NEETS Module 9 — Introduction to Wave Generation and WaveShapingPages
i  ix, 11 to 110,
111 to 120, 121 to 130,
131 to 140, 141 to 152,
21 to 210, 211 to 220,
221 to 230, 231 to 238,
31 to 310, 311 to 320,
321 to 330, 331 to 340,
341 to 350, 351 to 356,
41 to 410, 411 to 420,
421 to 430, 431 to 440,
441 to 450, 451 to 461, Index
Now, apply Ohm's law for
AC circuits:
Don't be confused by this high value of current. Our perfect, but impossible, circuit has no opposition
to current. Therefore, current flow will be extremely high. The important points here are that AT RESONANCE,
impedance is VERY LOW, and the resulting current will be comparatively HIGH. If we apply Ohm's law to the
individual reactances, we can figure relative values of voltage across each reactance.
E_{L} = I x X_{L}
E_{C} = I x X_{C}
These are reactive voltages that you have studied previously. The voltage across each reactance will be
comparatively high. A comparatively high current times 2580 ohms yields a high voltage. At any given instant, this
voltage will be of opposite polarity because the reactances are opposite in effect. E_{L} + E_{C}
= zero volts
WARNING THE INDIVIDUAL VOLTAGES MAY REACH QUITE HIGH VALUES.
ALTHOUGH LITTLE POWER IS PRESENT, THE VOLTAGE IS REAL AND CARE SHOULD BE TAKEN IN WORKING WITH IT.
Let's summarize our findings so far. In a seriesLC circuit with a resonantfrequency voltage applied,
the following conditions exist: · X_{L} and X_{C} are equal and subtract to zero.
· Resultant reactance is zero ohms. · Impedance (Z) is reduced to a MINIMUM value. · With
minimum Z, current is MAXIMUM for a given voltage.
· Maximum current causes maximum voltage drops across the individual reactances. All of the above follow
in sequence from the fact that X_{L} = X_{C} at the resonant frequency.
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How the Ideal SeriesLC Circuit Respond to a Frequency Below Resonance (100 kHz)
Given:
L = 2mH (2 x 10^{3}H) C = 300pF (300 x 10^{12}F) f_{r} = 205
kHz (at resonant frequency)
X_{L} = 1260 ohms (rounded off) (at 100 kHz) X_{C} = 5300 ohms (rounded off) (at 100kHz) ES
= 10 volts (at 100kHz) (as in the previous analysis, you are given
values that are possible for you to computer. If you do the
computations, remember that most values are rounded off.)
First, note that X_{L} and X_{C} are no longer equal. X_{C} is larger than it
was at resonance; X_{L} is smaller. By applying the formulas you have learned, you know that a lower
frequency produces a higher capacitive reactance and a lower inductive reactance. The reactances subtract but do
not cancel (X_{L}  X_{C} = 1260 5300 = 4040 ohms (capacitive)). At an input frequency of 100
kHz, the circuit (still resonant to 205 kHz)has a net reactance of 4040 ohms. In our theoretically perfect
circuit, the total opposition (Z) is equal to X, or 4040 ohms. As before, let's apply Ohm's law to the new
conditions.
The voltage drops across the reactances are as follows:
E_{L} = I x X_{L} E_{L} = .0025 A x 1260 Ω
E_{L} = 3 volts (approximately) E_{C} = I x X_{C} E_{C} = .0025 A x 5300
Ω E_{C} = 13 volts (approximately)
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In summary, in a seriesLC circuit with a source voltage that is below the resonant frequency (100 kHz
in the example), the resultant reactance (X), and therefore impedance, is higher than at resonance. In addition
current is lower, and the voltage drops across the reactances are lower. All of the above follow in sequence due
to the fact that X_{C} is greater than X_{L} at any frequency lower than the resonant frequency.
How the Ideal SeriesLC Circuit Responds to a Frequency Above Resonance (300 kHz)
Given:
L = 2mH (2 x 10^{3}H) C = 300pF (300 x 10^{12}F) f_{r} = 205
kHz (at resonant frequency) X_{L} = 3770 ohms (rounded off) (at 300kHz) X_{C} = 1770 ohms
(rounded off) (at 300 kHz) E_{S} = 10 volts (at 300 kHz)
Again, X_{L} and X_{C} are not equal. This time, X_{L} is larger than X_{C}.
