NONIDEAL BEHAVIOR OF COMPONENTS
The impedance characteristics of common circuit elements (resistors,
capacitors, inductors) utilized in circuit theory are simply low-frequency
asymptotes of the overall frequency responses of these components.
Since typical EMC problems are characterized by a wide frequency range
of interest (including high frequencies), the circuit theory impedance
relationships for these components are typically inadequate. Thus, the lowfrequency circuit component impedance models must be extended to higher
frequencies in order to accurately model these components in EMC
problems. These broadband circuit component models should accurately
define the frequency response of these components up to those frequencies
seen in typical EMC problems. But, these models should not be overly
complex such that they are difficult to implement.
Another low frequency approximation assumed in circuit theory
which is inadequate for EMC problems is the assumption that component
interconnections [wires, printed circuit board (PCB) lands, etc.] have
negligible impedance. At higher frequencies, these interconnections
typically have significant resistance and reactance. Thus, the effect of these
interconnections must be included when modeling an EMC problem. In
particular, the effect of discrete component leads should be considered.
The effect of these component leads can be minimized by using surface
mount technology (SMT) where lead length is minimized.
INTERNAL IMPEDANCE OF ROUND WIRES
The internal impedance of a round wire of radius a can be determined
by manipulating Maxwell’s equations into the governing differential
equation for the electric field (or current
density) within the wire. The current is
distributed over the cross-section of the wire
according to the phenomenon known as the
skin-effect. According to the skin-effect, the
current tends to crowd toward the outer
surface of the wire at high frequency.
We start with the phasor (frequencydomain) form of Maxwell’s curl equations
within the conducting wire as given by
Taking the divergence of ã gives
since
Taking the curl of â and inserting ã gives
If we define the wavenumber k as
then the electric field within the wire satisfies the following equation:
According to the vector identity,
the governing equation for the electric field becomes
Thus, the electric field within the conducting wire satisfies the vector wave
equation in ä. For the special case of a cylindrical conductor of infinite
length and radius a lying along the z-axis, the current density (and electric
field) has only a z-component. This electric field is axially-directed and
rotationally invariant. The wave equation for the electric field within the
wire (in cylindrical coordinates) becomes
The differential equation governing the wire electric field is Bessel’s
differential equation of order zero. The solution to the differential equation
may be written in terms of Bessel functions.
where Eo is the electric field at the wire surface and J0 is the Bessel function
of the first kind and order 0. The current density inside the wire is given
by the product of the wire conductivity and the electric field, such that
The current distribution over the wire cross-section is frequency dependent
according to the frequency dependence of the wavenumber k. At low
frequency, the current distribution is nearly uniform [exactly uniform at
zero frequency (DC)] while at high frequencies, the current tends to crowd
toward the outside surface of the wire.
The internal impedance Zi of the wire (resistance
plus reactance) is found by determining the ratio of
voltage to current. The phasor voltage V between the
ends of the conductor of length l is determined by
evaluating the line integral of the electric field along the
path L on the surface of the wire from point A to point B.
The total current I is found by integrating the current
density (J = óE) over the cross-section of the wire.
The internal impedance of the conductor becomes
The internal impedance per unit length of the conductor is
The real part of the per unit length wire impedance is the wire resistance
per unit length while the imaginary part (which is positive) is the wire
internal inductance per unit length. The internal impedance of the wire,
like the current distribution, is frequency dependent according to the
definition of the wavenumber k.
The low frequency and high frequency behavior of the wire internal
impedance can be determined by using the small argument and large
argument forms of the Bessel functions in the equation. The Bessel
function of the first kind can be written as a power series according to
Keeping the first two terms in the series gives the small argument forms of
the Bessel functions of the first kind of order 0 and 1.
where ka n1. Inserting the small argument forms into the wire internal
impedance formula gives
At low frequency in a good conductor (óoùå), the square of the
wavenumber is approximated by
which, when inserted into the low frequency wire internal impedance
equation gives
Thus, the low frequency resistance and inductance (per unit length) are
Note that the low frequency resistance is the standard DC resistance per
unit length formula while the low frequency inductance per unit length is
a constant value which is independent of the wire radius. Given a wire
made of nonmagnetic material (ì = ìo = 4ð × 10!7 H/m), the low frequency
inductance per unit length for the wire is 0.5 × 10!7 H/m = 50 nH/m [1.27
nH/inch].
