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4.1 BASIC INDUCTANCE AND REACTANCE Analysis of real inductor and transformer behavior requires a mastery of basic accircuit concepts. A condensed treatment of these topics is given in this section and the ones following, both as a review and as a ready reference. Inductance: If a current is run through a coil of wire, a magnetic field is set up whose strength is proportional to the current and the number of turns. This is the principle of the electromagnet. If a magnetic field is moved through a coil of wire, a voltage is induced across the coil whose magnitude is proportional to the speed of movement and the number of turns. This is the principle of the generator. Figure 41(a) and (b) illustrates these effects. Combining these facts, if ac is passed through a coil, a changing (moving) magnetic field is set up, and this field induces a voltage in the coil that caused it. The direction of this selfinduced voltage is always such as to oppose the initially applied voltage. Thus a coil (we call it an inductornever use one syllable when you can use three) opposes any change in current. Reactance: Notice that the inductor's opposition to ac is by means of a reverse voltage, not a restriction in the conductive path as in the case of a resistor. We call this kind of opposition reactance as opposed to resistance. The main difference is that reactance does not burn energyit merely stores it on one quartercycle and then bounces it back into the source on the next quartercycle. This bounceback results in a phase shift between current and voltage for an inductor. Current through the coil reaches a maximum just as the voltage across the coil falls back to zero. We say that current in an inductance lags voltage by 90°. This is shown in Fig. 41(c).
The amount of reactance that a coil presents to ac depends on the value of the inductance and the speed with which the current alternates (frequency): X_L=2 pi fL (41) ... where XL is the reactance in ohms, f the frequency in hertz, and L the inductance in henrys. Series and Parallel Reactances: Two inductors in series have a total reactance equal to their sum (just like series resistors), provided that their magnetic fields are not close enough to interact. This is illustrated in Fig. 42(a). Capacitors store energy in an electric field, and when connected to an ac source, simply bounce this energy back to the source on alternate quartercycles. Thus they present a reactance to ac, much like an inductor. However, there are two important differences. First, the current in a capacitance leads the voltage by 90°; just the opposite of the inductor. We give a negative sign to the reactance of a capacitor: XC 2pi fC where C is the value of the capacitance in farads. Second, as shown by equation 42, capacitive reactance varies inversely with frequency f, and capacitance C, again just the opposite of the inductor. This means that as frequency increases, inductive reactance goes up while capacitive reactance goes down. Calculations involving reactances are, therefore, valid only at the frequency for which reactance has been calculated. Capacitive and inductive reactances in series are subtractive, as illustrated in Fig. 42(c) and (d). Reactances in parallel are governed by the same formulas as resistors, namely, the product over the sum and the reciprocal of the reciprocals. In this case, however, attention must be paid to the signs. Figure 43 gives some examples. 4.2 BASIC CONCEPTS OF IMPEDANCE Series Resistance and Reactance: When resistance and reactance are placed in series, the result is termed impedance. Part of the energy delivered by the source is then dissipated as heat (in the resistance) and part is reflected back to the source (by the reactance). Current will lag applied voltage if the net reactance is inductive, and lead if the net reactance is capacitive, but by an angle between zero and 90°. The total value of the impedance Z and the exact phase angle 9 are determined by phasor addition of X and R, as shown in Fig. 44. The calculations proceed as follows: (43) Notice that the total circuit current is in phase with the resistor voltage and lags the applied voltage by 37°. A few trial calculations with equation 43 will show that, if Xs is three times Ra or more, the total impedance is only 5% greater than Xs. Likewise, if Rs > 