(If you don't know why, apply the formulas and review the past several pages.) The resultant reactance is 2000
ohms (X_{L}  X_{C} = 3770  1770 = 2000 ohms.) Therefore, the resultant reactance (X), or the
impedance of our perfect circuit at 300 kHz, is 2000 ohms. By applying Ohm's law as before:
I = 5 milliamperes E_{L} = 19 volts (rounded off) E_{C} = 9 volts
(rounded off)
In summary, in a seriesLC circuit with a source voltage that is above the resonant frequency (300 kHz in
this example), impedance is higher than at resonance, current is lower, and the voltage drops across the
reactances are lower. All of the above follow in sequence from the fact that X_{L} is greater than X_{C}
at any frequency higher than the resonant frequency. Summary of the Response of the Ideal
SeriesLC Circuit to Frequencies Above, Below, and at Resonance The ideal seriesresonant circuit
has zero impedance. The impedance increases for frequencies higher and lower than the resonant frequency. The
impedance characteristic of the ideal seriesresonant circuit results because resultant reactance is zero ohms at
resonance and ONLY at resonance. All other frequencies provide a resultant reactance greater than zero.
Zero impedance at resonance allows maximum current. All other frequencies have a reduced current because of the
increased impedance. The voltage across the reactance is greatest at resonance because voltage drop is directly
proportional to current. All discrimination between frequencies results from the fact that X_{L} and X_{C}
completely counteract ONLY at the resonant frequency.
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How the Typical SeriesLC Circuit Differs From the Ideal As you learned much earlier
in this series, resistance is always present in practical electrical circuits; it is impossible to eliminate. A
typical seriesLC circuit, then, has R as well as L and C. If our perfect (ideal) circuit has zero
resistance, and a typical circuit has "some" resistance, then a circuit with a very small resistance is closer to
being perfect than one that has a large resistance. Let's list what happens in a seriesresonant circuit because
resistance is present. This is not new to you  just a review of what you have learned previously. In a
seriesresonant circuit that is basically L and C, but that contains "some" R, the following statements are true:
· X_{L}, X_{C}, and R components are all present and can be shown on a vector diagram, each at
right angles with the resistance vector (baseline). · At resonance, the resultant reactance is zero
ohms. Thus, at resonance, The circuit impedance equals only the resistance (R). The circuit impedance can never be
less than R because the original resistance will always be present in the circuit. · At resonance, a
practical seriesRLC circuit ALWAYS has MINIMUM impedance. The actual value of impedance is that of the resistance
present in the circuit (Z = R). Now, if the designers do their very best (and they do) to keep the value
of resistance in a practical seriesRLC circuit LOW, then we can still get a fairly high current at resonance. The
current is NOT "infinitely" high as in our ideal circuit, but is still higher than at any other frequency. The
curve and vector relationships for the practical circuit are shown in figure 17.
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Figure 17.—Curves of impedance and current in an RLC series resonant circuit.
Note that the impedance curve does not reach zero at its minimum point. The vectors above and below
resonance show that the phase shift of the circuit at these frequencies is less than 90 degrees because of the
resistance.
The horizontal width of the curve is a measure of how well the circuit will pick out (discriminate) the one
desired frequency. The width is called BANDWIDTH, and the ability to discriminate between frequencies is known as
SELECTIVITY. Both of these characteristics are affected by resistance. Lower resistance allows narrower bandwidth,
which is the same as saying the circuit has better selectivity. Resistance, then, is an unwanted quantity that
cannot be eliminated but can be kept to a minimum by the circuit designers. More on bandwidth,
selectivity, and measuring the effects of resistance in resonant circuits will follow the discussion of parallel
resonance. Q3. State the formula for resonant frequency. Q4. If the inductor and capacitor
values are increased, what happens to the resonant frequency? Q5. In an "ideal" resonant circuit, what
is the relationship between impedance and current? Q6. In a seriesRLC circuit, what is the condition
of the circuit if there is high impedance, low current, and low reactance voltages?
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How the ParallelLC Circuit Stores Energy A parallelLC circuit is often
called a TANK CIRCUIT because it can store energy much as a tank stores liquid. It has the ability to take energy
fed to it from a power source, store this energy alternately in the inductor and capacitor, and produce an output
which is a continuous AC wave. You can understand how this is accomplished by carefully studying the sequence of
events shown in figure 18. You must thoroughly understand the capacitor and inductor action in this figure before
you proceed further in the study of parallelresonant circuits. In each view of figure 18, the waveform
is of the charging and discharging CAPACITOR VOLTAGE. In view (A), the switch has been moved to position C. The DC
voltage is applied across the capacitor, and the capacitor charges to the potential of the battery.
Figure 18A.—Capacitor and inductor action in a tank circuit.
In view (B), moving the switch to the right completes the circuit from the capacitor to the inductor and
places the inductor in series with the capacitor. This furnishes a path for the excess electrons on the upper
plate of the capacitor to flow to the lower plate, and thus starts neutralizing the capacitor charge. As these
electrons flow through the coil, a magnetic field is built up around the coil. The energy which was first stored
by the electrostatic field of the capacitor is now stored in the electromagnetic field of the inductor.
Figure 18B.—Capacitor and inductor action in a tank circuit.