At high frequencies, the arguments of the Bessel functions in the wire
impedance formula become large. Thus, we use the large argument forms
of the Bessel functions to find the high frequency asymptotes. The large
argument form of the Bessel function of the first kind of order n is
so that the large argument forms of J0(ka) and J1(ka) are
The wire internal impedance per unit length at high frequency becomes
For a good conductor at high frequencies such that (óoùå), the
wavenumber may be approximated by
Inserting the high frequency approximation for k into the wire impedance
formula, the first complex exponential terms in the numerator and the
denominator both approach zero at high frequency. This gives
The high frequency asymptotes for the wire internal impedance are then
Note that the wire resistance at high frequency increases as the square root
of frequency while the wire internal inductance decreases as the square root
of frequency.
The size of a round wire is typically defined according to the wire
gauge and the American Wire Gauge (AWG) is the most commonly used
definition. The radius or diameter of an AWG gauge wire is typically given
in the English units of mils where 1mil = 0.001 inch. For example, #12
AWG wire has a diameter of 80 mils (see Table 5.2 on page 302-303)
which corresponds to a wire radius of
HIGH FREQUENCY WIRE RESISTANCE
APPROXIMATION - SKIN DEPTH
The high frequency approximation for the wire resistance can be
interpreted in the same manner as the low-frequency (DC) resistance by
comparing the two equations. That is, as the current crowds toward the
outside surface of the wire at high frequency, we can define an equivalent
high frequency area AHF which, if a uniform current density is assumed,
would yield the same resistance.
Solving for AHF yields
where ä is defined as the skin-depth and given by
Note that the equivalent high-frequency
area AHF is the circumference of the wire
(2ða) times the skin-depth ä. Thus, the
actual high frequency resistance is obtained
by assuming a uniform current density over
the outermost portion of the wire crosssection to a depth of one skin depth.
The high frequency approximations for the wire resistance and
internal inductance may be written in terms of the low frequency
approximations and the skin depth.
Note that the high-frequency and low-frequency approximations are equal
when a = 2ä. Thus, the frequency at which the wire radius is equal to two
skin depths represents the break frequency fo (where the low and high
frequency asymptotes meet). Solving for the break frequency gives
A simple representation of the per-unit-length wire impedance frequency
response can be plotted using the low and high frequency asymptotes of r
and li vs. a logarithmic frequency scale. The low frequency asymptotes of
r and li are both constants. At high frequencies, r is directly proportional
with the square root of the frequency while li is inversely proportional to
the square root of the frequency. On a logarithmic frequency scale, the
high frequency asymptote of r increases at 10 dB/decade while the high
frequency asymptote of li decreases at 10 dB/decade.
The break frequency of a #12 AWG copper wire (ó = 5.8×107 S/m,
ì = 4ð×10!7 H/m, a = 1.016 mm) is
Below this frequency, the low frequency asymptotes for the resistance and
internal inductance are accurate, while above this frequency, the high
frequency asymptotes are accurate.
The high frequency resistance per unit length of a conductor of noncircular cross-section can be approximated by applying the skin depth
concept. The current density in any conductor tends to crowd toward the
outer surface of the conductor at high frequency. We may use the model
of one skin depth of uniform current density around the outer periphery of
the conductor to determine the high frequency resistance. For example, the
low and high frequency resistances per unit length of a PCB land of width
w and thickness t are
where a uniform current density is assumed for the low frequency
approximation.
The PCB land is normally specified by the weight of the copper
cladding which is etched away to form the land. The copper cladding is
normally designated in terms of “ounces”. This designation refers to the
weight of 1 square foot of the cladding. That is, one square foot of 1 ounce
copper cladding would weigh 1 ounce. The most common copper cladding
thicknesses are 1 and 2 ounce copper (t = 1.38 and 2.76 mils, respectively).
