NEETS Module 9  Introduction to Wave Generation and WaveShaping
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451 to 461, Index
CHAPTER 1
TUNED CIRCUITS
LEARNING OBJECTIVES
Learning objectives are stated at the beginning of each chapter. These learning objectives serve as a preview
of the information you are expected to learn in the chapter. The comprehensive check questions are based on the
objectives. By successfully completing the OCC/ECC, you indicate that you have met the objectives and have learned
the information. The learning objectives are listed below. Upon completion of this chapter, you will be
able to:
1. State the applications of a resonant circuit. 2. Identify the conditions that exist in a resonant
circuit. 3. State and apply the formula for resonant frequency of an AC circuit. 4. State the
effect of changes in inductance (L) and capacitance (C) on resonant frequency (fr). 5. Identify the
characteristics peculiar to a series resonant circuit.
6. Identify the characteristics peculiar to a parallel resonant circuit. 7. State and apply the
formula for Q. 8. State what is meant by the bandwidth of a resonant circuit and compute the bandwidth
for a given circuit. 9. Identify the four general types of filters.
10. Identify how the series and parallelresonant circuit can be used as a bandpass or a bandreject
filter.
INTRODUCTION TO TUNED CIRCUITS
When your radio or television set is turned on, many events take place within the "receiver" before you hear
the sound or see the picture being sent by the transmitting station. Many different signals reach the
antenna of a radio receiver at the same time. To select a station, the listener adjusts the tuning dial on the
radio receiver until the desired station is heard. Within the radio or TV receiver, the actual "selecting" of the
desired signal and the rejecting of the unwanted signals are accomplished by what is called a TUNED CIRCUIT. A
tuned circuit consists of a coil and a capacitor connected in series or parallel. Later in this chapter you will
see the application and advantages of both series and paralleltuned circuits. Whenever the characteristics of
inductance and capacitance are found in a tuned circuit, the phenomenon as RESONANCE takes place. You
learned earlier in the Navy Electricity and Electronics Training Series, Module 2, chapter 4, that inductive
reactance (XL) and capacitive reactance (XC) have opposite effects on circuit impedance (Z).
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You also learned that if the frequency applied to an LCR circuit causes X_{L} and X_{C}
to be equal, the circuit is RESONANT. If you realize that X_{L} and X_{C} can be equal
ONLY at ONE FREQUENCY (the resonant frequency), then you will have learned the most important single fact about
resonant circuits. This fact is the principle that enables tuned circuits in the radio receiver to select one
particular frequency and reject all others. This is the reason why so much emphasis is placed on XL and X C in
the discussions that follow. Examine figure 11. Notice that a basic tuned circuit consists of a coil and a
capacitor, connected either in series, view (A), or in parallel, view (B). The resistance (R) in the circuit is
usually limited to the inherent resistance of the components (particularly the resistance of the coil). For our
purposes we are going to disregard this small resistance in future diagrams and explanations.
Figure 11A.  Basic tuned circuits. SERIES TUNED CIRCUIT
Figure 11B.  Basic tuned circuits. PARALLEL TUNED CIRCUIT
You have already learned how a coil and a capacitor in an AC circuit perform. This action will be the basis of
the following discussion about tuned circuits. Why should you study tuned circuits? Because the tuned
circuit that has been described above is used in just about every electronic device, from remotecontrolled model
airplanes to the most sophisticated space satellite.
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You can assume, if you are going to be involved in electricity or electronics, that you will need to have a
good working knowledge of tuned circuits and how they are used in electronic and electrical circuits.
REVIEW OF SERIES/PARALLEL AC CIRCUITS
First we will review the effects of frequency on a circuit which contains resistance, inductance, and
capacitance. This review recaps what you previously learned in the Inductive and Capacitive Reactance chapter in
module 2 of the NEETS. FREQUENCY EFFECTS ON RLC CIRCUITS Perhaps the most often used control of a radio or
television set is the station or channel selector. Of course, the volume, tone, and picture quality controls are
adjusted to suit the individual's taste, but very often they are not adjusted when the station is changed. What
goes on behind this station selecting? In this chapter, you will learn the basic principles that account for the
ability of circuits to "tune" to the desired station. Effect of Frequency on Inductive Reactance
In an AC circuit, an inductor produces inductive reactance which causes the current to lag the voltage by 90
degrees. Because the inductor "reacts" to a changing current, it is known as a reactive component. The opposition
that an inductor presents to AC is called inductive reactance (X_{L}). This opposition is caused by the
inductor "reacting" to the changing current of the AC source. Both the inductance and the frequency determine the
magnitude of this reactance. This relationship is stated by the formula:
X_{L} = 2πfL
Where:
X_{L} = the inductive reactance in ohms
f = the frequency in hertz
L = the inductance in henries
π = 3.1416
As shown in the equation, any increase in frequency, or "f," will cause a corresponding increase of inductive
reactance, or "X_{L}." Therefore, the INDUCTIVE REACTANCE VARIES DIRECTLY WITH THE FREQUENCY. As you can
see, the higher the frequency, the greater the inductive reactance; the lower the frequency, the less the
inductive reactance for a given inductor. This relationship is illustrated in figure 12. Increasing values of X_{L}
are plotted in terms of increasing frequency. Starting at the lower left corner with zero frequency, the inductive
reactance is zero. As the frequency is increased (reading to the right), the inductive reactance is shown to
increase in direct proportion.
