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Stanford_ee344_ch1_components

Stanford_ee344_ch1_components - 12mm EE344 l—Hsla...

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Unformatted text preview: 12mm: EE344) l—Hsla Frequencb (4190mm , Set-mafonl Um‘v.) I‘MS. m. Mc whorlrer) D. Caterer, H.9Nd4n. agar! 1?? pxrmssmp. . Chapter 1 THE CHARACTERISTICS OF CIRCUIT ELEMENTS AT HIGH FREQUENCIES—M.MCWhorter SECTION 1-1 DUMPED ELEMENTS The purpose of this section is to acquaint the student with the behavior of."ordinary" circuit elements at the higher frequencies. A good place to begin is to define what is meant by high fre- quencies in the context of this course. To a mechanical engineer high frequencies might be a rela- tively few Hertz while to a microwave engineer high frequency might mean 10—30 GHz so some sort of definition for our purposes is needed. We are going to consider the frequency range where the "pure" circuit elements (R,L, and C) are hard to realize because of parasitic elements or where ‘ elements are best thought of in terms of distributed parameters like transmission lines or line res- onators. However, we shall not consider elements where modes other than the TEM (Transverse Electromagnetic Mode) are important. This range of frequencies includes frequencies where trans- mission line theory is useful, but we shall not consider waveguides or cavity resonators. The actual frequency limits depend upon the size of structures that can be built: for example, using hybrid circuits and chip transistors it is quite possible to consider frequencies of 3—5 GHz or more. Because of our test equipment we shall limit our work to <1300 MHzl. 1-1.1 RESISTORS AT HIGH FREQUENCIES A resistor made of a slab of resistive (or semi-conducting) material has the d—c resistance shown in Fig. 1—1. The ends are assumed to be equi—potentials and in practice might be a conduct— ing material fired or painted onto the resistive material. The d-c resistance of the resistor is given by the equation in Fig. 1—1. Such a slab resistor is similar to those actually used in hybrid circuits, and it or the cylindrical version shown in Fig. 1—2 are useful in visualizing the problems that are likely to exist at higher frequencies. Note that the current flowing through the resistor will set up ‘ magnetic lines of force around the resistor so that energy will be stored in the external magnetic - c d s rf ‘ 0" - u aces Fig. 1-1: Geometry of a slab resistor. The resistance R = r01/w-h where r = the resis- i tivity of the material in Q—cm. h ‘ w T-I/ field. This effect is normally accounted for by assuming a series inductance which accounts for the energy storage of a non—ideal resistor. In addition the current flowing through the resistor produces a voltage drop which in turn sets up an electrostatic field across the resistor. This field also stores energy, this time in proportion to the square of the voltage across the resistor. Such energy storage may be accounted for by considering the ideal resistor to have a parallel capacitor. Both the capacitor and the inductor are functions of the physical geometry of the resistor; e.g., making the resistor shorter reduces the inductance (see under the discussion of inductors.) The equivalent circuit for the resistor thus formed is shown in Fig. 1-3. The effects of the 1For an explanation of the TEM mode see Ref. 1, pp. 41-44. This reference also shows the effect of higher order modes. , 1—1 Fig. 1—2: Fields about an ordinary cylindrical 4 resistor [like a carbon composition resistor.] Fig. 1-3: An approximate equivalent circuit for a resistor including parasitic elements. - Q ”‘8 ‘?:-a 8 a ._. o w [— ._. O N Impedance Magnitude-Ohms _ 5 ,__O 105 106 107 108 10 Frequency—Hertz .— O “L I— o .u I ‘3» Fig. l—4; The impedance and phase of various values of a resistor vs frequency. The assumed shunt capacitance is 0.5 pF and the series inductance is 10 nl-I. parasitic elements L and C depend upon their values and the value of the resistor. A typical [/4 W resistor has about 10 nH of inductance (if the total length including the leads is about 1 cm) and 0.5 pF of capacitancel. Figure 1—4 shows the magnitude and phase of the impedance versus fre— quency for various values of R. Ideally 2 should not vary with frequency. Note that for high values of R the shunt capacitance predominates—for the case of the 1 M9 resistor the impedance is becoming capacitive below 100 KHz. (The magnitudes of the impedance shown assume that you don't worsen the situation by adding more capacitance or inductance from your wiring.) A low resistance like 10 ohms is not affected by the shunting capacitance until f > 1 GI-Iz, but the series inductance does increase the effective impedance at high frequencies. Note that for resis- tors in the range of 100 ohms the inductive and capacitive effects tend to balance and the impedance value stays relatively constant over a wide range of frequency. This is one reason wideband circuits tend to be built around this impedance level. Another