[SI-LIST] Re: Relationship between loss tangent and dielectric constant's frequency dependence
- From: "Howard Johnson" <howie03@xxxxxxxxxx>
- To: <si-list@xxxxxxxxxxxxx>
- Date: Thu, 1 Apr 2010 08:20:07 -0700
Dear Neo,
Hendrik Bode, in his seminal text, "Netowrk Analysis and Feedback Amplifier
Design", D. Van Nostrand Co., 1945, goes into a long discussion of the
relations between the real and imaginary parts of any causal linear network
function (complex electric permittivity is an example of such a function).
He devotes a whole chapter to the general relations, and then another
chapter on the "Graphical Computation of Relations Between Real and
Imaginary Components of Network Functions" -- in other words, he teaches you
how to just LOOK at a curve of lass tangent and predict the shape of the
dielectric loss curve. I included some parts of that discussion in my book,
"High-Speed Signal Propagation: Advanced Black Magic", pp. 105-115. The
most important, and practically useful, part of the discussion I think
appears on page 108, equation 2.88, where I point out that for a low-loss
material, a constant loss tangent equal to "tan theta" (that's the form of
worst-case specification for most dielectric materials) implies an
exponential decay in the dielectric constant, with a *very tiny* exponential
decay constant equal to omega (frequency) raised to the power of
(-2)(theta/pi).
Example: Let the loss tangent be one percent: "tan theta" = 0.01
For small angles, working in units of radians, theta approximately equals
"tan theta", so we have theta = 0.01
Assume you have Er = 4.00 at 1 GHz.
Suppose that you want to know Er at 5 GHz.
If the loss tangent is flat over a broad range extending well to either side
of both 1 GHz and 5 GHz, then the dielectric constant at 5 GHz must be
de-rated according to the rule:
Er[5 GHz] = Er[1 GHz] * (5GHz/1GHz)^^(-2)(theta/pi) [EQ 2.88]
(where the symbol "^^" means, "raised to the power of"
Evaluating the term (5GHz/1GHz)^^(-2)(theta/pi)
= (5)^^(-2)(0.01/3.1415926)
= (5)^^(-0.0063662)
= 0.9898
Multiplying that term times Er[1 GHz] gives:
Er[1 GHz] * 0.9898 = (4.00)*0.9898 = 3.959
Why this works is explained in Bode's book (if you read the whole thing) in
a fairly simple way. That was his genius.
Bode points out, and this may be familiar to you if you are a student of
electrical engineering, that flat, fixed phases are associated with each of
the following network functions:
|H1(f)| = f -- phase equals +90 degrees (a differentiator)
|H2(f)| = 1 -- phase equals 0 degrees (flat function - a constant)
|H3(f)| = 1/f -- phase equals -90 degree (an integrator)
In each case, the slope of the function (power of f) is proportional to the
phase:
|H1(f)| = f^^1 -- phase equals +90 degrees (a differentiator)
|H2(f)| = f^^0 -- phase equals 0 degrees (flat function - a constant)
|H3(f)| = f^^-1 -- phase equals -90 degree (an integrator)
----------------------------
If you are familiar with the skin effect, you know that it works like this:
|Hskin(f)| = f^^(-1/2) -- phase equals -45 degree (an integrator)
Note that the skin-effect relation seems to fit "in between" H2 and H3 in my
table.
----------------------------
Bode goes on to show how you can generalize his relation for any rational
power of frequency, so that:
|f^^a| must have a phase in degrees of (90 degrees times a)
In your case, the "loss tan" for a low-loss material represents an angle
given in units of radians. If we modify Bode's rule for radians we get this:
|f^^a| must have a phase in radians of (pi/2 times a)
If you supply the angle theta = (pi/2 times a), then the constant "a" must
equal (2/pi)*theta, which is where I got the rule for [EQ 2.88].
Rule [EQ 2.88] works we as an approximation for any low-loss material that
has a fairly flat loss tangent. From a simple specifiation of dielectric
constant and loss tangent at one frequency, you may use this "constant loss
tangent assumption" to evaluate the dielectric constant at every frequency,
in a way that is guaranteed to produce a causal network function.
I hope this brief note helps. Let me know if I've just made the subject
more, rather than less, confusing.
Best regards,
Dr. Howard Johnson, Signal Consulting Inc.,
tel +1 509-997-0505, howie03@xxxxxxxxxx
www.sigcon.com -- High-Speed Digital Design seminars, publications and films
(I'll be teaching in Portland May 3-6th)
-----Original Message-----
From: si-list-bounce@xxxxxxxxxxxxx [mailto:si-list-bounce@xxxxxxxxxxxxx] On
Behalf Of Neo
Sent: Tuesday, March 30, 2010 7:24 PM
To: List Si
Subject: [SI-LIST] Relationship between loss tangent and dielectric
constant's frequency dependence
Hi,
This email's title is a bit long but it is exactly a question annoying me.
I'm trying to find out whether there is a proven relationship between a
dielectric material's real part and imaginary part. Imaginary part is
dielectric loss tangent. And real part is the dielectric constant. For a
material like FR4 or Rogers, their dielectric constants all change over
frequency. And their loss tangents are also different. Is there any inner
relationship between the value of loss tangent and how the dielectric
constant (real part) changes over frequency?
Thanks,Neo
------------------------------------------------------------------
To unsubscribe from si-list:
si-list-request@xxxxxxxxxxxxx with 'unsubscribe' in the Subject field
or to administer your membership from a web page, go to:
//www.freelists.org/webpage/si-list
For help:
si-list-request@xxxxxxxxxxxxx with 'help' in the Subject field
List technical documents are available at:
http://www.si-list.net
List archives are viewable at:
//www.freelists.org/archives/si-list
Old (prior to June 6, 2001) list archives are viewable at:
http://www.qsl.net/wb6tpu
------------------------------------------------------------------
To unsubscribe from si-list:
si-list-request@xxxxxxxxxxxxx with 'unsubscribe' in the Subject field
or to administer your membership from a web page, go to:
//www.freelists.org/webpage/si-list
For help:
si-list-request@xxxxxxxxxxxxx with 'help' in the Subject field
List technical documents are available at:
http://www.si-list.net
List archives are viewable at:
//www.freelists.org/archives/si-list
Old (prior to June 6, 2001) list archives are viewable at:
http://www.qsl.net/wb6tpu
Other related posts:
