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Selfresonant
Frequency of an Inductor
or
Why does my LCR meter say the choke has "negative inductance"?
Revision History
23 August 2009. Original
24 August 2009 Added measured versus predicted for 100uH standard inductor;
added table of contents.
Table of Contents
Introduction
Distributed_Capacitance_and_SelfResonant_Frequency
How_an_Impedance_Meter_Determines_Inductance
Three_Element_RLC_Model_Impedance_versus_Frequenc
Measured_Data_Fastron_07MFG102J_1mH_RF_Choke
Measured_Data__Bourns_5800102_RF_Choke
Can_an_RF_Choke_be_used_above_the_SRF
Simple_ThreeComponent_Model_Compared_Against_Measured_100uH_Standard_Inductor
Introduction
A question was recently posed on the Agilent equipment
mailing list asking why a particular Agilent LCR meter showed an inductor with
"negative inductance."
The short answer to the question is that above the
selfresonant frequency of an inductor, it "looks like" a capacitor and since
capacitance is negative inductance, it all makes perfect sense. Although short
and correct, a more detailed discussion may be useful to those encountering for
the first time the selfresonant frequency of an inductor.
Inductors, like all nontheoretical electrical parts, are
not perfect. The schematic below shows a commonly used model of a real inductor,
of the type you might buy from a parts supplier or make from wire and a core. In
addition to the inductance L, the real part also has loss (modeled as a simple
resistance, shown in the model as R) and parasitic capacitance, shown as C. (It
is also possible, and desirable under some circumstances, to model the circuit
with parallel resistance. For simplicity, our discussion will consider the
series model only.)
This model has been criticized as being overly simplistic,
with a more detailed model being proposed at
http://www.edn.com/contents/images/159688.pdf.
For the purpose of this discussion, however, the
traditional threecomponent model suffices. And, although the model shows R, L
and C as "ideal" or theoretical components with no imperfections, in fact
in a real inductor R, L and C are anything but constant, perfect components. For
example, the skin effect causes the winding resistance to increase with
increasing frequency and, in an air core inductor, L will also change with
frequency as the distribution of current within the wires changes due to
proximity effect. If wound on a magnetic core, then the permeability of the core
will also have a frequency dependent component, which means L is a function of
frequency. And, the core will have a loss that is also frequency dependent,
which means another factor that changes R with frequency. 
Classical model of a real inductor, consisting of
inductance L, series resistance R and parasitic capacitance C. These values
are not constant with frequency in the general case.


Distributed
Capacitance and SelfResonant Frequency The classical description of C is that it represents the
turntoturn distributed capacitance of the inductor (and turntocore, etc.).
At some frequency, the "selfresonant frequency" or SRF, this turntoturn
capacitance resonates with the inductance L and the inductor becomes a parallel
resonant tuned circuit.
This traditional viewpoint
has been challenged and the inductor analyzed as a transmission line, with the
SRF frequency being determined as the frequency corresponding to the wire used
in L being a halfwave long. David Knight, G3YNH, makes a very persuasive
argument for this viewpoint at
http://www.g3ynh.info/zdocs/magnetics/appendix/selfres.html and I recommend
it to any interested in the subject. In fact, all of Dr. Knight's web site is
worth detailed consideration. I have made some measurements of air wound
inductors confirming G3YNH's SRF approach, and I hope to do more work in this
regard before reaching a conclusion. I believe, however, his analysis is worthy
of careful consideration.
For the purpose of measuring and describing a typical
store bought inductor, it makes little difference whether C is the turntoturn
capacitance or whether it is just a fictitious capacitance of a value computed
based on the SRF.

How an
Impedance Meter Determines Inductance First, we start by noting that modern laboratorygrade RLC
meters do not directly measure inductance or capacitance but rather measure
complex impedance, i.e., R+jX or Z and θ. (In some cases, admittance, the
inverse of impedance, is measured, but the concept is the same.) From the
complex impedance, the instrument can compute the inductance L and loss factor
Q, or capacitance and dissipation factor or the like.
If the instrument measures impedance in the form of Z and
phase angle θ, for example, and if the instrument is set to display L and Q,
then it computes L as:
L = X/ω
Q = X/R
where ω is the angular frequency in radians/sec or ω=2πf
where f is the frequency in Hz.
The table below, extracted from the instruction manual for
an HP 4192A impedance meter provides the formulas used for this conversion. The
series and parallel symbols in the top column rows indicate whether the 4192A is
making a series parameter (impedance) or parallel (admittance) measurement. 

