝛼c=0.072

√f wZ_o𝜆s

( 1+32

𝜋 tan⁻¹

[0.24Δ² 𝛿s²

])

dB∕𝜆s, (3.8)

which gives a higher predicted loss at millimetre-wave frequencies. Data published in [4] shows reasonably good agreement between measured and predicted loss forΔ/𝛿ratios up to 2, using the modified expression given in Eq. (3.8).

Another parameter that is often used when dealing with the performance of a conductor at high frequencies is that of surface impedance. This is essentially the impedance between two electrodes placed along opposite edges of a square on the surface of the conductor, and so has the units of Ohms per square, which is normally written in short-hand form asZ_◽. For smooth conductors the surface impedance is

Z_◽,smooth=1+j

𝜎𝛿_s , (3.9)

where the symbols have their usual meanings. Gold and Helmreich [5] showed how the concept of surface impedance could be used for modelling conductor roughness, by modifying Eq. (3.9) as

Z_◽,rough= 1

𝜎eff𝛿(𝜎eff)+j 1

𝜎bulk𝛿(𝜇_r,eff), (3.10)

where𝜎eff is an effective conductivity that represents the increased loss due to rough surfaces, and𝜇r,eff is an effective permeability that represents the effect that surface roughness has on the inductance of the surface and hence the propa- gation delay. Using this concept, Gold and Helmreich showed good agreement between simulation and measurement of transmission loss for a typical transmission line at millimetre-wave frequencies.