reason is the availability of coaxial lines in this impedance range.) At very high frequencies the series inductance predominates using this particular equivalent circuit; however, at these frequencies the circuit is inadequate because the elements are in actuality distributed. . The preceding analysis has two important defects that must be recognized: the first, as men— tioned, is that both L and C are really distributed elements, and the lumped circuit is an idealization. The second is that at higher frequencies the current does not flow uniformly throughout the body of the resistor. This manifestation of skin effect tends to increase the resistance with frequency, and will be discussed further in connection with transmission lines. The value of C can be measured with a bridge capable of measuring parallel components at high frequencies. The value of R can be measured using a r—f bridge capable of measuring series components or using the S—Parameter test set. 1-1.2 CAPACITORS AT HIGH FREQUENCIES A capacitor may be defined in terms of the ratio of charge stored (Q) to voltage across the cap- acitor (V) by C = Q/V. [If the capacitor is non—linear, it may be defined at some operating point V0 as C = dQ/dV.] An ideal capacitor stores energy in an electric field only and with no loss. The energy stored is W = 0.5'C-V2. At low frequencies the loss is often expressed as a power factor. If the loss is thought of as a series resistor (rs) then the power factor is given by: PF = cos[tan"(2-1t-f-rs-C)] =cos[tan‘1((o-rS-C)] [1—1] (O-I's'C [1-2] For lowloss capacitors where the power factor (5 the dissipation factor, also) is almost exactly l/Q where Q is the capacitor Q. Capacitor losses may also be represented by a parallel resistor, but this is usually done only at low frequencies. [Note that the value of the resistor for the two repre- sentations is very different] The loss in a capacitor arises from the resistance in the metallic plates and leads, and from the dielectric loss of the material between the plates. The dielectric loss of a wide range of materials is given in Ref. 8. Some representative values of the dissipation factor (D) for dielectrics widely used for insulators and capacitors are given in Table 1—1 where D is measured at 100 MHz and K Ill _—______——————-— 1Typical values of the parasitics for a small chip resistor are L = 1.2 nI-l, C = 0.03 pF. These values were measured at Hewlett Packard [HP]. Since this type of element is largely used with strip lines, the resistor may also be treated as a lossy transmission line. u u = a U a: o M Fig. 1-5: Reactance Chart 10 GHz Frequency The chart may be used to find the reactance of a given inductor or capacitor at a given frequency. or the res- onant frequency of a given inductance and capacitance. To Find Reactance: Enter the charts vertically from the bottom (frequency) and along the lines slanting upward to the left (inductance) or to the right (capac- itance). Corresponding scales (upper or lower) must be used throughout. Project horizontally to the left from the intersection and read reactance. To Find Resonant Frequency: Enter the slanting lines for the given inductance and capacity. Project downward from their intersecuon and read the resonant frequency from the bottom scale. Corresponding scales (upper or lower) must be used throughout. From the Radio Engineers Handbook. F. E. Terman Table 1—1 Air - Epoxy-glass Formica XXl Polyethylene Polystyrene is the dielectric constant. The capacitance of two parallel plates separated by a dielectric is given by C 0.0884-K-A/d [C in pF] [1—3] Where K: dielectric constant from Table 1—1, A = area of plates in cm2, and d = distance between plates in cm. The capacitance is in pF. This equation neglects the fringing fields and therefore gives a somewhat low value for C. II A 1 cm square section of copper on a 1/16" (0.159 cm) thick epoxy glass board with the aver- age dielectric constant of 5.0 gives 2.8 pF. At 100 MHz this represents only 570 ohms of capaci- tive reactance. [A convenient way to estimate reactances and resonant frequencies is to use the re- actance chart of Fig. 1—52.] Equations for the calculations of the capacitance of many other elec- trode configurations are given in F.E. Terman, Ref. 7, pp. 109-119. In addition to the loss of a non—ideal capacitor, there will be inductance due to the leads and perhaps the electrodes themselves if the capacitor is poorly made for h—f use. Therefore the equi- valent circuit for a non—ideal capacitor looks like Fig. 1—6 where the loss resistor (R), the lead inductance (L), and the desired capacitor (C) are all in series. The effect of L is to increase the apparent capacitance (i.e., lower the effective impedance) for frequencies less than the series reso- nant frequency, f0: f0 = [1—4] 1 2412-4 L-C At f0, 2 = R, and above fo the capacitor acts like an inductor approximately equal to the induc- tance of a wire connecting the two terminals of the capacitor (assuming little internal inductance in the capacitor which is usually true in a good h—f capacitor.) This reversal of the behavior of the capacitor at high frequencies can be very troublesome and is well illustrated in Fig. 1—7. The ideal capacitor shows a linear decrease in impedance with frequency (on log—log scales). The non—ideal capacitor in this example is: C = 10 nF, R = 0.5 Q, and L = 10 nH—values typical for a small ceramic disc capacitor with 1 cm total lead length3. The resonant frequency of this combination is 15.9 MHz, and the irnpedance is a minimum at this frequency. Above 15.9 MHz this capacitor and lFormica XX is also known as a phenolic-paper dielectric which is commonly used for cheap circuit boards. Its high loss makes it unsuitable at high frequencies. 