[35.05 and 70.10 ìm, respectively].
Just as the round wire has internal inductance, the PCB land also has
internal inductance. However, the computation of the rectangular
conductor internal inductance is complicated by the fact that the current
distribution is a complex function of two variables. In the following
section, it is shown that the external inductance for most conductor
configurations is much larger than the internal inductance. Thus, the
internal inductance of a conductor can be neglected in most EMC
applications. As shown for the round wire, the internal inductance of any
conductor decreases with frequency.
EXTERNAL INDUCTANCE, CAPACITANCE AND
CONDUCTANCE OF PARALLEL WIRES
The wire resistance and internal inductance determined in the
previous sections are quantities associated with a single wire. The external
inductance and capacitance, on the other hand, are associated with a pair of
wires such as that seen in a two-wire transmission line. The transmission
line current flows in one conductor, through the termination, and returns
through the opposite conductor. The external inductance represents the
magnetic flux linkage per unit current in the conductors. The capacitance
between the conductors represents the charge per unit voltage on the
conductors. If the conductors are closely spaced, the current and charge on
each conductor are influenced by the current and charge on the opposite
conductor. Under these conditions, the current and charge tend to crowd
in the region between the conductors. This phenomenon is known as the
proximity effect.
Consider a two-wire transmission line with conductors of radius a and
(center-to-center) spacing s in a homogeneous medium as shown below.
The per unit length capacitance c for the two-wire transmission line,
accounting for the proximity effect, can be shown to be
If the wires of the transmission line are sufficiently far apart (s $ 5a), the
current and charge distributions on the two wires are nearly uniform and the
per unit length capacitance between the wires may be approximated by
Given any two conductor uniform transmission line in a homogeneous
medium carrying the TEM mode (transmission line mode), the per unit
length capacitance, external inductance and conductance are related by
Thus, the per unit length external inductance and conductance for the twowire line (accounting for the proximity effect ) is
For sufficiently spaced conductors, the per unit length external inductance
and conductance for the two wire line may be approximated by
Note that c, le and g are all independent of frequency, unlike r and li.
Example (two-wire line / per unit length parameters)
Determine the per unit length parameters (r, li, c, le, and g) for
a two-wire air line consisting of #12 AWG copper wires with a
separation distance of s = 7mm at f = 1 MHz.
The transmission line is being operated well above the break
frequency for r and li for the #12 AWG copper wires (fo = 16.92 kHz
from previous results), thus we may use the high frequency
approximations.
The equations above for r and li are those for a single conductor. For
the two-wire line, we should multiply these terms by 2 to account for
both wires of the two-wire line. This gives
The wire spacing to radius ratio is roughly 7 so the c, li, and g
equations for widely spaced wires may be used.
Note that the per unit length external inductance of this two-wire line
is substantially larger than the internal inductance of the two
conductors. Thus, in most cases, the internal inductance of the wire
can be neglected when combined with the much larger external
inductance. Most transmission lines are constructed with very good
insulating materials between the conductors such that the per unit
length conductance for the transmission line can normally be assumed
to be zero.
For the special case of a two-wire transmission line with wires of
unequal radii (a1 and a2) and large conductor spacing (s$5a1 and s$5a2), the
per unit length capacitance, external inductance and conductance is
EXTERNAL INDUCTANCE, CAPACITANCE AND
CONDUCTANCE OF COAXIAL CONDUCTORS
The coaxial line is the most commonly used transmission line
configuration and is utilized in a wide variety of applications. Consider the
coaxial line with an inner conductor of radius a and an outer conductor
with an inner radius of b. Given an insulating medium between the
conductors characterized by material properties (ì, å, ó), the external
inductance, capacitance and conductance of a coaxial line are:
EXTERNAL INDUCTANCE AND CAPACITANCE
OF PRINTED CIRCUIT BOARD STRUCTURES
Various configurations of two conductor transmission lines can be
formed using standard PCB technology. Three of the these PCB twoconductor transmission line geometries are shown below. These
configurations are designated as (a.) microstrip (b.) coplanar strips and (c.)
opposite strips (d.) stripline.