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View (C) shows the capacitor discharged and a maximum magnetic field around the coil. The energy
originally stored in the capacitor is now stored entirely in the magnetic field of the coil.
Figure 18C.—Capacitor and inductor action in a tank circuit.
Since the capacitor is now completely discharged, the magnetic field surrounding the coil starts to
collapse. This induces a voltage in the coil which causes the current to continue flowing in the same direction
and charges the capacitor again. This time the capacitor charges to the opposite polarity, view (D).
Figure 18D.—Capacitor and inductor action in a tank circuit.
In view (E), the magnetic field has completely collapsed, and the capacitor has become charged with the
opposite polarity. All of the energy is again stored in the capacitor.
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Figure 18E.—Capacitor and inductor action in a tank circuit.
In view (F), the capacitor now discharges back through the coil. This discharge current causes the
magnetic field to build up again around the coil.
Figure 18F.—Capacitor and inductor action in a tank circuit.
In view (G), the capacitor is completely discharged. The magnetic field is again at maximum.
Figure 18G.—Capacitor and inductor action in a tank circuit.
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In view (H), with the capacitor completely discharged, the magnetic field again starts collapsing. The
induced voltage from the coil maintains current flowing toward the upper plate of the capacitor.
Figure 18H.—Capacitor and inductor action in a tank circuit.
In view (I), by the time the magnetic field has completely collapsed, the capacitor is again charged
with the same polarity as it had in view (A). The energy is again stored in the capacitor, and the cycle is ready
to start again.
Figure 18I.—Capacitor and inductor action in a tank circuit.
The number of times per second that these events in figure 18 take place is called NATURAL FREQUENCY or
RESONANT FREQUENCY of the circuit. Such a circuit is said to oscillate at its resonant frequency.
It might seem that these oscillations could go on forever. You know better, however, if you apply what you have
already learned about electric circuits. This circuit, as all others, has some resistance. Even the
relatively small resistance of the coil and the connecting wires cause energy to be dissipated in the form of heat
(I2R loss). The heat loss in the circuit resistance causes the charge on the capacitor to be less for each
subsequent cycle. The result is a DAMPED WAVE, as shown in figure 19. The charging and discharging action will
continue until all of the energy has been radiated or dissipated as heat.
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Figure 19.—Damped wave.
If it were possible to have a circuit with absolutely no resistance, there would be no heat loss, and
the oscillations would tend to continue indefinitely. You have already learned that tuned circuits are designed to
have very little resistance. Reducing I2R losses is still another reason for having low resistance. A
"perfect" tuned circuit would produce the continuous sine wave shown in figure 110. Its frequency would be that
of the circuit.
Figure 110.—Sine waveresonant frequency.
Because we don't have perfection, another way of causing a circuit to oscillate indefinitely would be to
apply a continuous AC or pulsing source to the circuit. If the source is at the resonant frequency of the circuit,
the circuit will oscillate as long as the source is applied. The reasons why the circuit in figure 18
oscillates at the resonant frequency have to do with the characteristics of resonant circuits. The discussion of
parallel resonance will not be as detailed as that for series resonance because the idea of resonance is the same
for both circuits. Certain characteristics differ as a result of L and C being in parallel rather than in series.
These differences will be emphasized. Q7. When the capacitor is completely discharged, where is the
energy of the tank circuit stored? Q8. When the magnetic field of the inductor is completely collapsed,
where is the energy of the tank circuit stored? PARALLEL RESONANCE Much of what
you have learned about resonance and seriesLC circuits can be applied directly to parallelLC circuits. The
purpose of the two circuits is the same — to select a specific frequency and reject all others. XL still equals
XC at resonance. Because the inductor and capacitor are in parallel, however, the circuit has the basic
characteristics of an AC parallel circuit. The parallel hookup causes
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Introduction to Matter, Energy, and Direct Current, Introduction
to Alternating Current and Transformers, Introduction to Circuit Protection,
Control, and Measurement, Introduction to Electrical Conductors, Wiring Techniques,
and Schematic Reading, Introduction to Generators and Motors,
Introduction to Electronic Emission, Tubes, and Power Supplies,
Introduction to SolidState Devices and Power Supplies,
Introduction to Amplifiers, Introduction to
WaveGeneration and WaveShaping Circuits, Introduction to Wave Propagation, Transmission
Lines, and Antennas, Microwave Principles,
Modulation Principles, Introduction to Number Systems and Logic Circuits, Introduction
to Microelectronics, Principles of Synchros, Servos, and Gyros,
Introduction to Test Equipment, RadioFrequency
Communications Principles, Radar Principles, The Technician's Handbook,
Master Glossary, Test Methods and Practices, Introduction to Digital Computers,
Magnetic Recording, Introduction to Fiber Optics
