Dielectric Loss Tangents

HIGH-SPEED DIGITAL DESIGN - online newsletter -
Vol. 4 Issue 5

*----------------------(QUESTION)---------------------*

DIELECTRIC LOSS TANGENTS

Michael Mirmak writes:

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 question is:

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?

*--------------(REPLY FROM DR. JOHNSON)---------------*

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:

C(f) = C0*[(jf/f0)^^k] [1]

--where C(f) is the value of capacitance as a function of frequency f

--where C0 is the value at one particular frequency, f0

--where j is the square root of minus one

--where the notation ()^^k means "raised to the power of k"

--where 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 C goes DOWN.

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

theta = RE(1/j2pifC)/IM(1/j2pifC) = IM(C)/RE(C).

Re-arranging equation [1] to pull out the j term, you get a phase term that looks like (j^^k), and a magnitude terms C0(f/f0)^^k. To evaluate the phase term it is helpful to recognize that exp(j*pi/2)=j, whereby we can express the phase term as [exp(j*pi/2)^^k], which gives us exp(j*(pi/2)*k), whose phase is immediately recognizable as (pi/2)*k. To get a loss tangent of size theta we must set (pi/2)*k = -theta, or equivalently,

k=-theta*2/pi

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

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

Let's try an 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]

--Where Er is the magnitude of the dielectric constant at 1 KHz, and

--where f0=1KHz

We can now plug in different operating frequencies f to see what happens. At an operating frequency f=1MHz, formula [2] yields:

Er(f)=Er0*[j^^-.006366]*[(1MHz/1KHz)^^-.006366]

=Er0*[.99995-j.01]*[.956978]

The term j^^k just gives us a loss tangent of 0.01 (we knew that), and the magnitude 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 a new value of 4.5.

At 1 GHz the dielectric constant deteriorates by a factor of

(10E6)^^-.006366=.9158, producing a new dielectric constant of about 4.3.

End example

The above example shows about how FR-4 works. The dielectric loss tangent is about 0.01 (although it isn't constant with frequency so manufacturers typically specify a worst-case value of 0.02), and the dielectric constant deteriorates from about 4.7 at 1KHz, to 4.5 at 1 MHz, to 4.3 at 1 GHz. Above 1 GHz I haven't got any solid information about the dielectric loss tangent for FR-4, but anecdotal evidence suggests it is close to 0.02 .

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:

H(f) = exp(-pi*f*T*theta)

Where pi is the constant 3.1415926...

Where f is the frequency of operation

Where T is the delay of the transmission line segment through which your signal has traveled

Where theta is the loss tangent of the material in the region near f

Where exp() is the exponentiation operation

The dielectric loss mechanism is (mostly) a zero- phase effect.

From measurements of loss using sinusoidal excitation at a particular frequency you can determine the product of skin effect and dielectric loss functions. 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", but not incorporated into any of the models in the back of the book.

Best Regards,
Dr. Howard Johnson

To Subscribe to this HSDD Newsletter, send an email to hsdd-request@freelists.org with 'subscribe' in the subject field.

All Publications by Dr. Howard Johnson except as noted.
Signal Integrity Training Classes taught exclusively by Dr. Howard Johnson - for full schedule, see www.sigcon.com
© 2001 Signal Consulting, Inc., Dr. Howard Johnson. All rights reserved. • site map