These transmission line geometries will not support a true TEM mode but
will support what are known as “quasi-TEM” modes. The quasi-TEM
modes have essentially the same transverse field structure as the true TEM
mode. For each PCB transmission line geometry, note that the fields of the
quasi-TEM mode travel through an inhomogeneous medium (the fields lie
partly within the PCB itself and partly in the surrounding air). The
inhomogeneous medium requires that the per unit length parameters be
computed numerically. Given the results of the numerical solutions,
empirical formulas can be formed for the per unit length parameters of the
PCB transmission lines.
The empirical formulas for the PCB transmission line parameters
normally include a quantity known as the effective dielectric constant
designated by årN. The per unit length parameters of the actual PCB
transmission line are equal to those of the same PCB conductor
configuration when located in a homogeneous medium characterized by årN.
The external inductance and capacitance of the actual PCB transmission
line can be accurately determined by assuming a lossless transmission line
(no conductor losses and no leakage current). Thus, the velocity of
propagation on the PCB transmission line is given by
while the characteristic impedance of the transmission line is given by
By combining these two equations, the per unit length capacitance and
external inductance can determined directly from the transmission line
characteristic impedance as
The characteristic impedance formulas developed for the four PCB
transmission line geometries (assuming zero thickness lands) considered
here are given below.
Microstrip
Coplanar Strips
K - Complete Elliptic Integral
of the First Kind
Opposite Strips
(w/h >1)
(w/h <1)
Stripline
NON-IDEAL BEHAVIOR OF RESISTORS
Resistors used as discrete circuit components can be classified into
three basic groups according to the resistor construction: (1) carbon
resistors, (2) wire-wound resistors, and (3) thin-film resistors.
Carbon resistors - the most common resistor type, a cylinder of
carbon material with wire leads connected at each end of the carbon
cylinder, cheap and easy to fabricate, low resistor tolerances of 510%.
Wire-wound resistors - resistive wire wound on an insulating tube
(for space reasons) which dissipates heat (normally porcelain), more
difficult to fabricate and more expensive than carbon resistors, higher
precision than carbon resistors, large inductive component.
Thin-film resistors - a thin metallic film (usually a meandering line of
film) is deposited on an insulating substrate with leads connected to
the conducting film, high precision, lower inductance than wire
wound resistors but more than carbon resistors.
Resistors also contain leakage capacitance due to charge leakage along the
resistor body. The equivalent model of a resistor must include the
dominant impedance components associated with the resistor construction
along with the effect of the component leads. The equivalent circuit for the
typical resistor is shown below.
The lead capacitance and the leakage capacitance can be combined in
parallel to form the total parasitic capacitance of the resistor component.
Using the parasitic capacitance definition, a simplified form of the resistor
equivalent circuit is shown below.
The impedance of the resistor equivalent circuit (using s = jù) is
Example
Plot the frequency response (impedance magnitude in dB vs. log
f ) of a 1 kÙ resistor with #20 AWG leads that are 0.75 inches long
and separated by a distance of 0.25 inch. Assume the leakage
capacitance is 1.2 pF.
#20 AWG Y a = 16 mils = 0.4064 mm
s = 0.25 in. = 6.35 mm, l = 0.75 in. = 19.05 mm
NON-IDEAL BEHAVIOR OF CAPACITORS
There are a wide variety of capacitor types with regard to their
construction. Some of the most commonly used capacitors are: (1) ceramic
capacitors, (2) mica capacitors, (3) plastic-film capacitors, (4) aluminum
electrolytic and (5) tantalum electrolytic.
Ceramic capacitors - small disk-shaped capacitors, can achieve only
small values of capacitance (1 pF to 0.1 ìF, typical), multilayer
ceramic capacitors up to 1 ìF, non-polarized, cheap and easy to
fabricate, low precision, relatively low leakage current, good high
frequency characteristics.
Mica capacitors - dielectric layer of mica coated with a conductor and
dipped in epoxy, large size, small values of capacitance (1 pF to 0.01
ìF, typical), non-polarized, high precision, relatively expensive, low
leakage current.
