IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 7, JULY 20063127
Wideband Measurement of the Dielectric
Constant of an FR4 Substrate Using a
Parallel-Coupled Microstrip Resonator
Eric L. Holzman, Senior Member, IEEE
Abstract—We have made a wideband measurement of the real
part of the dielectric constant of flame retardant #4 epoxy (FR4),
a common high-frequency printed-circuit-board insulator. We
designed a novel test circuit, an electrically long parallel-cou-
pled microstrip resonator, which was etched on a 0.014-in FR4
substrate, manufactured by NELCO, Melville, NY. We used a
computer model of the resonator to extract the dielectric constant
at the frequencies of zeroes in its measured transmission response.
By adjusting the model’s dielectric constant, we tuned the fre-
quency of each zero to match the measured frequency, yielding the
dielectric constant at that frequency. To validate our method and
results, we present a simple, but original proof that the frequencies
of zeroes in the resonator’s transmission response are insensitive
to input and output mismatches. Additionally, we compare the
measured and predicted response of a two-stub filter designed
with our measured data. The fabricated filter’s measured return
loss and insertion loss from 3 to 12 GHz are within 1% of the
predictions of Agilent Technology’s Momentum.
Index Terms—Dielectric materials, measurement, microstrip
resonators, permittivity measurement, printed circuits.
microwave printed circuit boards (PCBs). Its dielectric constant
is knowntovarywithfrequencyand manufacturer.FR4data
sheets generally do not list dielectric-constant data over a wide
frequency range, and we found only one set of broadband data
in the literature . Unfortunately, these data are presented in
a relative sense only (normalized to unity), not well validated,
and the vendors are not identified. Further to this, four different
measurement techniques were used to obtain the data. A simple
means to measure the real part of the dielectric constant is de-
sirable, particularly if it can serve as a process control monitor
Many methods for measuring the dielectric constant of mate-
rialshavebeendevelopedand usedsuccessfully. Fora PCBma-
terial such as FR4, a practical approach is to fabricate a circuit
mine the material’s dielectric constant. If such a circuit is mod-
eled accurately with computer-aided design (CAD) software,
one can determine the substrate’s dielectric constant by com-
paring the predictions of the software with the circuit’s mea-
LAME-RETARDANT #4 epoxy (FR4) is a low-cost di-
electric material that finds use as a substrate for RF and
Manuscript received February 2, 2006; revised April 7, 2006.
The author was with YDI Wireless, South Deerfield, MA 01373 USA. He is
now with Northrop Grumman Electronic Systems, Baltimore, MD 21240 USA
Digital Object Identifier 10.1109/TMTT.2006.877061
sured characteristics. The extracted dielectric-constant data can
then be used to design other circuits.
This type of empirical/analytical approach has been demon-
strated by a number of researchers in the microwave field.
Das et al. used two microstrip lines of unequal length to mea-
sure the effective dielectric constant of microstrip . With
a computer model of microstrip, they extracted the substrate
dielectric constant and were able to achieve a measurement
accuracy of 1% over a broad bandwidth. Their method required
care in assembling the test fixture, long microstrip lines, and
well-matched and repeatable coaxial transitions according
to Lee and Nam . Shimin , and Verma and Verma 
used a microstrip patch antenna as the test circuit, and by
comparing the resonant frequency predicted by an analytical
model with the measured resonant frequency, they determined
the dielectric constant of the substrate. For the best results, the
substrate had to be 3 –4
larger than the patch. Akhavan and
Mirshekar-Syahkal replaced the patch with a microstrip fed slot
antenna to overcome some of the limitations of the resonant
patch method . In both cases, a different test circuit was
required for each frequency of interest. Bernard and Gautray
used a ring resonator fabricated on alumina as their test circuit
. They placed a test sample of the material of interest on
top of the ring resonator. The ring’s resonant frequency was
perturbed by the sample, enabling the authors to determine the
dielectric constant of the material using an analytical model
of the ring. Measurements of several substrates were within
15% of those from a cavity resonator. Similarly, Kantor used
microstrip, stripline, and disk resonators to determine the di-
electric constant of several microwave PCB materials . Yue
et al. measured the characteristic impedance of the stripline,
and determined the dielectric constant of the substrate from
equations for the impedance . Their technique required a
precision coaxial load to terminate one end of the stripline and a
full two-port calibration of the vector network analyzer making
the measurements. Gruszczynski and Zaradny made measure-
ments of a sample of dielectric of fixed width, metallized on
both sides . The primary source of error in their technique
was also the coaxial transition.
