Download Quality factor of components and approximate analysis of high

1
Quality factor of components and approximate analysis of high-Q circuits
Models of inductor and capacitor. The preceding sections examined circuits consisting on
inductance and capacitance. These elements do not accurately account for the behavior of
circuits with practical (i.e., non-ideal) components. Recall that an ideal voltage source in series
with an internal resistance models a practical voltage source. In the same fashion, a practical
inductor –coil or wire (or a practical capacitor) is modeled by an inductor (or a capacitor),
together with some other "parasitic" elements to
account for losses and coupling effects. Figure la
illustrates a fairly accurate model of a practical
L
inductor. The primary parameter is, of course, the
L
Cp
inductance L. The remaining elements, Rs, Rp, and
Rp
Cp are viewed as "parasitic." Since an inductor
usually consists of a coil of wire, the model uses Rs
R
s
s
to represent the wire's resistance. Also, a
capacitance always exists between any adjacent
(a)
(b)
turns of wire. For convenience, this is represented
by Cp across the two end terminals of L. The
Fig. 1 Two models of an inductor: for high
resistance Rp accounts for the energy loss in the
frequencies (a) and for low to medium
magnetic core material (if present). Such complex
frequencies (b)
models, although important, if used for every
inductor in a circuit, would make the analysis of even a simple tuned circuit a horrendous task.
Fortunately, for low to medium frequencies of up to a few megahertz, the simpler model of
figure 1b is sufficiently accurate. This simpler model will underlie the analysis that follows.
Figure 2a shows a fairly accurate model of a practical capacitor. Again, the primary parameter
here is the capacitance C, whereas Rp, Rs, and Ls are "parasitic." The resistance Rp, often called a
leakage resistance, accounts for the energy loss in the dielectric; the inductance Ls and resistance
Rs are due mainly to the connecting wires of the capacitor. At frequencies above 1 / LS C , the
C
C
R
L
s
Rs
(a)
R
p
(b)
Fig 2. Two models of a
capacitor:
for
high
frequencies (a) and for
low
to
medium
frequencies (b)
capacitor actually behaves as an inductor! For low to
medium frequencies of up to a few megahertz, the
simpler model shown in figure 2b is sufficiently
accurate. As in the case of an inductor, this simpler
model of a capacitor will underlie the analysis that
follows. Very often, we will even omit Rp.
If the two-terminal network is a practical
inductor or capacitor, then the reactance X or
susceptance B are the primary parameters of
concern. Resistance R and conductance G will
represent the "parasitic effect" that we would rather
not have, but cannot avoid. In such a case, the
magnitudes of X and B are usually much greater than
the magnitude of R and G.
The Concept of Q. To evaluate quality of reactive components the term Q is frequently
encountered. The Q of a component is a measure of ratio of its energy-storing capability to
energy dissipation as heat. Specifically, the Q of a circuit component is defined as
2
Q=
reactive _ power
real _ power
(1)
Since energy is stored in the electric field of a capacitor or in the magnetic field of an inductor,
we associate Q with reactive elements. Reactance of these elements shows capability to storage
energy and having resistance in a circuit causes energy to by dissipated as heat. For a reactive
element inductance in series with resistance (figure 1a)
QL =
reactive _ power I L2 X L
= 2
.
real _ power
I R RL
Here IL and IR – currents in the inductor and resistor. Since IL=IR
QL =
X LS ϖLS
=
.
RL
RL
(2)
Similarly, it maybe shown that for capacitor and resistor in parallel (figure 2a)
reactive _ power VC2 / X C
QC =
= 2
real _ power
VR / RC
Her VC and VR – voltage drop across capacitor and resistor. Because VC=VR,
QC =
RC
= ω C P RC .
X CP
(3)
The quality factor QL of a coil varies with the operating frequency ω and remains unchanged
irrespective of the location of the coil in the circuit. Higher QL implies a better quality coil in the
sense that the energy loss in the component is smaller. Infinite QL represents an idealized loss-less
inductor modeled by a pure inductance L. In the audio frequency range, QL may vary from 5 to
20, whereas in the radio frequency range, it may reach above 100 for practical circuits. Although
RS also varies with frequency (due to the so-called skin effect), we shall, at the level of this text,
treat RS as a constant independent of ω.
Just like QL, the quality factor Qc varies with the operating frequency ω and remains
unchanged, no matter where the capacitor is located in the circuit. A higher QC, of course,
implies a better quality capacitor, again in the sense that the energy loss in the device is smaller.
Infinite Qc represents an idealized lossless capacitor modeled by a pure capacitance. In practice,
Qc is usually much greater than QL, and its effect on the performance of a circuit is often
neglected (i.e., Qc is assumed to be infinite). The reciprocal of Qc is called the dissipation
factor of the capacitor and is denoted by dC. A lower dissipation factor means a better quality
capacitor.
