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Unformatted text preview: 12mm: EE344) l—Hsla Frequencb (4190mm , Setmafonl 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 11 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. 11.1 RESISTORS AT HIGH FREQUENCIES A resistor made of a slab of resistive (or semiconducting) 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 dc 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. 11: Geometry of a slab resistor. The resistance R = r01/wh where r = the resis
i tivity of the material in Q—cm. h ‘ w TI/ 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. 13. The effects of the 1For an explanation of the TEM mode see Ref. 1, pp. 4144. 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. 13: An approximate equivalent circuit for a resistor including
parasitic elements.  Q ”‘8
‘?:a 8
a ._.
o
w [— ._.
O
N Impedance MagnitudeOhms _
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 nlI. 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 GIIz, 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. 11.2 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'CV2. 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"(21tfrsC)] =cos[tan‘1((orSC)] [1—1]
(OI's'C [12] 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 nIl, 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. 15: 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
 Epoxyglass Formica XXl
Polyethylene
Polystyrene is the dielectric constant. The capacitance of two parallel plates separated by a dielectric is given by
C 0.0884KA/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. 109119. 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
24124 LC
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 phenolicpaper 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
MI
0
g
'E
a?
z 104 . . . . . . .
102 103 10" 105 10*5 107 108 109
FrequencyHertz Fig. 17: The impedance of four different nonideal 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 highK 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 = L12/2 where W
is the stored energy in Watt—seconds or Joules. Any current carrying conductor has a magnetic
ﬁeld around it and therefore has selfinductance. The conductors may be geometrically arranged to
increase the L over that of straight wire. The selfinductance 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 = 0.00508L[Ln(4L/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 r2n2 .
L 9r+lOL “ m “H1 ' [1—6]
where L is in pH and the dimensions are in inches. The formula is accurate to 1% if L > 0.8r, 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 _ 21tf.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. 17 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. [17] Rs L
Fig. 18: 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 Impedance01mm 101 ............. a ................. 3 .................. =. ................. a .................. .................. a .....
102 103 10‘ 105 105 10" 108 109 FrequencyIIertz
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. 19 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 rf 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. 19 should help design such a choke. The stability of an aircore
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. 110: 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 selfresonant 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. l10b 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 elementsin the case of
the capacitors—by using types especially designed to have low inductance to ground. Certain feed
through ...
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