3XS, Z is greater than Rs by only 5%. We will often find it helpful to neglect the lower of Xs or Rs in these cases, so ZmRs if Xs is small, or ZsmX3 if Rs is small. Parallel Resistance and Reactance: When resistance and reactance appear in parallel, the equivalent impedance is always less than the smaller of the two, but never smaller than 0.707 times the lower one. The phase shift is characteristic of the reactance involved: total current lagging applied voltage for inductance, leading for capacitance. The exact impedance and phase angle can be calculated: (46) In the parallel circuit it can be shown that, if Xp is at least three times Rp (or vice versa), the impedance of the parallel combination is less than the lower value (Xp or Rp) by only 5%. Here, again, we will often find it appropriate and helpful to disregard the higher parallel value (Xp or Rp ), so ZmXp if Rp is large, or ZznRp if Xp is large. Such simplifications can be of great help in analyzing real inductor and transformer equivalent circuits. A few examples are given in Fig. 45. The exact Z, by equation 45, Is 195 ohm. Equivalent Series and Parallel Circuits: Any parallel RX circuit can be replaced by an equivalent series RX circuit using the following equations: R=X„ X=Rr Xp Rp , X? + R>p X p Rp Conversely, any series RX circuit can be replaced by an equivalent parallel circuit using the equations: (410) Of course, these equivalencies are valid only at the particular frequency for which the reactance has been calculated. Also note that all of the reactances are to be placed into these formulas as absolute (unsigned) quantities. Although capacitive reactance has been termed negative, it can be subtracted only from inductive reactance ( XL  Xc valid), not from resistance (R  Xc invalid). Using the reactance combination methods illustrated in Figs. 42 and 43 and the equivalency formulas 47 through 410, it is possible to reduce most circuits containing numerous R, XL, and Xc components to a single series or parallel RX equivalent. EXAMPLE 41 Figure 46(a) shows a seriesparallel RLC circuit. It happens to represent a real transformer, but we’ll get to that later. For now, determine the power delivered to R2, which represents the load. Solution First it is necessary to calculate the reactances: 663 a XL2 and Xc can be parallelcombined: +i5AQ Xli + Xc 75.4  663
Now the parallel combination of R2 and Xp can be converted to its series equivalent: The equivalent circuit now stands as in Fig. 46(b), and consists of a total series resistance and inductive reactance: RsT=Ri +Rs= 12 + 39.1 =51.1 ohm X,t = Xli +XS 22.6 + 25.7 = 48.3 l ZT = Jx?T+ R1,t = V48.32 + 51.12 =70.3 ohm The total current is … This current a flows through the impedance of C, R2, and L2 in parallel, producing a voltage across R2: 4.3 BASIC RESONANCE CONCEPTS This section discusses the effects of resonance and tuned circuits. After that we we’ll be equipped to start talking about real inductors. Series Resonance: You may have noticed that in the example of Fig. 42(d), the inductive and capacitive reactances exactly canceled, leaving a net reactance of zero. This shortcircuit condition prevails at only one frequency, called the resonant frequency, which is given by ... [411] ... is in hertz, L is in henrys, and C is in farads. At lower frequencies the capacitive reactance predominates and at higher frequencies the inductive reactance predominates. We therefore have a series tuned circuit which can be placed in series with a line to pass only the resonant frequency or in shunt across the line to trap out the resonant frequency. Series Circuit Q: Of course, the series tuned circuit is not a perfect short circuit at resonance. The coil has some resistance, perhaps increased by the skin effect (Section 1.4) if the frequency is high, and at very high frequencies the capacitor may have an appreciable dissipation factor, which can be represented as additional series resistance according to equation 34. The ratio of the reactance of either the coil or the capacitor at resonance to the total effective series resistance is termed the quality factor or Q of the series tuned circuit: … at resonance (412) When a series tuned circuit is used in series with a load resistance and / or a source resistance, the entire series resistance must be included as Rs in determining Q. This is illustrated in Fig. 47(a).