Plastic-film capacitors - a dielectric layer of thin plastic (polystyrene,
polyester, polycarbonate, polyethylene and others), moderate values
of capacitance (1 nF to 1 ìF, typical), non-polarized, moderate
precision, leakage current characteristics depend on type of plastic
material used, inexpensive, coiled or multi-layer geometry.
Aluminum electrolytic capacitors - aluminum electrodes separated by
an electrolyte, an extremely thin layer of oxide (dielectric) is
deposited on one aluminum electrode, large values of capacitance (1
ìF to 10 mF, typical) in small size, polarized, low precision, more
expensive, high leakage current.
Tantalum electrolytic capacitors - same as the aluminum electrolytic
capacitor except the electrodes are made of tantalum, electrode can be
wet or dry, low precision, large values of capacitance (0.1 ìF to 100
ìF) in very small size, expensive, low leakage, better high frequency
characteristics than aluminum electrolytic capacitors.
In EMC applications where radiated or conducted emissions are to be
suppressed using capacitors, the capacitor characteristics drive the selection
of the capacitor type to be used. Since ceramic capacitors offer near-ideal
capacitor behavior up to higher frequencies, ceramic capacitors are
typically used for the suppression of radiated emissions. Tantalum
electrolytic capacitors are typically used in conducted emission suppression
problems because of their large values of capacitance and small size.
The equivalent model of a realistic capacitor must include the
resistance of the conducting plates (Rplates) and the resistance of the
dielectric (Rdielectric) in addition to the element capacitance (C). The
dielectric resistance should model both the ohmic losses in the dielectric
and heating losses in the dielectric due polarization losses. Combining
these capacitor components with those components that model the effect of
the capacitor leads (Clead, Llead) yields the capacitor equivalent model shown
below.
The dielectric resistance Rdielectric is typically so large that it may be
modeled as an open circuit while the capacitance of the connecting leads
is typically very small in comparison to the element capacitance such that
the lead capacitance can be neglected. These approximations yield a simple
series RLC circuit model for the capacitor including the lead inductance,
the plate resistance and the element capacitance.
The impedance of the capacitor equivalent circuit is
Note how the impedance equation of the capacitor equivalent circuit
compares with that of the ideal capacitor. At low frequencies, the ideal
capacitor term in the capacitor equivalent circuit model is dominant. In
fact, the frequency response of the capacitor equivalent circuit can be
represented by the product of the ideal capacitor impedance and a second
order term representing the non-ideal behavior of the capacitor.
Example
Plot the frequency response (impedance magnitude in dB vs. log
f ) of a 0.1 ìF capacitor with #20 AWG leads that are 0.75 inches
long and separated by a distance of 0.25 inch. Assume the plate
resistance is 1 Ù.
From the previous results for the non-ideal resistor (the same
lead dimensions were assumed),
s = 6.35 mm, l = 19.05 mm
The resonant frequency fo in the capacitor equivalent circuit is
referred to as the self-resonant frequency of the capacitor element. Note
that the self-resonant frequency represents the critical frequency below
which the capacitor operates with near-ideal characteristics while above the
self-resonant frequency, the capacitor acts like an inductor. Also note that
increasing the capacitance in this model reduces the self-resonant frequency
which reduces the bandwidth over which the capacitor acts like a capacitor.
NON-IDEAL BEHAVIOR OF INDUCTORS
The common characteristic of all inductors with regard to their
construction is the geometry of coiled conductors in order to concentrate
the magnetic field. The resistance of the coils is considered a parasitic
component of the inductor impedance and designated as Rparasitic. The
proximity of the adjacent inductor coils introduces a parasitic capacitance
component into the inductor equivalent impedance. This parasitic
capacitance, designated as Cparasitic, increases significantly when spacesaving winding techniques (such as multiple layers of coils) are employed.
The equivalent circuit for the inductor is shown below, including the
impedance of the connecting leads.