When considered in the rectangular or Cartesian form,
impedance Z = R +jX where: R is the resistance in
ohms
X is the reactance in ohms
j is square root of 1.
X can be positive or negative. By convention, positive X
is considered inductive reactance and negative X is considered capacitive
reactance.
Our formula for computing inductance, however, has no
requirement that it is valid only for X > 0. Hence, a circuit with a measured
impedance of, say, 100 j100 ohms at an angular frequency of 1 radian/sec can
equally be interpreted as being a resistor of 100 ohms in series with an
inductor of
100 Henries or a capacitance of 0.01 Farads. Which is displayed depends on
whether the "show L" or "show C" function is selected on the instrument control
panel.
Of course, the radio parts catalog does not show negative
value inductors, but mathematically, at the measured frequency, an inductor with
a value of 100H behaves identically with a capacitor of value 0.01F.

Three Element RLC
Model Impedance versus Frequency Returning to the threeelement inductor model, at frequencies below the SRF, the
model appears to be inductive; at frequencies above the SRF it appears to be
capacitive and at the SRF it is resistive, as the inductive and capacitive
reactance are equal in magnitude but opposite in sign and thus cancel.
The
impedance Z of the model inductor can be shown to be (in R+jX form):
Let's look at two real inductors and see how they behaves when measured with
an HP 4192A impedance meter.
The two inductors are both 1mH nominal value, a
Fastron
07MFG102J shielded RF choke and a
Bourns (J.W.
Miller) 5800102. Neither has a selfresonant frequency identified in the
data sheets. I use both of these parts in various kits I sell, including the
Z10040B Norton amplifier.



Bourns 5800102 choke plugged into the 16047A test
fixture, attached to the 4192A impedance meter.


Measured Data Fastron
07MFG102J 1mH RF Choke
The plot below shows the inductance of the Fastron choke as reported by the
4192A. It rises sharply around 1500 KHz and then abruptly switches to a large
negative value and declines.
This is the classical signature of a resonant circuit.
As resonance is approached from the low frequency side, the parallel resonant
circuit apparent inductance increases. This can be seen from the equation with a
small bit of manipulation. (In order to avoid cluttering up this analysis with
even more algebra, I've made a number of simplifying assumptions in the
following discussion.) At resonance the capacitive and inductive reactance
are equal, so ω^{2}LC = 1. But suppose we are at a
lower frequency ω' such that ω' = X_{L} = 1.1 X_{C}.
At ω' therefore ω'^{2} LC = 1.1. Substituting ω'^{2} LC =
1.1 into the equation for Z, we find (approximately) that L' ≈ 10L where
L' is the apparent inductance or measured inductance computed from the Z
measurement. Likewise, if we let ω' = X_{L} = 0.9 X_{C} (above
resonance) we find L' ≈ L/10.
At resonance, the capacitive and inductive reactance are equal, so
ω^{2}LC = 1 and Z reduces to:
Z = R/ωC j/ωC. For the high Q case, j/ωC is negligible
and Z is resistive with a value of R/ωC. In essence, the resistance is
"magnified" by 1/ωC. (This is the principle of the Qmeter, after all.)
Thus we expect to see the apparent or measured L increase
near resonance, then reach zero at resonance (only a resistive component is
measured at resonance) and then switch to a high negative value just above
resonance, dropping with increasing frequency.


In order to estimate the distributed capacitance, it's handier to measure the
impedance in terms of Z and the associated phase angle, theta. At resonance,
theta is zero degrees, and it's usually easier to measure phase accurately than
look for an impedance peak.
(In fact, there are three definitions for resonance in a parallel LRC
circuit. Fortunately for high Q, the definitions are numerically close, so we
can choose the phase = 0 version of resonance.)
The Fastron 07MFG102 inductor displays a selfresonant frequency of 1575
KHz. 

At 100 KHz, far away from resonance, the measured inductance
is 0.9856mH.
If, and it's a major if, the inductance at 1575 KHz
remains the same as measured at 100 KHz, we may compute the distributed
capacitance:
ω^{2} = 1/LC or
C=1/ω^{2}L
C = 1/(2π*1.575X10^{6})^{2}*0.9856*10^{3}
= 10.4 pF
This is a plausible value.
Whether L is still around 1mH at 1.5 MHz is open to
serious question, of course, as ferrites are dispersive, i.e., the
permeability varies with frequency. Some ferrite materials exhibit a relatively
constant permeability with frequency, but the particulars of the ferrite used in
the 07MFG102J are unidentified. 
Measured Data  Bourns
5800102 RF Choke
The Bourns 5800102 RF choke exhibits a slightly lower SRF, at 1277 KHz. Based
on a 100 KHz measured inductance of 1.008mH, the computed distributed
capacitance is 15.4 pF. This figure, of course, is subject to the same caution
as in the 07MFG102J case.