Each of the above techniques, to a varying degree, depends
on having well-matched coaxial transitions attached to the sub-
strate sample under test. With increasing frequency, such tran-
sitions become difficult to produce, and it is at higher frequen-
cies that accurate knowledge of the dielectric constant of most
substrates is most critical and often is not known. A measure-
able. Toward that end, Amey and Curilla  and Peterson and
Drayton  used the transmission response of microstrip and
0018-9480/$20.00 © 2006 IEEE
3128 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 7, JULY 2006
Fig. 1. Two-port network situated between a source and load. (a) With imped-
ances ? . (b) With impedances ?
and ? .
coplanar lines with series stubs to extract the dielectric constant
that their measurement is insensitive to impedance mismatch at
the transitions. Another advantage of the stub is that a single
circuit has multiple transmission zeroes over a wide frequency
band, with each zero yielding a value of the substrate dielec-
tric constant. A limitation of the coplanar version is that higher
order modes are excited at the tee junction.
In this paper, we extend the work of Peterson and Drayton,
first by presenting in Section II a simple proof that theoretically
validates their observation that the frequencies of the transmis-
sion zeroes of a passive two-port circuit are independent of port
mismatch. In Section III, we describe an alternative to the tee
circuit, the parallel-coupled resonator, which inherently is well
matched. We use this resonator to measure the real part of the
dielectric constant of NELCO FR4 over a broad range of fre-
quencies. In modeling our test circuit, we take advantage of the
high level of accuracy that commercial circuit simulators can
achieve. In particular, we use Agilent Technology’s Advanced
Design System (ADS), which includes a standard circuit sim-
ulator, based on analytical models, and Momentum, which is
based on the method of moments (MoM). We know the dielec-
tric constant of FR4 sufficiently well to design the test circuit.
We then fabricate it, measure its insertion response accurately,
and compare the data with the predictions of ADS. Due to our
confidence in the simulator, we can attribute any difference be-
tween the measured and predicted performance primarily to the
error in our knowledge of the dielectric constant. With a rela-
circuit element such as our microstrip resonator,
we can accurately adjust the dielectric constant in the simulator
until its prediction matches the data at the zero frequencies. In
Section IV, we use our measured FR4 dielectric constant data
to design an evaluation circuit. We perform a precision thru-re-
flect-line (TRL) calibration to enable us to measure the
rameters of the circuit at its microstrip inputs, and compare the
results with the predictions. Our measured and predicted reso-
nantfrequencyagreementis within1%,whichis excellent,con-
substrate thickness, and dielectric constant typical of most PCB
Fig. 1 shows a generic two-port network embedded between
and assign the two-port network a scattering matrix S, normal-
ized to the port impedance
at which the
determined [see Fig. 1(a)]. S’ is the generalized scattering ma-
trix of the same two-port network situated between a source and
We can write the transmission response or insertion loss as
If the source and load impedances are equal to
is perfectly matched,
Now let us assume
has a zero at a frequency
, the two-port
and , and that
. At , the bracketed term in (1)
which is finite valued. Thus,
zeroes that appear in
are those appearing in
quencies are dependent on the substrate dielectric constant. If
we build and test such a circuit, the frequencies of those zeroes
will be insensitive to port mismatches. A calibration of the test
equipment should not even be necessary, as verified empirically
by Peterson and Drayton . We can use an accurate model of
the circuit to extract the value of the dielectric constant at each
measured zero frequency.