2The chart may be used to find the reactance of both inductors and capacitors as well as the resonant frequency of their combination. The directions for use of the chart on on the bottom of the figure. 3Typical values of the parasitic elements for a NPO ceramic chip capacitor are 2 nH and 0.2 (1 [HP communication]. For a 100 pF capacitor this results in a self resonant frequency of 360 MHz. 1—5 Fig. [~6: An equivalent circuit for a capacitor. 0 I E 5:! S. E M-I 0 g 'E a? z 104 . . . . . . . 102 103 10" 105 10*5 107 108 109 Frequency-Hertz Fig. 1-7: The impedance of four different non-ideal capacitors. The assumed series resistance and inductance are the same for both: R = 0.5 Q and L = 10 nH. its leads behave like a 10 nH inductor. Note that using a larger capacitor [0.1 uF] decreases the impedance at low frequencies but not at high, assuming the same lead lengths. Sometimes the res- onance can be used to advantage to obtain a very low impedance (as a bypass capacitor, for exam- ple) at one frequency. The performance at other frequencies may be worse, however. In addition to these effects on the capacitor one should also know parameters such as the tem- perature coefficient, etc. These considerations are discussed in Ref. 8, but in general the best capacitors for stable h—f use are made with air [or vacuum], mica, low temp. coefficient. ceramic (NPO), or polystyrene dielectrics. Bypass capacitors which need only present a low impedance over wide ranges of frequency can be made of high—K ceramics which are usually not very tem- perature stable. The high-K ceramic capacitors have the advantage of being physically small for a given C; therefore their parasitic inductance can be kept small. 1—1.3 INDUCTORS AT HIGH FREQUENCIES Inductance is that property of an electric circuit that opposes the change of current through the circuit. An inductor stores energy in a magnetic field. The energy stored is W = L-12/2 where W is the stored energy in Watt—seconds or Joules. Any current carrying conductor has a magnetic field around it and therefore has self-inductance. The conductors may be geometrically arranged to increase the L over that of straight wire. The self-inductance of a straight wire (which must be a part of a closed path so that current can flow) at high frequencies where the current flows primarily on the surface of the wire is: L = 0.00508-L-[Ln(4-L/d)—l] [L in pH] [1—5] where L is in pH, L is the length of the wire in inches, and d is the wire diameter in inches. Some typical values for small lead wires are given in Table 1—2. (The 1 mil = 0.001 inch = 0.025 mm 1—6 Table 1—2 Wire Diam. (mils) 32 Wire au e #20 diameter wire is typical for the internal lead into an integrated circuit.) In general, the inductance of a lead wire is minimized by keeping the wire as short and broad as possible. To make a high quality air core inductor a common form is a cylinder wound with wire in a uniform spiral. If the radius of the wire spiral (measured to the center of the wire) is (r), the winding length (L), and the number of turns (n), the approximate inductance is r2-n2 . L 9-r+lO-L “- m “H1 ' [1—6] where L is in pH and the dimensions are in inches. The formula is accurate to 1% if L > 0.8-r, i.e., the coil is not too short. The inductance of many other configurations is given in Terman, Ref. 7. A real inductor typically is less ideal than a good capacitor. The resistance of the wire produces loss which varies with frequency because of skin effect. The effect of the loss can be approxi- mated by adding a series or shunt resistor to the equivalent circuit for the inductor. The quality factor or Q or the inductor is given by Q _ 2-1t-f.L _ g1; ‘ Rs ' Rs where R5 is the effective resistance in series with the inductor‘. High Q is often desirable and is maximized by making the inductor large physically and using large diameter wire. Spacing the turns on the order of the wire diameter tends to increase the Q (and lowers the distributed capacity.) The Q of an inductor is very much a a function of frequency because of the (o in Eq. 1-7 and because R5 is proportional the square root of frequency due to skin effect. These two effects tend to make inductor Q rise with frequency which is true up to some frequency where other effects such as losses due to radiation cause the Q to decrease with frequency. There is, therefore, some frequency where the Q is a maximum for any inductor. [1-7] Rs L Fig. 1-8: An equivalent circuitfor an inductor. Cp In addition to resistance the inductor also has capacitance between each turn and groups of turns. This distributed capacity can approximated by a single capacitor across the entire equivalent circuit giving the equivalent circuit shown in Fig. 1—8. 