The lead inductance is typically much smaller than the element inductance
such that we may neglect Llead. Also, the parasitic capacitance of a typical
inductor is significantly larger than the lead capacitance, under most
circumstances. Thus, the equivalent model of the inductor can be
approximated by a series combination of the element inductance and the
parasitic resistance in parallel with the parasitic capacitance of the inductor.
Note that the effect of the element leads is much less critical for the
inductor than for the resistor or the capacitor. Nonetheless, a highly
accurate model of the inductor frequency response would require that the
lead impedance be included in the model.
The simplified version of the inductor equivalent circuit (neglecting
the lead inductance and the lead capacitance) is shown below.
According to the approximate inductor equivalent circuit, the impedance
of the inductor is given by
According to the inductor equivalent circuit impedance expression,
the inductor does not operate as an ideal element at very low frequencies.
In fact, the inductor acts like a resistor at very low frequencies. According
to the form of the numerator expression in the inductor impedance, there is
a critical frequency at f1 = Rparasitic/(2ðL) where the impedance of the
inductor is equal to that of the resistor. Above this frequency, the inductor
impedance dominates that of the resistor and the normal low-frequency
approximation for the inductor is valid until the frequency nears the selfresonant frequency of the inductor. This self-resonant frequency is given
by
Above the self-resonant frequency of the inductor, the impedance of the
capacitor becomes small in comparison to that of the element inductance
and the parasitic resistance. Thus, at sufficiently high frequencies, the
inductor behaves like a capacitor.
Example
Plot the frequency response (impedance magnitude in dB vs. log
f ) of a 100 ìH inductor. Assume the parasitic resistance is 1 Ù and
the parasitic capacitance is 1 pF.
The critical frequencies for the inductor model are:
Inductors are frequently wound on ferromagnetic cores. A
ferromagnetic material is one with a large relative permeability ìr. The
relative permeability is a measure of how much magnetization occurs in the
material. Ferromagnetic materials are highly nonlinear. This means that
the relative permeability of the material is not actually constant but varies
with the magnitude of the applied magnetic field. Ferromagnetic materials
have that property that the relative permeability decreases as the size of the
applied field increases. Thus, in an inductor with a ferromagnetic core, as
the inductor current increases, the magnetic field applied to the core
increases, and the relative permeability of the core decreases. Since the
inductance is directly proportional to the relative permeability, we find that
the inductance of the component L decreases as the current is increased.
NOISE SUPPRESSION WITH CAPACITORS AND INDUCTORS
Capacitors and inductors can be used effectively for the suppression
of noise signals under certain circumstances. In general, the low
impedance of the capacitor at noise signal frequencies can be used to shunt
noise currents away from a particular path while the high impedance of the
inductor can be used to block noise currents from a particular path.
However, several factors must be considered when selecting the noise
suppression component. Included in these factors are:
(1) the circuit impedance characteristics at the location where
the noise suppression is needed,
(2) the frequency spectrum of the operational and noise
signals in the circuit at the noise suppression location,
(3) the size of the noise suppression component, and
(4) the self resonant frequency of the noise suppression
component.
Consider the following scenario for noise suppression. A pair of
lands on a PCB carry an operational signal current plus a noise current
given by
where the frequency content of the noise signal is assumed to be higher
than that of the operational signal. As shown below, a noise suppression
capacitor (Co) is to be placed between the conductors to shunt the noise
signal and pass the operational signal. A simple equivalent circuit can be
determined by replacing the terminated PCB land pair by its equivalent
input impedance.
The equivalent circuit is shown below along with the current division
expressions for the current components.
In a practical sense, the capacitor current should be as close to the noise
current as possible making the output current approximately equal to the
signal current. This current distribution is achieved if
which occurs when
Thus, we must carefully select the magnitude of the noise suppression
capacitance to yield the proper impedance characteristics at the signal and
noise frequencies. These impedance values depend on the impedance of
the circuit at the location of the noise suppression component placement.
Note that the shunt noise suppression capacitor is most effective when
placed in a circuit at a high impedance location.
