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Q & ESR EXPLAINED
A JOHANSON TECHNOLOGY PRIMER
November 24, 2004
One of the most important parameters in evaluating a high frequency chip capacitor is the Q
factor, or the related equivalent series resistance (ESR). In addition to providing excellent
performing RF components, JTI strives to provide our customers with accurate and complete data.
Towards this end, a more detailed discussion of Q & ESR measurement issues follows.
In theory, a “perfect” capacitor would exhibit an ESR of 0 (zero) ohms and would be purely
reactive with no real (resistive) component. The current going through the capacitor would lead the
voltage across the capacitor by exactly 90 degrees at all frequencies.
In real world usage, no capacitor is perfect, and will always exhibit some finite amount of ESR.
The ESR varies with frequency for a given capacitor, and is “equivalent” because its source is
from the characteristics of the conducting electrode structures and in the insulating dielectric
structure. For the purpose of modeling, the ESR is represented as a single series parasitic element.
In past decades, all capacitor parameters were measured at a standard of 1 MHz, but in today’s
high frequency world, this is far from sufficient. Typical values for a good high frequency
capacitor of a given value could run in the order of about 0.05 ohms at 200 MHz, 0.11 ohms at 900
MHz, and 0.14 ohms at 2000 MHz.
The quality factor Q, is a dimensionless number that is equal to the capacitor’s reactance
divided by the capacitor’s parasitic resistance (ESR). The value of Q changes greatly with
frequency as both reactance and resistance change with frequency. The reactance of a capacitor
changes tremendously with frequency or with the capacitance value, and therefore the Q value
could vary by a great amount. See Equations 1 and 2.
Eq. 1 :
|Xc| = 1 / { 2 (π) (f) (C) } ; where |Xc| is the absolute value of the reactance in Ohms;
f is the frequency in Hertz;
C is the capacitance In Farads
Eq. 2 :
Q = |Xc| / ESR ; where Q is a dimensionless number meaning 'Quality Factor';
|Xc| is the absolute value of the reactance in Ohms;
ESR is the Equivalent Series Resistance in Ohms
Johanson Technology measures ESR and Q on a Boonton 34A resonant line. The capacitor
under test is resonated with an inductive line of an accurately characterized impedance and Q.
From the resultant data (the center frequency and bandwidth of the resulting peak), the Q, ESR,
and capacitance value of the device is output. This method is a long-standing industry standard
for the measurement of Q and ESR at RF frequencies. Since this method depends on the frequency
accuracy of a signal generator (which can be measured with extreme precision), the data taken in
this manner is quite accurate. As the ESR of modern capacitors goes ever lower, the accuracy,
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even of this method, will approach +/- 10% (reference Boonton 34A Instruction Manual, page
2.2). This is still the most accurate method for measuring Q and ESR at RF frequencies presently
available. See Figure 1 for a block diagram of the test set-up. See Figures 2 & 3 for pictures of the
setup.
Computer
Freq. Synthesizer
RF Millivoltmeter
D.U.T. goes here
Boonton Model 34A Resonant Coaxial Line
Figure 1
Using S-Parameter data collected from a Vector Network Analyzer to derive ESR is not
acceptable. The predominant reason is that the accuracy of this data is limited by the accuracy of
the network analyzer in a 50 ohm system (a typical amplitude calibration of ± .05 dB is not
accurate enough in the low-loss region of a capacitor which could be .01 dB). Using Q / ESR data
collected from an LCR meter is also not acceptable for these high-Q devices. When the device Q is
very high, the LCR meter just cannot resolve an extremely small resistance (R) against an extremly
large reactance (X). For this reason, ESR and Q parameters presented in Johanson Technology’s
published catalog, and in the MLCSoft® program are based on the industry standard resonant line
test method.
Figure 2
Figure 3
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Camarillo, California • Tel 805.389.1166 • Fax 805.389.1821 • www.johansontechnology.com
CAPACITOR RF CURRENT & POWER
APRIL 1, 1999
This note discusses the relationship of current handling, power dissipation, applied voltage and
the maximum rated voltage of a multi-layer ceramic capacitor. Either the maximum voltage
rating or the maximum power dissipation of the part limits the maximum current through a
capacitor. The capacitance value and the operating frequency determine which of the two
becomes the governing limit. For lower values of capacitance at a given frequency, or lower
frequencies at a given capacitance, the voltage limitation is generally reached before the power
dissipation limitation.
The voltage limitation depends on the capacitors reactive impedance (Xc), where as the
power dissipation depends on the resistive impedance or equivalent series resistance (ESR) of the
capacitor.
Xc is calculated by Xc=1/[2FC], where F= frequency in hertz and C = Capacitance in farads.
The maximum allowable peak current flow through a capacitor (without exceeding the capacitors
rated voltage) is calculated by I=Er/Xc, where Er is the capacitors rated voltage, and I = the peak
current flow in amperes.
The actual current flow through a capacitor is calculated by I=Ea/Xc, where Ea is the applied or
working voltage.
The following examples illustrate how these variables effect the voltage/current limitation for
some specific capacitors at a given frequency.
EXAMPLE 1: A 0.1 pF, 500 Volt capacitor is to be used at 1000 MHz:
Xc=1/[2(3.14)(1000x10 6)(0.1x10 -12)]=1591 ohms;
Ipeak=500/1591=0.315 Apeak or 0.22 Arms.
If you exceed this current, you will exceed the voltage rating of the capacitor.
EXAMPLE 2: A 1.0 pF, 500 Volt capacitor is to be used at 1000 MHz:
Xc=1/[2(3.14)(1000x10 6)(1.0x10 -12)]=159 ohms;
Ipeak=500/159= 3.15 Apeak or 2.2 Arms.