has a zero at the same fre-
. At frequencies away from , the denomi-
, and we
III. TEST CIRCUIT DESIGN AND MEASUREMENT
A. Test Circuit
Circuits fabricated on FR4, a relatively lossy material, typi-
cally have passbands that do not extend above 6 GHz, but they
may have reject requirements at higher frequencies. Thus, it
would be useful to have accurate dielectric-constant data from
approximately 2 to 12 GHz. We know that FR4’s dielectric con-
with half a dozen transmission zeroes over that bandwidth, we
will have sufficient data to interpolate values at other frequen-
cies with good accuracy. Fig. 2 shows such a circuit, i.e., a mi-
crostrip parallel-coupled resonator. This particular example has
zeroes in transmission starting at approximately 2.7 GHz, and
repeating approximately every 2.7 GHz. To select the resonator
dimensions, we assumed the dielectric constant of the FR4 sub-
strate is 4.5 for all frequencies. Fig. 3 plots the insertion loss
of the resonator as predicted by ADS’s circuit simulator and by
Momentum. All Momentum analyses used a mesh with at least
15 cells/wavelength at the highest frequency of simulation. Mo-
mentum’s edge mesh feature was enabled also. We generated a
photo-mask and printed the filter on 14-mil FR4. We confirmed
the filter dimensions to be within 0.5 mil of the design and ad-
justed our model’s dimensions accordingly. The only important
circuit dimension is the resonator length, which, along with the
HOLZMAN: WIDEBAND MEASUREMENT OF DIELECTRIC CONSTANT OF FR4 SUBSTRATE3129
Fig. 2. Microstrip parallel-coupled resonator. Dimensions are in inches.
FR4 substrate thickness ? ????? in. Metallization thickness ? ????? in
Fig. 3. Insertion loss of microstrip parallel-coupled resonator—ADS circuit
separation of the resonator and main transmission line only af-
fects the depth of the transmission zero at each frequency.
Our test setup consisted of a Hewlett-Packard 8510 vector
network analyzer, a Wiltron 3680 K Universal Test Fixture with
FR4 substrate metallized with the test circuit shown in Fig. 2.
Since calibration is not critical, we only calibrated the analyzer
with a K-connector coaxial calibration. We then placed the test
circuit in the test fixture and measured its transmission response
Fig. 3 plots the measured insertion loss, and it is obvious that
our assumed value of 4.5 for the dielectric constant is in error,
with the error increasing with increasing frequency.
B. Dielectric-Constant Computation
To extract the correct frequency-dependent dielectric con-
stant, we adjust manually its value in our ADS circuit simulator
and Momentum models at each of the measured reject frequen-
cies until the predicted zero matches the measured zero. We
then know the dielectric constant at the reject frequency null.
Fig. 4 shows an example at 11.21 GHz. In this case, a dielectric
constant of 4.00 in the circuit simulator and 4.03 in Momentum
matched the frequencies of the zeroes predicted by the models
to the measured results. The values differ slightly because the
two analytical methods are different.
We extracted the real part of the dielectric constant in this
manner at every measured zero frequency through 16.7 GHz,
and the results are summarized in Table I. The rise in dielectric
constant above 14 GHz, though surprising, has been observed
by others .
Each set of data can be fit to a third-order polynomial. The
polynomial for the circuit simulator, which can be inserted di-
rectly into the ADS MSUB block, is
Fig. 4. Adjustment of substrate dielectric constant (ADS—4.00, MoM—4.03)
to match predicted and measured insertion loss at 11.21 GHz.