For a shunt noise suppression capacitor to satisfy the required
impedance characteristics, the self-resonant frequency of the capacitor
should be sufficiently high relative to the noise frequency to ensure nearideal capacitance characteristics at the noise signal frequency. The selfresonant frequency of a capacitor was previously shown to be
In order to place a shunt noise suppression capacitor at a low impedance
location in a circuit, we must satisfy the relationship that
which requires the use of a large value of capacitance C. This large value
of capacitance results in a lower value for the capacitor self-resonant
frequency, causing the capacitor to become ineffective when its selfresonant frequency is located below the noise frequency.
A noise suppression capacitor is placed in parallel with the signal
conductors in order to shunt the noise currents located on the conductors.
A noise suppression inductor must be placed in series with the signal
conductors in order to block the noise currents. The connection of a series
noise suppression inductor (Lo) is shown below.
The voltage between the PCB land pair is assumed to be the
superposition of an operational signal voltage and a noise signal voltage.
Prior to the introduction of the series noise suppression inductor, the
current into the PCB land pair may be written as
The equivalent circuit after the introduction of the series noise suppression
inductor is shown below along with the resulting current relationship.
The signal current before and after the introduction of the noise suppression
inductor should be approximately equal while the noise current should be
essentially eliminated by the introduction of the inductor. This relationship
is achieved if
Just as with the shunt noise suppression capacitor, we must carefully select
the magnitude of the noise suppression inductance to yield the proper
impedance characteristics at the signal and noise frequencies. These
impedance values depend on the impedance of the circuit at the location of
the noise suppression component placement. Note that the series noise
suppression inductor is most effective when placed in a circuit at a low
impedance location.
For a series noise suppression inductor to satisfy the required
impedance characteristics, the self-resonant frequency of the inductor
should be sufficiently high relative to the noise frequency to ensure nearideal inductance characteristics at the noise signal frequency. The selfresonant frequency of an inductor was previously shown to be
In order to place a series noise suppression inductor at a high impedance
location in a circuit, we must satisfy the relationship that
which requires the use of a large value of inductance L. This large value
of inductance results in a lower value for the inductor self-resonant
frequency, causing the inductor to become ineffective when its selfresonant frequency is located below the noise frequency.
Summary (Noise suppression with capacitors and inductors)
Circuit locations with high impedance Y use shunt capacitor
Circuit locations with low impedance Y use series inductor
COMMON MODE AND DIFFERENTIAL MODE CURRENTS
Given a realistic system that must meet EMC standards, the currents
encountered on parallel conductors in these systems exhibit characteristics
that cannot be described using circuit theory alone. The general currents
on a parallel conductor system can be written as the superposition of two
types of current: common-mode currents and differential-mode currents.
Differential-mode currents, as predicted by circuit theory for closed
loops, are equal currents that flow in opposite directions (such as those
predicted by transmission line theory). The differential-mode currents
normally represent the functional currents in the system.
Common-mode currents, which cannot be defined by circuit theory,
are equal currents that flow in the same direction. Common-mode currents
are sometimes called antenna-mode currents. The common-mode currents
normally represent the noise currents in the system. The common-mode
currents in a given system are typically much smaller than the differentialmode currents.
The parallel conductor currents can be expressed in terms of the
differential-mode and common-mode currents as
Solving for differential-mode and common-mode currents gives
The orientation of the differential-mode and common-mode currents dictate
how efficiently these currents radiate electromagnetic waves. Differentialmode currents, being closely-spaced currents flowing in opposite
directions, radiate inefficiently. Common-mode currents, which flow in the
same direction, radiate much more efficiently even though the commonmode current amplitude may be much smaller than the differential-mode
current amplitude (assuming the length of the conductor pair is sufficiently
long to radiate effectively). Thus, common-mode currents are a much more
significant source of radiated emissions than differential-mode currents.
Example (Common-mode and differential mode currents)
Determine the common-mode and differential-mode current levels
given the measured currents in the two-wire system below.