Bandwidth Bw is defined as the span between the 3dB (0.707 voltage) points. Bandwidth: The current through a series tuned circuit is maximum at resonance, and decreases above or below that frequency as net inductive or capacitive reactance appears in the circuit. Seriescircuit analysis, as illustrated in Fig. 44, can be used to determine the current response at any frequency, and a graph of the results appears in Fig. 47(b). Of particular interest are the frequencies at which the response falls by 3 dB (to 0.707 of maximum). These are commonly termed the lower (f1) and upper (f2) cutoff frequencies or halfpower points. The frequency span between them is called the bandwidth and is approximated by Bw = f_r/Q (413) The approximation grows better as Q goes higher, being about 5% off when Q = 5. Bandwidth is easy to measure experimentally, and narrow bandwidth (high (?) is generally a desirable feature in a tuned circuit. At frequencies beyond f1 and f2, the response can be determined from the following table: Voltage Rise at Resonance: Notice that in the circuit of Fig. 47(a) the total impedance seen by the source at resonance is 125.6 Q purely resistive, and the current is V/R = 126.5 V / 126.5 ohm =1 A. This current flows through Xc and XL, producing voltages across each of them, even though their reactances cancel. The voltages produced are: VXL = Vxc = IXc = 1 A X 628 =628 V These voltages are larger than the source voltage by exactly a factor of Q (five times in this case). In general, Vxl=Vxc=QVc (414) … for a seriesresonant circuit, where VG is the opencircuit generator voltage and Q is calculated using the entire circuit resistance. Notice that with Q = 50 and VG = 100 V, the voltage rating required of the capacitor becomes 5000 V. This is no idle theory! The voltage rise at resonance must be considered when selecting components for seriesresonant circuits. Of course, since VXL leads the circuit current by 90° and Vxc lags by 90°, the two voltages are 180° out of phase and exactly cancel each other. Thus the voltages across the circuit elements Rc, C, L, Rc, and RL in Fig. 47(a) add up to 125.6 V, which is the generator voltage. Parallel Resonance: In the circuit of Fig. 43(c), equal capacitive and inductive reactances were placed in parallel, and the resulting reactance rose to infinity. This was because the two currents (IL = V/XL and Ic = V/Xc) were equal and exactly 180° out of phase, resulting in zero total current. Figure 48(a) shows a real parallelresonant circuit with the inevitable series resistance of the inductor. Equations 49 and 410 can be used to change Ls and Rs to their parallel equivalent, but if Q is reasonably high, we can operate with the much simpler approximations: (415) (416) The error incurred with Q_coil = 5 is 4%, dropping to 1% at Q = 10.
Qp Rp(tot)/Xp. The Q of the entire circuit can now be determined from the equivalent circuit of Fig. 48(b) with Rc, RL, and Rp appearing in parallel: The maximum output voltage at resonance is: (418) The curve of Fig. 47(b) can be used to determine the output at other frequencies. Notice that on highimpedance lines (Rc and RL above 100 i or so) the highest Q is obtained with a parallelresonant circuit across the line, while low impedance lines are more sharply tuned with a seriestuned circuit in series with the line. Also notice that with the parallel circuit, a low L/C ratio (small inductance, large capacitance) will produce lower reactances at resonance, and hence higher Q, according to equation 417, while the series circuit will tune more sharply with a high L/C ratio, according to equation 412. Some of the advantage of increasing L in the series circuit is offset by the inevitable increase in Rs of the coil when its number of turns increases. However, L increases by the square of the turns, while Rs increases linearly, so substantial benefits may be realized. Resonant Traps: Figure 49(a) and (b) show, respectively, a seriesresonant trap across the line and a parallelresonant trap in the line, together with formulas for their Q and minimum output at the null point. Again, these formulas are approximations whose accuracy falls off markedly below a Q of 5. It is also assumed that Rs is much less than either Rc or RL. Notice that the depth of the null is completely dependent upon Rs, whereas the sharpness of the null is dependent upon the ratio of reactance at resonance to line impedance. As with the peaking tuned circuits, seriesresonant traps tune most sharply with a high L/C ratio on a lowimpedance line, whereas parallelresonant traps have highest Q with a low L/C ratio on a highimpedance line. EXAMPLE 42 Design a tuning circuit with a  3 dB passband from 110 to 140 kHz. The source resistance is 40 kOhm and the load impedance is essentially infinite. A 2.5mH coil with a Q of 8.0 at 125 kHz is available. Solution Solving equation 411 for C gives the required capacitance: The Q required if the parallelresonant circuit of Fig. 48(a) is used is obtained from equation 413: = 4.17 FIGURE 49 Resonant traps: (a) series tuned across the line; (b) parallel tuned in series with the line; (c) response curve showing bandwidth and null depth. The parallel resistance needed to produce this is obtained by equation 417: XL  2 pi fL= 1960 ohm Rp = QX = 4. 17 X 1960 = 8.17 k Ohm The coil's series resistance and its parallel equivalent, by equation* 412 and 416, are: The effective parallel resistance of Rc and Rp is 40 k ohm  15.68 k ohm, or 11.26 k Ohm. A shunt resistance, sufficient to bring the parallel total down to the necessary 8. 