If you exceed this current, you will exceed the voltage rating of the capacitor.
EXAMPLE 3: A 10 pF, 500 Volt capacitor is to be used at 1000 MHz:
Xc=1/[2(3.14)(1000x10 6)(10x10 -12)]=15.9 ohms;
Ipeak=500/15.9=31.5 Apeak or 22.2 Arms.
This current level is actually never reached because at this frequency, values over
approximately 1.5 pF are limited by the power dissipation of the capacitor.
IMPORTANT NOTE: The power dissipation limited data assumes an infinite heat sink at the
capacitor terminations. The heat transferred to the air is ignored.
The 10pF, 500V capacitor at 1000 MHz would actually be power limited to about 7Apeak
which is about 5 Arms. At this current level the temperature of the capacitor would rise to about
125 degrees C. For reliable operation, it would be practical to limit the current to 2 Arms with a
decent heat sink, or even 3Arms if there is a very good heat sink at the terminations.
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SRF & PRF AND THEIR RELATION TO
RF CAPACITOR APPLICATIONS
JAN 17, 1999
Capacitors exhibit both series and parallel resonant frequencies. There is a frequency for a
capacitor with a given physical size/construction and a given capacitance value at which the
component looks like an inductive impedance. Indeed, above this frequency one can have
essentially a “DC blocking inductor”. This frequency is called the Series Resonant Frequency
(SRF). The magnitude of the transmission impedance the capacitor presents will be extremely low
from the bottom frequency end determined by the capacitance value, right through the SRF until
one approaches the first Parallel Resonant Frequency (PRF). The first PRF is frequently called
simply the PRF, although in reality there is a second PRF, third PRF, and so on. A rough rule of
thumb puts the PRF at twice the frequency as the SRF. The magnitude of the transmission
impedance dip differs significantly at each PRF frequency, and with the various values of
capacitors.
Exactly at the SRF, one will have the lowest possible impedance, and therefore, a capacitor is
commonly used for narrow band RF bypass applications by choosing a capacitor whose SRF is at
the required bypass frequency. In other words, at the SRF, the capacitive impedance of the device
equals the parasitic inductive impedance of the device. The user is only left with the ESR (no
reactive component) at that frequency.
For applications where low impedance is desired, but it is unimportant if the capacitor looks
either capacitive or inductive, a capacitor could be used up to a frequency close to the PRF.
Above the SRF, the user essentially has a “DC blocking inductor”. At the PRF, the transmission
impedance goes relatively high, and the capacitor is virtually useless around this frequency. There
is a “trick” to effectively extend the usable frequency range in this application. That is to mount
the capacitor in a vertical position so that the capacitor terminations both still meet the pads
where it’s mounted, and so the width of the capacitor is now effectively the height. Doing this
will eliminate all odd-order PRFs including the first PRF.
0
Insertion Loss (db)
-1
-2
-3
-4
-5
Parallel Electrode Mount
-6
0
1
2
3
4
5
6
7
8
9
10
Frequency (GHz)
Figure A
Figure A illustrates normal mounting where the electrodes are parallel to the substrate surface
and the electrical effect for a given capacitor value. Figure B illustrates vertical mounting where
the electrodes are perpendicular to the substrate surface and the electrical effect for the same
given capacitor value.
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0
Insertion Loss (db)
-1
-2
-3
-4
-5
Vertical Electrode Mount
-6
0
1
2
3
4
5
6
7
8
9
10
Frequency (GHz)
Figure B
The PRF effects are due to very complex mechanisms related to the different phase paths to
the various electrodes of the multi-layer capacitor. At JTI, we typically measure the S-parameters
in a normal orientation showing the first PRF, allowing the user to optionally mount it vertically
which will eliminate the first PRF.
Of course, for matching applications, it is extremely important as to whether the component
looks like a capacitor or an inductor. In this application, a designer will almost always want to be
well below the SRF of the capacitor. We can say that, for example, a 10pF capacitor could look
like a 1000pF capacitor as the SRF is approached.
CP
Figure C:
Equivalent circuit of
a high frequency
ceramic chip capacitor
RS
CS
LS
RP
Typical Values
Cs = 4.8 pF
Ls = 0.5 nH
Rp = 78 Kohms
Cp = 5.2 pF
Rs = 0.11 ohms
See Figure C for Johanson Technology’s SPICE equivalent circuit for a capacitor. The values
shown are for a typical 10 pF capacitor at 900 MHz. This model tends to give a better SPICE
simulation than the more common model where Cs represents the essential C at low frequencies,
and Cp (if a Cp is even in the model) represents a fraction of Cs, even though this more common
model is more conceptually easy to understand. Keep in mind that even our improved model’s
accuracy is not ideal at RF frequencies because the elements of this model themselves have a given
dependence on frequency. In our model, at low frequencies, the parallel equivalent of Cs and Cp
give the essential capacitance. Cs and Ls in our model are what give us the SRF effect. The
network of Ls in series with Cs; which is in parallel with Cp is what gives us the first PRF effect.
Note also that our model (and the common model) has no provision for simulating a second PRF.
Rp is in our model for a more accurate simulation of what the capacitor looks like at RF
frequencies, so don’t put an ohmmeter across a capacitor looking for Rp at DC where the
Insulation Resistance will be typically greater than 10,000 Ohms! This model (as well as a more
“traditional” model) is reflected in our MLCSoft® SPICE simulation software which is available
on CD-ROM or from our web page.
IMPORTANT NOTE: Our value for Ls, when using the improved JTI model, may seem
slightly (about 50%) higher than a competitor’s advertised Ls because of the modeling difference
noted above. Be cautious not to compare Ls directly without realizing the difference in modeling.
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