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Figure 12.  Effect of frequency on inductive reactance.
Effect of Frequency on Capacitive Reactance In an AC circuit, a capacitor produces a
reactance which causes the current to lead the voltage by 90 degrees. Because the capacitor "reacts" to a changing
voltage, it is known as a reactive component. The opposition a capacitor presents to AC is called capacitive
reactance (X_{C}). The opposition is caused by the capacitor "reacting" to the changing voltage of the AC
source. The formula for capacitive reactance is:
Where:
X_{C} = the capacitive reactance in ohms
f = the frequency in hertz
C = the capacitance in farads
π = 3.1416
In contrast to the inductive reactance, this equation indicates that the CAPACITIVE REACTANCE VARIES INVERSELY
WITH THE FREQUENCY. When f = 0, X_{C} is infinite (∞) and decreases as frequency increases. That is, the
lower the frequency, the greater the capacitive reactance; the higher the frequency, the less the reactance for a
given capacitor. As shown in figure 13, the effect of capacitance is opposite to that of inductance.
Remember, capacitance causes the current to lead the voltage by 90 degrees, while inductance causes the current to
lag the voltage by 90 degrees.
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Figure 13.  Effect of frequency on capacitive reactance.
Effect of Frequency on Resistance In the expression for inductive reactance, X_{L}
= 2πfL, and in the expression for capacitive reactance,
both contain "f" (frequency). Any change of frequency changes the reactance of the circuit components as
already explained. So far, nothing has been said about the effect of frequency on resistance. In an Ohm's law
relationship, such as R = E/I no "f" is involved. Thus, for all practical purposes, a change of frequency does not
affect the resistance of the circuit. If a 60hertz AC voltage causes 20 milliamperes of current in a resistive
circuit, then the same voltage at 2000 hertz, for example, would still cause 20 milliamperes to flow.
NOTE: Remember that the total opposition to AC is called impedance (Z). Impedance is the combination of inductive
reactance (X_{L}), capacitive reactance (X_{C}), and resistance (R). When dealing with AC
circuits, the impedance is the factor with which you will ultimately be concerned. But, as you have just been
shown, the resistance (R) is not affected by frequency. Therefore, the remainder of the discussion of AC circuits
will only be concerned with the reactance of inductors and capacitors and will ignore resistance.
AC Circuits Containing Both Inductive and Capacitive Reactances AC circuits that contain both an
inductor and a capacitor have interesting characteristics because of the opposing effects of L and C. X_{L}
and X_{C} may be treated as reactors which are 180 degrees out of phase. As shown in figure 12, the
vector for X_{L}
should be plotted above the baseline; vector for X_{C}, figure 13, should be plotted below the baseline.
In a series circuit, the effective reactance, or what is termed the RESULTANT REACTANCE, is the difference between
the individual reactances. As an equation, the resultant reactance is:
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X = X_{L}  X_{C}
Suppose an AC circuit contains an XL of 300 ohms and an XC of 250 ohms. The resultant reactance
X = X_{L}  X_{C} = 300  250 = 50 ohms (inductive)
In some cases, the X_{C} may be larger than the X_{L}. If X_{L} = 1200 ohms and X_{C}
= 4000 ohms, the difference is: X = X_{L}  X_{C} = 1200  4000 = 2800 ohms (capacitive). The
total carries the sign (+ or ) of the greater number (factor). Q1. What is the relationship between
frequency and the values of (a) X_{L}, (b) X_{C}, and (c) R?
Q2. In an AC circuit that contains both an inductor and a capacitor, what term is used for the
difference between the individual reactances?
RESONANCE
For every combination of L and C, there is only ONE frequency (in both series and parallel circuits) that
causes X_{L} to exactly equal X_{C}; this frequency is known as the RESONANT FREQUENCY. When the
resonant frequency is fed to a series or parallel circuit, X_{L} becomes equal to X_{C}, and the
circuit is said to be RESONANT to that frequency. The circuit is now called a RESONANT CIRCUIT; resonant circuits
are tuned circuits. The circuit condition wherein X_{L} becomes equal to X_{C} is known as
RESONANCE.
Each LCR circuit responds to resonant frequency differently than it does to any other frequency. Because of this,
an LCR circuit has the ability to separate frequencies. For example, suppose the TV or radio station you want to
see or hear is broadcasting at the resonant frequency. The LC "tuner" in your set can divide the frequencies,
picking out the resonant frequency and rejecting the other frequencies. Thus, the tuner selects the station you
want and rejects all other stations. If you decide to select another station, you can change the frequency by
tuning the resonant circuit to the desired frequency. RESONANT FREQUENCY
As stated before, the frequency at which X_{L} equals X_{C} (in a given circuit) is known as
the resonant frequency of that circuit. Based on this, the following formula has been derived to find the exact
resonant frequency when the values of circuit components are known:
There are two important points to remember about this formula. First, the resonant frequency found when using
the formula will cause the reactances (X_{L} and X_{C}) of the L and C components to be equal.