17 k ohm, must now be added in the R L position of Fig. 48(a). The value is calculated by the product over the difference: The practice of shunting a tuned circuit to lower its Q and widen the bandwidth is in common use. 4.4 REAL INDUCTORS For the sake of discussion and analysis we will place inductors into four categories, all of which are shown in Fig. 410. AirCore SolenoidWound Coils: These may commonly range in value from 0.1 uH to several mH, having one to several hundred turns. They may be selfsupporting in air, or wound on a ceramic, plastic, or cardboard form. Their diameter depends upon inductance required and power level, but commonly ranges from 0.5 to 5 cm, with still larger sizes appearing in highpower transmitters. When used in frequencysensitive tuning and filtering circuits, they are called coils, and their exact inductance and its stability are likely to be critical. When used simply to block high frequencies they are called rf chokes, and wider tolerances and lower Q, s are generally acceptable. Coils with OpenPath Magnetic Cores: These are wound on forms commonly between 0.5 and 2 cm in diameter, and have powdered iron or metal oxides (ferrites) in the core to increase the inductance. The core may be an adjustable slug on screw threads, providing variable inductance. Their values range to a few hundred millihenrys. Coils with ClosedPath Magnetic Cores: These are wound on a form consisting of strips (laminations) of iron, as shown in Fig. 414, and are often enclosed (potted) in a steel cover. They contain several hundred to several thousand turns of wire, and common values range from a few millihenrys to several hundred henrys. Common sizes are from 2 to 10 cm on a side, although much larger units are available for highpower applications. They are generally used to block the flow of relatively lowfrequency ac (either audio or linefrequency ripple in a power supply) while passing dc. Tolerance is generally rather broad (  20, + 50% typical) and Q is relatively low. They can be designed with an adjustable air gap in the core to provide variable inductance. Toroidal Inductors: Toroids are wound on donutshaped cores of magnetic materials specially formulated for temperature stability, low loss, and high permeability. Nonmagnetic cores may be used for microhenry values. Because the turns cannot be wound on a simple rotating drum or bobbin, toroids are considerably more expensive than af chokes of equal inductance. However, by proper selection of core material and winding, they can achieve a remarkably high Q, often above 100 in the frequency range 1 kHz to 1 MHz. Their tolerance, linearity, temperature stability, and Q are generally far superior to inductors wound on laminated iron cores. Predicting Inductance: The inductance of an aircore coil of known dimensions can be predicted to within about 10% by the formula [419] where L is in microhenries, d is diameter in centimeters, l is length in centimeters, and N is the number of turns. The equivalent with d and l in inches is (420) Figure 411 illustrates the application of the formula. The formula is most accurate for single layer coils of 10 turns or more, but is still useful for smaller or multilayer coils. For ironor ferritecore coils, be sure to see Section 4.6 on core effects.
Turn spacing center to center is designated t. Usually, the question is not "calculate the inductance of a given coil, "but rather " wind me a coil of a given value." Solving the foregoing equations for N with tN substituted for /, gives the somewhat unwieldy turnswinding formula for singlelayer closewound coils: (421) where N is the number of turns required, L is the desired inductance in microhenries, d is coil diameter in centimeters, and t is the wire thickness (including insulation) in centimeters. The same formula, with d and t in inches, becomes: (422) For multilayer coils wound on a bobbin of a given length, the required number of turns is given by direct transposition of equations 419 and 420: (423) (424) Stray Capacitance and Resonance: Adjacent turns of wire in a coil have a certain amount of capacitance between them, the total effect being very much as if a single capacitor were placed across an ideal inductor, as in Fig. 412(a). This, of course, gives the coil a natural parallelresonant frequency, above which its reactance is capacitive and decreases with frequency. Using a coil at onefifth of its resonant frequency will result in an apparent inductance increase of 4%. At onethird the resonant frequency the increase is 11%, and at 0.5 fr it is 25%. The coil is still useful as a choke up to and beyond parallel resonance, however, since its impedance is still very high. As frequency is increased above the parallelresonant point, the impedance (capacitive) of the coil will drop until a point is reached at which the stray capacitance from the bulk of the coil becomes seriesresonant with the first turn or so on the end of the coil. Impedance at this natural seriesresonant frequency drops to near zero as shown in Fig. 412(b). There may be several more seriesresonant points above the first one, but the coil is useless even as a choke above the first one. When a wide range of frequencies is to be blocked, two chokes in series may be used. The first, placed at the "hot" end, is small and has a very high seriesresonant point. The second, placed at the "cold" or ac ground end, is large and has its seriesresonant point at a frequency where the reactance of the first choke is high.
