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Self-resonant 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_Self-Resonant_Frequency
How_an_Impedance_Meter_Determines_Inductance
Three_Element_RLC_Model_Impedance_versus_Frequenc
Measured_Data_Fastron_07MFG-102J_1mH_RF_Choke
Measured_Data_-_Bourns_5800-102_RF_Choke
Can_an_RF_Choke_be_used_above_the_SRF
Simple_Three-Component_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 self-resonant 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 self-resonant frequency of an inductor.

Inductors, like all non-theoretical 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 three-component 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 Self-Resonant Frequency

The classical description of C is that it represents the turn-to-turn distributed capacitance of the inductor (and turn-to-core, etc.). At some frequency, the "self-resonant frequency" or SRF, this turn-to-turn 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 half-wave long. David Knight, G3YNH, makes a very persuasive argument for this viewpoint at http://www.g3ynh.info/zdocs/magnetics/appendix/self-res.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 turn-to-turn 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 laboratory-grade 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 three-element 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 07MFG-102J shielded RF choke and a Bourns (J.W. Miller) 5800-102. Neither has a self-resonant frequency identified in the data sheets. I use both of these parts in various kits I sell, including the Z10040B Norton amplifier.
 

 
Bourns 5800-102 choke plugged into the 16047A test fixture, attached to the 4192A impedance meter.
 

Measured Data Fastron 07MFG-102J 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 ω2LC = 1. But suppose we are at a lower frequency ω' such that ω' = XL = 1.1 XC.   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' = XL = 0.9 XC (above resonance) we find L' ≈ -L/10.

At resonance, the capacitive and inductive reactance are equal, so ω2LC = 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 Q-meter, 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 07MFG-102 inductor displays a self-resonant 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:

ω2 = 1/LC or

C=1/ω2L

C = 1/(2π*1.575X106)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 07MFG-102J are unidentified.


Measured Data - Bourns 5800-102 RF Choke

The Bourns 5800-102 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 07MFG-102J 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 07MFG-102J 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 07MFG-102J'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 multi-aperture 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 5800-102 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 Q-spoiler 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 Q-spoiling 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 Three-Component 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 103A-21 and computed the "apparent" inductance using a simple three element model. This inductor is a clone of the Boonton 103-series "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 103A-21 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 self-resonant 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 10-fold 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 103A-21 standard inductor

The image below shows that over the 100 KHz - 10 MHz range, this simple three-element model fits reasonably well to the measured data.
 
Looking in greater detail at the critical range around the self-resonant 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 5000-7000 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 impedance-based instrument.