Second, any change in the value of either L or C will cause a change in the resonant frequency. An increase
in the value of either L or C, or both L and C, will lower the resonant frequency of a given circuit. A decrease
in the value of L or C, or both L and C, will raise the resonant frequency of a given circuit.
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The symbol for resonant frequency used in this text is f. Different texts and references may use other
symbols for resonant frequency, such as f_{o}, F_{r}, and fR. The symbols for many circuit
parameters have been standardized while others have been left to the discretion of the writer. When you study,
apply the rules given by the writer of the text or reference; by doing so, you should have no trouble with
nonstandard symbols and designations. The resonant frequency formula in this text is:
Where:
f_{r} = the resonant frequency in Hertz
L = the inductance in Heries
C = the capacitance in Farads
π = 3.1416
By substituting the constant .159 for the quantity
the formula can be simplified to the following:
Let's use this formula to figure the resonant frequency (f_{r}). The circuit is shown in the practice
tank circuit of figure 14.
Figure 14.  Practice tank circuit.
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Given:
L = 2mH (2 x 10^{3} H)
C = 300pF (300 x 10^{12} F)
Solution:
The important point here is not the formula nor the mathematics. In fact, you may never have to compute a
resonant frequency. The important point is for you to see that any given combination of L and C can be resonant at
only one frequency; in this case, 205 kHz. The universal reactance curves of figures 12 and 13 are joined
in figure 15 to show the relative values of X_{L} and X_{L} at resonance, below resonance, and
above resonance.
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Figure 15.  Relationship between X_{L} and X_{C} as frequency increases.
First, note that f_{r}, (the resonant frequency) is that frequency (or point) where the two curves
cross. At this point, and ONLY this point, X_{L} equals X_{C}. Therefore, the frequency indicated
by f_{r} is the one and only frequency of resonance. Note the resistance symbol which indicates that at
resonance all reactance is cancelled and the circuit impedance is effectively purely resistive. Remember, AC
circuits that are resistive have no phase shift between voltage and current. Therefore, at resonance, phase shift
is cancelled. The phase angle is effectively zero. Second, look at the area of the curves to the left of f_{r}.
This area shows the relative reactances of the circuit at frequencies BELOW resonance. To these LOWER frequencies,
X_{C} will always be greater than X_{L}. There will always be some capacitive reactance left in
the circuit after all inductive reactance has been cancelled. Because the impedance has a reactive component,
there will be a phase shift. We can also state that below f_{r} the circuit will appear capacitive.
Lastly, look at the area of the curves to the right of f. This area shows the relative reactances of the circuit
at frequencies ABOVE resonance. To these HIGHER frequencies, X_{L} will always be greater than X_{C}.
There will always be some inductive reactance left in the circuit after all capacitive reactance has been
cancelled. The inductor symbol shows that to these higher frequencies, the circuit will always appear to have
some inductance. Because of this, there will be a phase shift.
RESONANT CIRCUITS
Resonant circuits may be designed as series resonant or parallel resonant. Each has the ability to
discriminate between its resonant frequency and all other frequencies. How this is accomplished by both series
and parallelLC circuits is the subject of the next section. NOTE: Practical circuits are often more
complex and difficult to understand than simplified versions. Simplified versions contain all of the basic
features of a practical circuit, but leave out the nonessential features. For this reason, we will first look at
the IDEAL SERIESRESONANT CIRCUIT  a circuit that really doesn't exist except for our purposes here.
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THE IDEAL SERIESRESONANT CIRCUIT The ideal seriesresonant circuit contains
no resistance; it consists of only inductance and capacitance in series with each other and with the source
voltage. In this respect, it has the same characteristics of the series circuits you have studied previously.
Remember that current is the same in all parts of a series circuit because there is only one path for current.
Each LC circuit responds differently to different input frequencies. In the following paragraphs, we will
analyze what happens internally in a seriesLC circuit when frequencies at resonance, below resonance, and above
resonance are applied. The L and C values in the circuit are those used in the problem just studied under
resonantfrequency. The frequencies applied are the three inputs from figure 16. Note that the resonant
frequency of each of these components is 205 kHz, as figured in the problem.
Figure 16.  Output of the resonant circuit.
How the Ideal SeriesLC Circuit Responds to the Resonant Frequency (205 kHz)
Given:
Note: You are given the values of X_{L}, X_{C}, and f_{r} but you can apply the formulas
to figure them. The values given are rounded off to make it easier to analyze the circuit. First, note that
X_{L} and XC are equal. This shows that the circuit is resonant to the applied frequency of 205 kHz.
X_{L} and X_{C} are opposite in effect; therefore, they subtract to zero. (2580 ohms  2580
ohms = zero.) At resonance, then, X = zero. In our theoretically perfect circuit with zero resistance and zero
reactance, the total opposition to current (Z) must also be zero.
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NEETS Table of Contents
 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