FERRITES AND COMMON MODE CHOKES
The elimination of common-mode currents can be achieved through
the use of an element known as a common-mode choke. This noise
suppression device is designed to pass the desired differential-mode current
but block the unwanted common-mode current. The connection of the
common-mode choke in a two conductor system is shown below. The
current-carrying conductors are wrapped around a ferromagnetic core
where the orientation of the coils is critical to the operation of the commonmode choke.
The ferromagnetic core can be considered to be a “conductor” of
magnetic fields. That is, whatever magnetic flux is generated in the core
tends to follow the core and not leak out into the surrounding medium,
which is assumed to be air. Note that, given the orientation of the
conductor currents and the associated coils, the total magnetic flux terms
due to each current (ø1 and ø2) are in the same direction within the
ferromagnetic core. The common-mode introduces series self-inductances
into each of the conductors along with a large mutual inductance between
the two coils as the two coils are tightly coupled by the ferromagnetic core
of the common-mode choke.
The equivalent circuit of the common-mode choke within the twoconductor system is shown below where the device is assumed to be
symmetric. Each coil has a self-inductance L in series with each conductor
and the mutual coupling between the coils is defined by M.
The net series impedance introduced into the two conductors is
For common currents, we have
while for differential-mode currents we find
If the coupling between the conductors is assumed to be ideal (all of the
magnetic flux stays within the ferromagnetic core with no losses in the core
or conductors), then L = M. This yields a differential-mode impedance of
zero per conductor and a common-mode impedance of 2L per conductor.
Thus, the common-mode choke will pass the differential-mode signal and
block the common-mode signal if the value of L is chosen properly.
The common-mode impedance (~2L) of the common-mode choke
should be made as large as possible for effective attenuation of commonmode signals. This requires that the self-inductance of each coil in the
common-mode choke be large. Since the self-inductance is directly
proportional to the relative permeability of the core material, the relative
permeability of the core material must be sufficiently large at all current
levels of interest (a saturated core has a smaller relative permeability) and
all frequencies of interest. Also, if the relative permeability of the core
material is reduced, the mutual coupling between the two coils of the
common-mode choke is reduced which results in more leakage flux into the
air surrounding the core. This further reduces the effectiveness of the
common-mode choke.
The cores used in common-mode chokes are typically ferrimagnetic
materials (also know as ferrites). Ferrites have magnetic properties similar
to ferromagnetic materials (large relative permeability) but have much
lower conductivity. Thus, ferrites generate much lower eddy current losses
than do ferromagnetic materials. The combination of high relative
permeability and low conductivity is only found in compounds. Typical
ferrite core materials are nonconductive ceramics such as manganese zinc
(MnZn) and nickel zinc (NiZn). The frequency characteristics of the
relative permeability for these two materials are shown in Figure 5.28
(p.343). Note that while MnZn has a significantly larger value of relative
permeability than NiZn at low frequencies, the relative permeability of
MnZn drops rapidly after its peak value at approximately 100 kHz. At
higher frequencies, the relative permeability of NiZn is significantly larger
than that of MnZn. Thus, the selection of the material used in the core of
the common-mode choke is driven by the spectral characteristics of the
common-mode signals to be blocked. Based on the frequency
characteristics of MnZn and NiZn, it is clear that NiZn is the preferable
material for the core of a common-mode choke to be used for the
suppression of radiated emissions given the frequency range of interest.
While the common-mode choke is used to block common-mode
signals on a two conductor system, a ferrite bead is used to add a high
frequency impedance (inductance) in series with the conductors it encloses.
The basic geometry of the ferrite bead is shown below.
The ferrite bead increases the external inductance of the conductor passing
through the bead when compared to the external inductance of the same
conductor in air (the external inductance of a conductor in a homogenous
region is proportional to the permeability of the medium). In addition to
increasing the external inductance of the conductor, the ferrite bead also
provides significant magnetic losses in the form of heat. These magnetic
losses are more pronounced at higher frequencies. Thus, the impedance
introduced by the ferrite bead can be written as
Typical ferrite beads yield impedance magnitudes on the order of 100 Ù at
frequencies above approximately 100 MHz. Given this relatively low
impedance, ferrite beads are most effective when applied to low impedance
circuits.