October 1963 Electronics World
Table of Contents
People old and young
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'Q' is an often used term to describe the elctrical 'quality'
of a circuit or component, and for the most part anyone engaged
in the conversation (verbally or via reading) understands the
concept. However, having a firm grasp on the technical ramifications
is required if you happen to be a circuit or system designer
and need to conform to certain specifications. 'Q' can be good
or bad, depending on your needs. If, for example, you need a
narrowband receiver to reject adjacent signals or you are designing
a high stability and spectrally clean oscillator, then you want
all the 'Q' you can get. On the other hand, if your goal is
to receive a spread spectrum signal or generate white noise
across some bandwidth, then a lower 'Q' is what you want. This
article provides a brief look at 'Q' in series and parallel
circuits and relates it to energy stored in the circuit versus
energy dissipated. BTW, 'Q' is also used in mechanical
and optical system descriptions.
What is Q?
By Joseph Tusinski,
Senior Technical Instructor,
Technical Institute, Norfolk College of William and Mary
A review of an important circuit concept for the technician
who wants to brush up on his theory.
The concept of "Q" is probably one of the most misunderstood
in electronics and yet it might be considered just as important
as Ohm's Law. The average technician will state that it is a
"figure of merit" or possibly the ratio of inductive reactance
to the resistance of a circuit. The first definition is, of
course, true in all situations; however, the second definition
will hold only for a specific case.
The restrictive definition of "Q" is brought about by the
early training a technician receives in the study of tuned circuits.
In this study the resistance of a circuit is distributed in
the inductive branch of the circuit, as shown in the equivalent
circuit of Fig. 1B. This is normally the condition that would
exist in a conventional tuned circuit, and other possibilities
that could exist are disregarded either temporarily or permanently.
The resistance is depicted as a lumped unit in series with
the remainder of the secondary seriesresonant circuit. The
secondary is considered a seriesresonant circuit by virtue
of the induced voltage, which is considered to be in series
with the inductance.
Energy & Power
If we may think of "Q" in terms of the energystoring ability
of a circuit instead of X_{L}/R, then a feeling for
what "Q" actually means will result. It may be stated that "Q"
is a ratio of the energy stored in the circuit to the energy
that is being lost in the circuit. The power that is lost is
usually in the form of heat.
The reader may have noticed the use of the words "power"
and "energy" in the preceding statements. It should be remembered
that energy is the ability to do work; whereas power is the
time rate of doing work. The reactive elements have the ability
to do work if some load (resistive) is applied to them. The
resistance of the circuit dissipates this energy in the form
of heat, i.e., the energy lost is the power it takes to produce
the heat. When dealing with very low voltages this heat, of
course, is of a magnitude that may not be felt with the fingers.
Referring to Fig. 1B, it is obvious that the only factor
common to all of the elements in the circuit is the current,
where the voltage is in series with the circuit. Then applying
our definition to the circuit we may say that the "Q" of the
series circuit is the ratio of the reactive power to the real
power or
"Q" = reactive power/real power.
Fig. 1  Tuned circuit and its equivalent.
The reactive power may be expressed as I^{2}X_{L}
and the real power may be expressed as I^{2}R for the
inductive branch. The "Q" of the capacitance branch may also
be stated as I^{2}X_{C}/I^{2}R, but
in this case the resistance of the capacitive circuit at the
lower radio frequencies may be considered zero. Therefore:
"Q" = (I^{2}X_{L}) / (I^{2}R)
= X_{L}/R and: (I^{2}X_{C}) / (I^{2}R)
= X_{C}/R = ∞.
It can be seen that the expression is the very familiar X_{L}/R,
but what about the capacitive element? Actually this situation
may be likened to two resistors in parallel. This is shown in
Fig. 2.
The "Q" of the capacitive branch is considered to be infinite
at the lower frequencies where the resistance of the circuit
is low. The "Q" of the inductive branch is considered to be
finite and therefore is the controlling factor in respect to
the energystoring ability of the circuit.
Fig. 2  Parallelcircuit equivalents.
Let us now examine the case of a parallel circuit in which the
voltage is applied across both Land C. Let us assume that the
resistance of the circuit appears in parallel with the circuit
as shown in Fig. 2B.
In this case, the resistance of the inductor, which is still
present, will be considered to absorb negligible power compared
to that absorbed by the parallel resistance. This time the voltage
is common to all elements of the circuit and we will use E^{2}/Rand
E^{2}/X to determine the power and "Q" of the circuit.
In this case, either X_{L} or X_{C} may be
used to determine the reactive power (energy) for in a resonant
circuit the total energy is either stored in the magnetic field
of the inductor or onehalf cycle later it will be all stored
in the electric field of the capacitor. Then:
"Q" = reactive power/real power
=
(E^{2}/X_{L}) / (E^{2}/R).
Inverting and then multiplying,
"Q" = (E^{2}/X_{L}) x
(R/E^{2}) = R/X_{L}.
From this result it may be seen that the definition stated
earlier for the case of a series resistor is just the inverse
of the definition developed for the case of a parallel resistor.
Note further, that for maximum "Q", the resistor value should
be as low as possible in the series case and as high as possible
in the parallel case.
The "Q" of a circuit may be determined by measuring its effectiveness
in a circuit. A high"Q" circuit will respond very sharply at
its resonant frequency and the voltage or current will be "Q"
times the applied voltage or current. Typical values of "Q"
may average about 100 at the lower frequencies, and may reach
300 or more in some transmitter tank circuits. One reason for
the higher "Q's" in transmitter tank circuits is the use of
large conductors which have a lower a.c. resistance.
The "Q" of a typical resonant circuit may vary over a band
of frequencies as shown in Fig. 3.
Fig. 3  Variation of "Q" with frequency.
As mentioned, the "Q" may be determined by a measurement
of the circuit response, that is, how sharply the circuit responds
at its resonant frequency. The "Q" may be determined by measuring
the halfpower (3 db) response of a circuit, i.e., where the
current or voltage of the circuit drops to 0.707 of the current
or voltage at resonance. Between the halfpower points a certain
range of frequencies has been covered and, by definition this
is termed the "bandwidth" of the circuit. This is shown in Fig.
4.
Fig. 4  Higher "Q's" mean narrow bandwidths.
From the response of the circuit, the "Q" may be expressed
as: "Q" = resonant frequency/bandwidth or "Q" = f_{r}
/ (f_{2}  f_{1})·
For example, if a circuit would be assumed to be tuned to
1000 kc. and the halfpower points were measured at 1005 kc.
and 995 kc., the "Q" of the circuit would be: 1000 / (1005 
995) = 1000 / 10 = 100.
A great number of commercial "Q" meters base their operation
on this simple fact, while others work on a precalibrated method
of measuring the degree of response of a circuit or component
on the resonant rise of voltage appearing across the circuit
or component.
Posted March 9, 2015