Inductors and capacitors with finite QL and Qc are termed lossy components. Of course, all
real-world components are lossy. The determination of QL and Qc requires the specification of
the operating frequency ω. However, if the value of ω is unspecified, the analysis of a tuned
circuit proceeds under the assumption that QL = QL(ω0) and QC=QC(ω0), where, as
3
usual ω 0 = 1 / LC . We will use these ideas shortly in more complicated resonant circuits.
When one uses the inductor and capacitor models of figures 1b and 2b, the circuit models for the
parallel- and series-tuned circuits have the configurations of figures За and 3b, respectively.
These models have neither the parallel RLC circuit structure nor the series RLC circuit structure.
From a topological point of view, the circuits of figure 3 are called series-parallel circuits,
meaning that the input impedance "seen" by the source consists of a sequence of series
connections and parallel connections of simpler networks. Exact analysis of such series-parallel
circuits is extremely complicated and is usually not worth the effort.
There is, however, a simpler, more efficient method widely used by engineers to obtain
approximate solutions. This method employs two different, but equivalent, representations of an
impedance Z(jω) or admittance Y(jω), as explained next. Let us follow the steps used to define
the Q of a practical inductor and capacitor and write a
RS
complex impedance of the two-terminal network as
VS
L
R
RC
C
Z ( jω ) = RS + jX S .
Load
(4)
RL
This suggests the representation of Z(jω) as a reactance XS
in series with a resistance RS as shown in figure 4. The
admittance of the same network has a similar expression,
(a)
RS
VS
L
RL
Y ( jω ) = G P + jB P
RLoad
RC
C
(5)
which leads naturally to the representation of Y(jω) as a
susceptance BP in parallel with a conductance GP as per
figure 4.
B
(b)
Fig. 3 Parallel- (a) and series- (b) tuned
circuits using practical inductor and
capacitor models
Since Y(jω) = [Z(jω)]-1, manipulating equation 4 into the
form of equation 5 produces
Y ( jω ) = G P + jB P =
R − jX S
1
= S2
.
RS + jX S RS + X S2
(6)
Equating the real and imaginary parts of equation 6 identifies the following relationships among
the four parameters RS, XS, RP, and BP:
B
R
S
GP
GP =
RS
;
R + X S2
BP = −
XS
;
R + X S2
(7)
RS =
GP
;
G + B P2
XS = −
BP
.
G + B P2
(8)
B
2
S
2
S
P
X
S
Fig. 4. Conversion of a series RSXS
combination to a parallel GPBP
combination
B
2
P
2
P
4
These formulas can be used to obtain approximate parallel circuits for given series
circuits or approximate series circuits for given parallel ones. By a sequence of approximations,
a complicated circuit can be reduced to either an approximate series or an approximate parallel
RLC circuit, amenable for calculating circuit parameters and characteristics. To see this, consider
the model of the practical inductor given in figure 1b. The preceding formulas make it possible
to convert the series connection to an approximate parallel connection, as illustrated in figure 5a.
To find Lp and Rp in the parallel representation, we use equations 7 and express the results in
terms of QL as defined by equation 2. Specifically, for fixed nonzero ω, we may apply equation 7
to obtain
ωLS
1
−
=− 2
ωL P
RS + ω 2 L2S
and apply equation to obtain
R
1
= 2 S 2
R P RS + X S
LS
LP
R
P
C
P
RP
From these two equations, at fixed ω
one readily obtains
RS
CS
R
S
(a)
(b)
Fig.5. Conversion of a inductor model from a series connection
to a parallel connection (a) and conversion of a capacitor model
from a parallel connection to a series connection (b)
LP = LS (1 +
1
)
QL2
(9)
and
RP = RS (1 + QL2 )
(10)
Then QL>10, LP ≈ Ls and RP ≈ QL2 RS = QLωLS .
Similarly, the model for a capacitor, given in figure 2b, can be converted from a parallel
connection to a series connection at a given frequency as illustrated in figure 5b. To find CS and
RS in the series representation, we use equations 8 and express the results in terms of QC, as
defined by equation 3. The results are
1
)
QP2
(11)
1
(1 + QC2 )
(12)
C S = C P (1 +
and
RS = RP
For QC>10
C S ≈ C P and RS ≅ RP / QC2 = 1 /(QC ωC )
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With the foregoing formulas, we can easily convert the parallel-tuned circuit of figure 3 to an
approximate parallel RLC circuit and convert a series-tuned circuit to an approximate series RLC
circuit valid in a neighborhood of a given frequency. Consequently, it becomes possible to
analyze the circuit with the more simple techniques. Of course, all of the results based on these
converted circuits will be approximate primarily because we have assumed that XS = ωL, and Bp
= 1/ω)LP or XS = l/ωCS and Bp = ωCP in equations 7-8 which is legitimate at a particular nonzero
frequency. The approximation arises in two ways: (1) from neglecting the 1 / QS2 term in
equations 9 and 11 and (2) from ignoring the dependence of QZ on ω, i.e., using the same value
of QZ calculated at ω = ω0 for all frequencies near ω0. For high-Q circuits (say, Qcir > 10) an
approximate analysis using this conversion technique ordinarily yields quite satisfactory results
(an error of less than 1%).