1The value of Rs takes into account the effects of skin effect on wire loss. any losses due to radiation from the inductor. and losses due to dielectrics in the electric field around the inductor. Therefore its calculation is very difficult. If the resistor accounting for the loss is considered to be in parallel with the inductor, then Q = lip/(tn -L). l—7 Magnitude of Impedance-01mm 10-1 ............. a ................. 3 .................. =. ................. a .................. .................. a ..... 102 103 10‘ 105 105 10" 108 109 FrequencyI-Iertz Fig. 1—9: The impedance of atypical inductor with frequency. [R5 = 0.5 9, CI, = 5 pF] The impedance and phase of a typical inductor versus frequency are shown in Fig. 1—9 for several values of inductance, but the same series resistance [Rs = 0.5 Q] and parallel capacitance [Cp = 5 pF.] In Fig. 1-9 the rising straight line is shows the region where the inductor is acting like his supposed to. Note at low frequencies the real inductor becomes purely resistive—proba- bly not as badly as shown because the d—c resistance is far less than Rs at 10 MHz. The dis— tributed capacitance which is assumed to be 5 pF causes a parallel resonance at the frequencies where the impedance has a maximum. Note that the effect of C is to increase the effective induc- tance below the resonant frequency. Far above the resonance the impedance of the "inductor" is essentially determined by the shunting capacitance. If the coil has a magnetic core, it usually will determine the shape of the Q vs. frequency curve. Note that some inductors are used to carry d—c currents to parts of a circuit. These are usually called r-f chokes. Such a choke is often needed to carry d—c bias voltages to a transistor. In the operating frequency range a choke should provide a high impedance, although not necessarily inductive, so as not to interfere with the operation of the rest of the circuit. Knowledge of a typical impedance curve as in Fig. 1-9 should help design such a choke. The stability of an air-core inductor is primarily determined by its physical dimensions. Therefore winding the coil on a dimensionally stable form is helpful, and the wire should be under some tension so that it does not move when the temperature changes. Inductors need not be air—cored even at high frequencies. Various magnetic materials may be used to increase the inductance obtained for a given physical size and number of turns. The mater- ials chosen must minimize losses due to hysteresis and eddy currents in the core material. In the range of 10 KHz to about 30 MHz powdered iron cores may be used. Ferrites are usually better and may be obtained with a variety of properties that allow them to be selected for the frequency range in question. The proper ferrite can give good results from audio frequencies through hun- dreds of MHz. In general, the ferrites useful at high frequencies do not have large permeabilities, however. An interesting problem that is caused by parasitic elements is encountered when one tries to Cparasitic WW: L C i C C Cparasitic C q . - Lparasitic Lparasitic Lparasitic (a) (b) (c) Fig. 1-10: Effect of parasitic elements on a presumed lowpass filter structure. (a) The circuit as designed. (b) The circuit with the parasitic elements added. [Resistances are not shown since they do not affect the argument] (c) The effective circuit at frequencies above the self-resonant frequencies of the elements L and C. build a low pass filter useful over very wide bands. Such an application might be filtering the power supply leads to an oscillator that must be very well isolated from the outside environment. The normal solution is shown in Fig. l—lOa where an ordinary LC lowpass filter is shown. The selection of proper elements is crucial. however, as is shown in the Fig. l-10b where the equiva- lent circuit is shown including the parasitic elements for the L's and C‘s (The loss resistors are not included because the problem will become evident without them.) If the frequency to be rejected is above the resonant frequencies of the elements, the equivalent circuit becomes as shown in Fig. l— 10c. You will note the circuit has become a highpass filter now, and the rejection of these high frequencies will be extremely poor. To avoid the problem we must operate at frequencies below the resonance of the elements of the filter. This may be done by raising the resonant frequency of individual elements-in the case of the capacitors—by using types especially designed to have low inductance to ground. Certain feed through ...
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