A griddip meter coupled to the opencircuited coil will detect the parallel resonant point. With the terminals shorted, the dip meter will respond to the seriesresonant points. The shorting wire may have to be looped once around the dip meter's coil to provide adequate coupling. Figure 412(c) shows four techniques for reducing stray capacitance and hence raising the natural resonant frequency of a coil. The first is simply to keep the turns spaced apart, since capacitance varies inversely with conductor spacing. Of course, this spacing also decreases the inductance by increasing length (see equation 419), but if the spacing is limited to onehalf the wire diameter or less, significant improvement can be realized. Spacing can be achieved by using specially grooved forms, by winding a string along with the wire to hold the space, or by simply using Tefloninsulated wire in place of the usual thin enamel insulation. Common PVC (polyvinyl chloride) insulated hookup wire should never be used for winding coils. PVC has severe dielectric loss at certain frequencies which are a function of temperature, and serious reduction in the Q of the coil will result. Bakelite coil forms should be avoided at radio frequencies for similar reasons. The other techniques for raising f_s illustrated in Fig. 412(c) are (2) keep the starting and finishing coils of a twolayer coil at opposite ends of the form, (3) use a diagonalpitched winding, alternating direction with each layer so the wires are close only at the crossing points, and (4) wind the coil in separate sections (called pies), and build up the required inductance by seriesconnecting the required number of these lowvalue high/. coils. This last technique has the drawback of requiring more wire than a single bulk coil because there is no mutual inductance between windings of different sections. The result is higher winding resistance and lower Q. Estimating the stray capacitance of a coil is difficult because it depends upon so many factors: insulation thickness and dielectric constant, coil spacing, layering, winding pattern, wire thickness, and coil diameter primary among them. However, for singlelayer closewound coils using No. 32 to No. 18 enameled wire on 1 to 3cm forms, capacitance will generally be in the vicinity of 0.05 pF/turn. Multi layered coils may have less than half that capacitance. Spaced winding and crosspitch winding can reduce the capacitance by about 25%. Skin Effects and Q: In the absence of nearby objects, such as shields or other components, the Q of an aircore coil depends almost entirely on the resistance of the wire at the frequency in question. The formulas for skineffect resistance of copper wire are therefore reproduced from Section 1: (14) ... where 8 is skin depth in centimeters, t is wire diameter (thickness) and fI is wire length, both in the same units, and / is in hertz. In winding coils for high frequencies it is necessary to consider the proximity effect as explained in Section 14. Some examples from the laboratory will serve to demonstrate the utility of the inductance and skineffect formulas. EXAMPLE 43 The core was removed from an audio choke. The remaining air winding had an average diameter of 3.0 cm and a length of 2.2 cm, and was wound of wire with a thickness measured as 0.014 cm. The dc resistance of the coil was 460 ohm. Predict the coils inductance and Q at 1000 Hz. Solution Interpolating from the wire table (Fig. 11), this wire has a resistance of about 0.011 ohm/cm. The resistance of one turn is then = 3.14 X 3 X 0.011  0.104 ohm The number of turns is then found from the total resistance: N = 4420 turns R1 0.104 The inductance, from equation 419, is _ 32 X 44202 _0j 4lBH 46d+ 101/ (46 x 3) + (101 x 2. 