NONIDEAL BEHAVIOR OF COMPONENTS

Transcription

1 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.

2 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 gives since Taking the curl of and inserting gives If we define the wavenumber k as then the electric field within the wire satisfies the following equation: According to the vector identity,

3 the governing equation for the electric field becomes Thus, the electric field within the conducting wire satisfies the vector wave equation in. 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 E o is the electric field at the wire surface and J 0 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.

4 The internal impedance Z i 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

5 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 1. Inserting the small argument forms into the wire internal impedance formula gives

6 At low frequency in a good conductor (ó ùå), 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 7 made of nonmagnetic material (ì = ì o = 4ð 10 H/m), the low frequency 7 inductance per unit length for the wire is 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 J 0(ka) and J 1(ka) are

7 The wire internal impedance per unit length at high frequency becomes For a good conductor at high frequencies such that (ó ùå), 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

8 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 = inch. For example, #12 AWG wire has a diameter of 80 mils (see Table 5.2 on page ) which corresponds to a wire radius of

9 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 A HF which, if a uniform current density is assumed, would yield the same resistance. Solving for A HF yields where ä is defined as the skin-depth and given by Note that the equivalent high-frequency area A HF 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.

10 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 f o (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 l i vs. a logarithmic frequency scale. The low frequency asymptotes of r and l i are both constants. At high frequencies, r is directly proportional with the square root of the frequency while l i 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 l decreases at 10 db/decade. i

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12 7 The break frequency of a #12 AWG copper wire (ó = S/m, 7 ì = 4ð 10 H/m, a = 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.

13 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 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.

14 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

15 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, l and g are all independent of frequency, unlike r and l. e i

16 Example (two-wire line / per unit length parameters) Determine the per unit length parameters (r, l i, c, l e, 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 l i for the #12 AWG copper wires (f o = khz from previous results), thus we may use the high frequency approximations. The equations above for r and l i 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 i The wire spacing to radius ratio is roughly 7 so the c, l, and g equations for widely spaced wires may be used.

17 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 (a 1 and a 2) and large conductor spacing (s 5a 1 and s 5a 2), the per unit length capacitance, external inductance and conductance is

18 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:

19 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.

20 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 år. 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 år. 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

21 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

22 Opposite Strips (w/h >1) (w/h <1) Stripline

23 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 5-10%. 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.

24 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

25 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 a = 16 mils = mm s = 0.25 in. = 6.35 mm, l = 0.75 in. = mm

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28 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.

29 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 (R plates) and the resistance of the dielectric (R dielectric) 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 (C lead, L lead) yields the capacitor equivalent model shown below. dielectric The dielectric resistance R 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.

30 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.

31 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 = mm

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33 o The resonant frequency f 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.

34 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 R parasitic. The proximity of the adjacent inductor coils introduces a parasitic capacitance component into the inductor equivalent impedance. This parasitic capacitance, designated as C parasitic, 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 L lead. 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.

35 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

36 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 f 1 = R parasitic/(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:

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38 Inductors are frequently wound on ferromagnetic cores. A ferromagnetic material is one with a large relative permeability ì r. 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.

39 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 (C o) 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.

40 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.

41 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 (L ) is shown below. o

42 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

43 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 use shunt capacitor Circuit locations with low impedance use series inductor

44 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.

45 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.

46 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.

47 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 (ø 1 and ø 2) 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.

48 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.

49 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.

50 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.