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Sec. 4–8 Classification of Filters and Amplifiers 255
transfer function that is realized, such as the Butterworth or Chebyshev response (defined
subsequently). These two classifications are discussed in this section.
Filters use energy storage elements to obtain frequency discrimination. In any physical
filter, the energy storage elements are imperfect. For example, a physical inductor has some
series resistance as well as inductance, and a physical capacitor has some shunt (leakage)
resistance as well as capacitance. A natural question, then, is, what is the quality Q of a circuit
element or filter? Unfortunately, two different measures of filter quality are used in the
technical literature. The first definition is concerned with the efficiency of energy storage in a
circuit [Ramo, Whinnery, and vanDuzer, 1967, 1984] and is
2p(maximum energy stored during one cycle)
Q =
(4–38)
energy dissipated per cycle
A larger value for Q corresponds to a more perfect storage element. That is, a perfect L or C
element would have infinite Q. The second definition is concerned with the frequency selectivity
of a circuit and is
Q = f 0
(4–39)
B
where f 0 is the resonant frequency and B is the 3-dB bandwidth. Here, the larger the value of Q,
the better is the frequency selectivity, because, for a given f 0 , the bandwidth would be
smaller.
In general, the value of Q as evaluated using Eq. (4–38) is different from the value of Q
obtained from Eq. (4–39). However, these two definitions give identical values for an RLC
series resonant circuit driven by a voltage source or for an RLC parallel resonant circuit driven
by a current source [Nilsson, 1990]. For bandpass filtering applications, frequency selectivity
is the desired characteristic, so Eq. (4–39) is used. Also, Eq. (4–39) is easy to evaluate from
laboratory measurements. If we are designing a passive filter (not necessarily a single-tuned
circuit) of center frequency f 0 and 3-dB bandwidth B, the individual circuit elements will each
need to have much larger Q’s than f 0 B. Thus, for a practical filter design, we first need to
answer the question. What are the Q’s needed for the filter elements, and what kind of
elements will give these values of Q? This question is answered in Table 4–2, which lists filters
as classified by the type of energy storage elements used in their construction and gives
typical values for the Q of the elements. Filters that use lumped † L and C elements become
impractical to build above 300 MHz, because the parasitic capacitance and inductance of the
leads significantly affect the frequency response at high frequencies. Active filters, which use
operational amplifiers with RC circuit elements, are practical only below 500 kHz, because
the operational amplifiers need to have a large open-loop gain over the operating band. For
very low-frequency filters, RC active filters are usually preferred to LC passive filters because
the size of the LC components becomes large and the Q of the inductors becomes small in this
frequency range. Active filters are difficult to implement within integrated circuits because
the resistors and capacitors take up a significant portion of the chip area. This difficulty is
reduced by using a switched-capacitor design for IC implementation. In that case, resistors
† A lumped element is a discrete R-, L-, or C-type element, compared with a continuously distributed RLC
element, such as that found in a transmission line.