## Dielectric Loss Tangents

Looking through tool vendor manuals and technical presentations, I have come across mention of loss tangent for dielectrics several times. This value appears to be an expression of the frequency dependence of the relative dielectric constant, Er.

My questions are:

1) How is this value derived? In other words, if I were to take laboratory measurements of Er versus edge rate (frequency) using capacitive or TDR techniques, how would the value of loss tangent "fall out" of the lab observations?

2) In impedance and propagation delay equations such as the ones at the end of your book "High Speed Digital Design," are loss tangents taken into account in the Er values used?

Thanks for your interest in High-Speed Digital Design.

For a capacitor formed from a lossy dielectric material, the loss tangent is the ratio at any particular frequency between the real and imaginary parts of the impedance of the capacitor. A large loss tangent means you have a lot of dielectric absorption.

If you construct a capacitor C from a lossy dielectric, the dielectric absorption causes the value of C to change with frequency. For a good dielectric, the value of C will very slowly deteriorate with frequency. For a crummy dielectric with higher dielectric loss, the decay in capacitance with frequency will be more pronounced. The rate of deterioration in capacitance is directly linked to the loss tangent. If the loss tangent is known, you can model the effect on the complex-valued capacitance C like this:

^{k}[1]

**Where:**

- C(f) is the value of capacitance as a function of frequency f,
- C0 is the value at one particular frequency, f0
- j is the square root of minus one, and
- k is a tiny
**negative**constant that sets the rate of deterioration in C per decade of frequency.

At frequency f0 there is (by construction of the formula) no deterioration.

As you go higher in frequency, the term (f/f0)^{k} gets smaller and smaller, making C go DOWN.

The constant k also controls the loss tangent,
where the loss tangent is *defined *as

Re-arranging equation [1] to isolate the j term produces a product of two terms: a pure phase term that looks like (j^{k}) and a
magnitude term equal to C0(f/f0)^{k}. To evaluate the phase
term it is helpful to remember Euler's identity: exp(jπ/2)=j. With that substution you can express the phase term as
[exp(jπ/2)^{k}], which reduces to exp(j(π/2)·k),
whose phase I hope you recognizable as
(π/2)·k. To model a loss tangent of size theta you must set (π/2)·k = –theta, or equivalently,

The negative sign gives you the correct behavior of capacitance DECREASING with frequency and power being ABSORBED by the FR-4.

Now you have a model for C(f), which you can plug into any frequency-domain simulation. If you replace the term C(f) by an expression for the complex-valued dielectric constant Er(f), the same logic applies.

Example.Assume you have a material with a dielectric constant of 4.7 at 1 KHz, and a constant loss tangent of theta=0.01. What's the dielectric constant at 1 MHz? How about at 1 GHz? To solve this problem we first evaluate k=- theta*(2/pi) = -.006366. The Er() model for FR-4 therefore looks like this:

Er(f) = Er0·(jf/f0)^{-.006366}[4]

Where:

- Er is the magnitude of the dielectric constant at 1 KHz, and
- f0=1KHz.
You can now plug in different operating frequencies f to see what happens. At an operating frequency f=1MHz, formula [4] yields:

Er(f)=Er0·j^{-.006366}?(1MHz/1KHz)^{-.006366}[5]=Er0·(.99995-j.01)·(.956978)[6]The term j^{k}gave us a loss tangent of 0.01 (represented in the equation as (.99995-j.01)). The overall magnitude of Er(1MHz) works out to 4.7·0.956978=4.5. Starting with a dielectric constant of 4.7 at 1 KHz, the effective dielectric constant at 1 MHZ has deteriorated by a factor of 0.956978 to its new value of 4.5. By the time you get to 1 GHz the dielectric constant has deteriorated by another factor of (1E6)^{-.006366}=.9158, producing a new dielectric constant of about 4.3.

The above example shows about how FR-4 works. The dielectric loss tangent is about 0.01. Although loss tangent isn't exactly constant with frequency, it's close. Manufacturers of FR-4 materials often specify a worst-case value of 0.02. With a loss tangent of 0.01 the dielectric constant deteriorates from about 4.7 at 1KHz, to 4.5 at 1 MHz, to 4.3 at 1 GHz.

Because, for good dielectrics, the dielectric constant is such a slowly moving function of frequency we generally assume it's just constant over the band of interest. I usually just use the value of dielectric constant applicable at the HIGHEST frequency of interest. For digital signals the highest frequency of interest F depends on your rise/fall time Tr, with F approximately equal to 0.5/Tr.

When you propagate a high-speed signal through a PCB trace the portion of your signal loss attributable to dielectric absorption depends on the frequency f, the trace delay T, and the loss tangent theta. A reasonable approximation for magnitude H of the dielectric-loss effect would be:

**Where:**

- pi is the constant 3.1415926...,
- f is the frequency of operation,
- T is the delay of the transmission line segment through which your signal has traveled,
- theta is the loss tangent of the material in the region near f, and
- exp() is the exponentiation operation.

### Final notes

Because its phase remains fixed at a very small value throughout the frequency spectrum, you can think of the dielectric loss mechanism as a (mostly) zero- phase effect. That means you should expect roughly equal-sized precursor and postcursor artifacts from this effect.

Measurement of loss using sinusoidal excitation at a particular frequency gives you the product of skin effect and dielectric loss functions (that's the same as the sum of the skin effect and dielectric losses in units of dB). If you have an independent means of assessing the skin-effect loss, the portion of loss attributable to dielectric losses may be used to estimate the value of theta at the test frequency.

A combined chart showing both skin effect and dielectric losses in units of percent signal loss per inch appears in: dielectricloss.pdf.

Dielectric losses are discussed in a general way on
page 159 of *High-Speed Digital Design: A Handbook of Black Magic*,
but not
incorporated into any of the models in the back of

Best Regards,

Dr. Howard Johnson