2) The skin depth at 1000 Hz is calculated from equation 13: = 0.21 cm This is 30 times the radius of the wire, so skin effect is negligible and the dc resistance is used: _ X 2nfL 6.28 X 1000 X 0.488 , _ V = R=R460 f f7 The inductance and Q of the actual coil, as measured on a commercial bridge at 1000 Hz, were 0.47 H and 6.6, respectively. EXAMPLE 44 Find the inductance and Q at 3 MHz of a 30turn closewound coil of No. 22 enameled wire on a 1.9cmdiameter plastic form. Solution: The thickness of the wire, from the table in Fig. 11, is 0.064 cm. Allowing 0.002 cm for the enamel, the length occupied by 30 turns is calculated: l = Nt = 30 X 0.068  2.04 cm The inductance, from equation 419, ... The skin depth, by equation 13, is ... This is 84 times less than the wire radius, so the skineffect resistance is calculated by equation 14: Wire length= lw [...] Checking the table in Section 14, we make a nottooconfident estimate of the proximityeffect resistance as ten times the calculated skineffect resistance, or 4 ohm. The measured values for L and Q of the actual coil at 3 MHz were 9.6 /tH and 41, respectively.  FIGURE 413 Actual reactance, resistance, and Q for a 1mH aircore threepie crosspitch choke. R holds at its dc value until skin effect causes it to rise as Vf . X increases as f until parallel resonance (9 MHz) brings it up and series resonance (35 MHz) brings it down. Q rises as f until skin effect begins, whereupon Q rises as Vf until parallel resonance, where it is maximum. Above f, the 1mH coil becomes swamped by the stray effects and is no longer accessible. Q readings above fr therefore apply to the stray effects and not to the coil as such.  The Q of a coil is not a constant, but varies with frequency in a generally predictable way, as shown in Fig. 413. At low frequencies, XL is low, although the resistance of the wire is equal to its dc value. Q is therefore relatively low. Q increases linearly with frequency until the skin effect begins to raise the value of R. Since XL goes up as /, while R goes up only as V/ . Q now rises as V?: Q =  cc  = Vf R Vf At the selfresonant frequency, Q is supposedly at a maximum, since the reactance of XL and Xc in parallel approaches infinity. However, the coil is not purely inductive at this point. The maximum usable Q should generally be read at 0.5 to 1/5 the frequency of the peak at fr. 4.6 MAGNETICCORE INDUCTORS Core Effect on Inductance: A magnetic field can pass much more readily through a magnetic metal such as iron or nickel than through air. This fact is demonstrated by the common tick of holding up a sting of a dozen straight pins from a single magnet at the top. We say that the magnetic metals have a higher permeability than air. To the extent that we can provide a highpermeability medium within and around a coil, we can increase its inductance for a given number of turns. Although the amount of increase depends upon the exact shape and composition of the core, we can estimate that a slug of magnetic material in the center of the coil will provide an inductance increase of seven times over an air core. Brass slugs are occasionally used in rf coils to provide an inductance decrease of up to 25%. Calculating Core Effects: The highest inductances are obtained by providing a complete magnetic flux path through and around the coil. Figure 414 shows the common methods for achieving this. The CI and Elshaped stacks of siliconiron strips insulated with varnish have long been popular and still prevail at powerline and audio frequencies, while the newer ferrite materials in the pot core and toroid ring structures generally offer superior performance in the ranges above a few kilohertz. The inductance of a cored coil with no air gap can be calculated with reasonable accuracy from the formula: (425) where N is the number of turns, n the relative permeability of the material, A the crosssectional area of the core in cm^2 , and I the effective magnetic path length in centimeters. Note that in the EI structure there are two parallel paths if the coil is wound on the center post, so the effective length is: WX + [XZ + ZY+ YW] / 2
For the potcore structure, the effective length and area are generally given by the manufacturer. AirGap Considerations: All of the structures in Fig. 414 except the toroid contain an air gap in the flux path. It may be less than 0.01 cm, due primarily to the insulating varnish coat on the laminations; or it may be intentionally made much larger, or made variable to allow adjusting inductance. The inductance of a coil having a magnetic core with an air gap can be calculated from ... (426) ... where l_g is the length of the gap in centimeters, L_c the effective core length in centimeters, and the other terms are as above. Values of u for various popular core materials are given in the table of Fig. 415.