Can an RF
Choke be used above the SRF? A logical follow on question to these measurements is "how
can an RF choke be used above its SRF"?
The answer
to this question should follow from examining the impedance magnitude versus
frequency in the plot below.


Over the measured frequency range 100 KHz to 10 MHz, the
07MFG102J maintains a useful impedance. Below the SRF, the impedance is
inductive and above the SRF it is capacitive. For many purposes, it matters not
whether the impedance is inductive or capacitive and hence the RF choke may be
used without regard for parallel resonance due to distributed capacitance.
Above the SRF, it can be considered to be an odd type of
capacitor�one that conducts DC, but with a capacitive reactance that is useful
in the same way an inductor is used, e.g. to isolate RF but allow DC to pass.
There are limits, of course, and as the frequency
increases above 10 MHz, the 07MFG102J's impedance will drop further, which may
be an issue in some applications.
And, there are applications where the circuit needs an
inductive impedance so that operation above a choke's SRF is unsatisfactory.
Where a broadband, inductive choke is needed, it is
possible to series connect several inductors so that at least one appears
inductive over the frequency range in question. In this regard, ferrite beads
can be quite useful to extend the frequency range upward. Careful selection of
the ferrite bead will be necessary, however, if it is to provide useful
impedance in the low MHz range. A multiaperture core will be useful in this
frequency range, if it can be found in the proper core material.
It's also possible to temper the large impedance and phase
excursions seen in these examples with a parallel resistor to work as a "Q
spoiler" i.e., to intentionally add loss. The plot below shows the
effects of adding a 1K ohm parallel resistor across the Bourns 5800102 RF
choke. The impedance now varies less than 2:1 over the frequency range 100 KHz 
10 MHz and the phase excursions are also reduced. The price paid is a lower
impedance than possible with the RF choke alone, of course. The Qspoiler does
not alter the SRF nor the capacitive nature of the impedance above the SRF, but
there are occasions when an RF choke with a relatively flat frequency
characteristic is desired. In those cases, a Qspoiling resistor can be
beneficial. It goes without saying, of course, that
the reason one does simply use the 1K resistor by itself is that the DC pass
characteristic of the RF choke is preserved.


Simple ThreeComponent Model Compared Against Measured 100uH Standard Inductor
Returning to the question to the adequacy of a simple
inductor model, I measured a standard inductor, Bellaire Electronics model
103A21 and computed the "apparent" inductance using a simple three element
model. This inductor is a clone of the Boonton 103series "working inductors"
apparently made under US government contract for the Department of Defense. Its
specifications are:
Working frequency: 800 KHz  2000 KHz
Inductance: 100µH
Distributed Capacitance: 6pF (marked 6µµF, of course.)
Approximate Q: 200
The test fixture I used to connect the 103A21 inductor to
the HP 4192A adds another 0.80pF, so the total parallel capacitance is 6.8 pF.
For the nameplate inductance of 100μH and 6.8pF distributed capacitance the
selfresonant frequency may be calculated as 6.10 MHz.
The simplest inductor model is illustrated below. The
major flaw in this simple model is the series resistance which I have based upon
a Q of 200 at 1 MHz. The series resistance will be the least constant of these
three parameters and may well increase by a factor of 10 or so from 100 KHz to
10 MHz. Skin effect resistance is proportional to the square root of the
frequency, so over this 100:1 frequency range skin effect alone will account for
a 10fold change in conductor resistance. In addition, the other loss elements
lumped into the series resistor, such as the loss in the coil former, are not
constant over this wide frequency range. Still, we see that a very simple model
with these flaws provides a more than acceptable view of the inductor's
performance.

Model for Bellaire Electronics 103A21 standard inductor


The image below shows that over the 100 KHz  10 MHz range, this simple
threeelement model fits reasonably well to the measured data.


Looking in greater detail at the critical range around the
selfresonant frequency (which we compute at 6.1 MHz), the three element model
shows quite decent fit to the measured data. The main divergence is related to
our simplistic fixed resistance loss. 

By adjusting the series resistance to 50 Ohms, we can
make the simple model fit the measured data almost perfectly over the frequency
range 50007000 KHz. In fact what we need is a model in which R is a function of
frequency, but even the simple fixed R model works surprisingly well over a
limited frequency range when it comes to predicting the apparent inductance, as
measured by an impedancebased instrument. 