DIELECTRIC CONSTANT OF FR4 VERSUS FREQUENCY FOR ADS’S CIRCUIT
SIMULATOR AND MOMENTUM. FR4 MANUFACTURER: NELCO
After designing a preliminary circuit with the circuit sim-
ulator, one should perform an analysis in Momentum, which
models the circuit more accurately. Since Momentum does not
allow parameterization of the dielectric constant as a function
of frequency, one must analyze the circuit over frequency bands
narrow enough such that the dielectric-constant variation is
small. For instance, if we design a circuit to operate from 3 to
8 GHz, we might use two frequency bands based on the data in
Table I for analysis in Momentum, say, from 3 to 5.5 GHz and
from 5.5 to 8 GHz. Over these bands, the dielectric-constant
variation will be no more than 0.06, approximately 1(1/2)%.
For those designers who want to interpolate the Momentum
data in Table I, we have generated a third-order fit
is the frequency in gigahertz.
It is important to keep in mind that (2) and (3) and the data in
NELCO. New data should be measured.
IV. VALIDATION CIRCUIT
To confirm the accuracy of our dielectric-constant data, we
designed a microstrip two-stub reject filter on FR4. This filter
was designed to pass the band at 5.7–5.9 GHz while rejecting
signals at 3.3 and 11.5 GHz. The design was optimized with
3130 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 7, JULY 2006
Fig. 5. Microstrip two-stub reject filter for validating the measured dielectric
constant of FR4. All dimensions are in inches. Substrate: NELCO 14-mil FR4.
Fig. 6. Microstrip reject filter—comparison of measured and predicted (Mo-
mentum) rejection, return loss, and insertion loss.
ADS’s circuit simulator using (2) for the dielectric constant. It
of the filter is shown in Fig. 5.
We fabricated the filter along with TRL calibration standards
covering the 2–12-GHz frequency range. With these standards,
we deembedded our Wiltron test fixture’s coax-to-microstrip
transitions and microstrip lines up to the input and output ports
of the filter. As shown in Fig. 6, the measured insertion loss and
return loss are within 1% of the performance predicted by Mo-
FR4’s known variability is best managed with a circuit-board
process control monitor. The efficient shape and noncritical test
didate. Its insertion response can be an important part of a spec-
ification provided to a circuit-board vendor. These resonators
can be placed on the edge of or between the circuits on a stan-
the frequency response of the filter to determine if the dielec-
tric constant of the substrate is sufficiently close to the desired
value by comparing the frequencies of the transmission zeroes
with the specification. The verification test can be used to de-
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Eric L. Holzman (S’86–M’89–SM’95) received the
B.S., M.S., and Ph.D. degrees from the University of
California at Los Angeles (UCLA), in 1984, 1987,
and 1989, respectively, all in electrical engineering.
In 2004, he joined Northrop Grumman Electronic
Systems, Baltimore, MD, as a Consulting Engineer
with the Advanced RF Product Technology Depart-
ment. His research involves design and analysis of
active arrays and other antennas operating from UHF
to millimeter-wave frequencies. From 1999 to 2004,
less, South Deerfield, MA, where he designed antennas and transceiver cir-
cuits for a variety of fixed wireless applications. From 1993 to 1999, he was
a Principal Engineer and Manager with Lockheed Martin Government Elec-
tronic Systems, where he was involved in the design of advanced, solid-state
phased arrays. He began his career designing power oscillators, low-noise am-
approximately 35 publications. He authored Essentials of RF and Microwave
Grounding (Artech House, 2006) and Solid-State Microwave Power Oscillator
Design (Artech House, 1992). He holds seven patents in the microwave field.
He is listed in Who’s Who in Young America (1992).
Dr. Holzman a member of Tau Beta Pi and Eta Kappa Nu. He is a reviewer
for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He
is past chairman of the Philadelphia Chapter of the IEEE Antennas and Prop-
agation (AP)/Microwave Theory and Techniques (MTT) societies. He was a
member of the Organizing Committee for the Benjamin Franklin Symposium
(1995–1997). He was the recipient of the 1997 Lockheed Martin Engineer of
the Year award for his research on antennas and transmit/receive modules. He
is a former Howard Hughes Fellow.