Besides lowering the inductance, the presence of an air gap has the important effect of linearizing the core. Notice from Fig. 415 that for most materials p varies markedly from low to highlevel magnetization. This is further illustrated in the magnetization curve of Fig. 416(a). The air gap adds a constant and relatively large reluctance (the magnetic equivalent of resistance) to the magnetic path, tending to swamp out the reluctance changes of the core material. Nevertheless, it is common to find that an ironcore inductor measures 1 H at low signal levels and 2 H at higher levels. Waveform distortion can also be expected due to the nonlinear response of magneticcore inductors. Core Saturation: There is a limit to the degree of magnetization (molecular alignment) that a material can obtain, and this places a limit on the peak current that can pass through a magneticcore coil without causing saturation. If this limit is exceeded with ac, the coil suddenly stops generating inductive reactance to the ac and becomes (almost) a simple piece of wire. The result could be excessive current, overheating, and the destruction of the coil. The slope of the magnetization curve, Fig. 416(a), equals permeability, which is proportional to inductance. Notice that this slope drops to near zero as current I is increased. The value of flux density B in the core of an acexcited coil can be calculated from B =  (427) ... where B is in gauss, V is in volts, f is in hertz, N is turns, and A is the crosssectional area in cm^2. Notice that u and l do not enter this equation, since their effect on flux density via reluctance is exactly offset by a reciprocal effect on current via inductance. The value of B must be kept less than the saturation value given in Fig. 415. The flux density for a dcexcited coil is 1. 267V/ t . + Uv (428) where B is in gauss, N is turns, / is dc amperes, tg is airgap length in centimeters, lc is effective core length, and u is core permeability. EXAMPLE 45 An inductor is wound on a CIshaped core of siliconiron laminations. The legs of the lamination stack are 2 cm wide by 2 cm thick, and the overall dimensions of the core are 6 cm X 6 cm. An air gap of 0.05 cm is left in the core. The winding consists of 900 turns of AWG 24 enameled wire. Calculate the inductance at high and low excitation levels, and determine the maximum dc current before saturation. Solution Lowlevel n is 400, highlevel n is 40,000, lc is 4 x 4 or 16 cm, and A is 4 cm^2. Using equation 426, we obtain ... Transposing equation 428 to find current for the 15,000G saturation level yields [...] Measuring Saturation: Figure 4 16(b) shows an experimental circuit for measuring the dc saturation current of a choke. LT is the inductor under test, and the saturation current is measured on dc ammeter I. R must have a value of at least 3 X_lt and Xc must be 1/3 XLT or less. L1 must have a higher current rating than LT and XL1 must be 3XLT or more. L1 can be replaced by a resistor if these conditions are met. In operation, Vac is set to produce a small voltage, say 1 V rms, across L_T. Vdc is then increased until the measured ac voltage drops by a specified amount (usually 10%), indicating that has dropped by 10%. Useful Saturation: Core saturation can be used to advantage in some applications. A swinging choke in a power supply is designed to saturate at maximum dc load current. The resulting decrease in inductance allows more charging current to reach the first filter capacitor, thus largely offsetting the output voltage drop under heavy load. The magnetic amplifier uses a relatively small dc current through a manyturn coil to saturate a core upon which is wound an inductor carrying a heavy ac current. A small increase in the dc can thus cause a large increase in the ac. Core Effect on Q: The type of core material used in an inductor must be selected for the frequency range of intended use. Thin strips (laminations) of silicon iron are used at powerline and audio frequencies, with powdered iron, and then various compositions of metal oxides (ferrites) being preferred at progressively higher frequencies. The reason for the frequency dependence of the core is this: In magnetizing the core with ac, we are requiring the molecules of the core material to realign their positions every halfcycle. It takes a rather specially composed material to follow these realignments into the high kilohertz and megahertz ranges, and there is always some frictional loss (due to hysteresis) in the process. The Q of a magnetically cored coil is therefore likely to be higher than its aircore counterpart because of its higher XL, but only up to the maximum frequency of the core material, beyond which Q drops sharply due to hysteresis loss. Predicting Q for a magneticcore coil is quite difficult, since permeability and hysteresis vary with frequency and material composition. Measurement at the intended frequency is the most practical recourse.

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