 1. LIEC (todo)
 2. Navy (todo)
 2.1. Resonance
 2.2. Resonant Circuits
 2.2.1. The Ideal SeriesResonant Circuit
 2.2.1.1. How the Ideal SeriesLC Circuit Responds to the Resonant Frequency (205 kHz)
 2.2.1.2. How the Ideal SeriesLC Circuit Respond to a Frequency Below Resonance (100 kHz)
 2.2.1.3. How the Ideal SeriesLC Circuit Responds to a Frequency Above Resonance (300 kHz)
 2.2.1.4. Summary of the Response of the Ideal SeriesLC Circuit to Frequencies Above, Below, and at Resonance
 2.2.1.5. How the Typical SeriesLC Circuit Differs From the Ideal
 2.2.1.6. How the ParallelLC Circuit Stores Energy
 2.2.2. Parallel Resonance
 2.2.1. The Ideal SeriesResonant Circuit
 2.3. Bandwidth
1. LIEC (todo)
1.1. An Electric Pendulum
Capacitors store energy in the form of an electric field, and electrically manifest that stored energy as a potential: static voltage. Inductors store energy in the form of a magnetic field, and electrically manifest that stored energy as a kinetic motion of electrons: current. Capacitors and inductors are flipsides of the same reactive coin, storing and releasing energy in complementary modes. When these two types of reactive components are directly connected together, their complementary tendencies to store energy will produce an unusual result.
If either the capacitor or inductor starts out in a charged state, the two components will exchange energy between them, back and forth, creating their own AC voltage and current cycles. If we assume that both components are subjected to a sudden application of voltage (say, from a momentarily connected battery), the capacitor will very quickly charge and the inductor will oppose change in current, leaving the capacitor in the charged state and the inductor in the discharged state: (Figure below)
Capacitor charged: voltage at (+) peak, inductor discharged: zero current.
The capacitor will begin to discharge, its voltage decreasing. Meanwhile, the inductor will begin to build up a "charge" in the form of a magnetic field as current increases in the circuit: (Figure below)
Capacitor discharging: voltage decreasing, Inductor charging: current increasing.
The inductor, still charging, will keep electrons flowing in the circuit until the capacitor has been completely discharged, leaving zero voltage across it: (Figure below)
Capacitor fully discharged: zero voltage, inductor fully charged: maximum current.
The inductor will maintain current flow even with no voltage applied. In fact, it will generate a voltage (like a battery) in order to keep current in the same direction. The capacitor, being the recipient of this current, will begin to accumulate a charge in the opposite polarity as before: (Figure below)
Capacitor charging: voltage increasing (in opposite polarity), inductor discharging: current decreasing.
When the inductor is finally depleted of its energy reserve and the electrons come to a halt, the capacitor will have reached full (voltage) charge in the opposite polarity as when it started: (Figure below)
Capacitor fully charged: voltage at () peak, inductor fully discharged: zero current.
Now we’re at a condition very similar to where we started: the capacitor at full charge and zero current in the circuit. The capacitor, as before, will begin to discharge through the inductor, causing an increase in current (in the opposite direction as before) and a decrease in voltage as it depletes its own energy reserve: (Figure below)
Capacitor discharging: voltage decreasing, inductor charging: current increasing.
Eventually the capacitor will discharge to zero volts, leaving the inductor fully charged with full current through it: (Figure below)
Capacitor fully discharged: zero voltage, inductor fully charged: current at () peak.
The inductor, desiring to maintain current in the same direction, will act like a source again, generating a voltage like a battery to continue the flow. In doing so, the capacitor will begin to charge up and the current will decrease in magnitude: (Figure below)
Capacitor charging: voltage increasing, inductor discharging: current decreasing.
Eventually the capacitor will become fully charged again as the inductor expends all of its energy reserves trying to maintain current. The voltage will once again be at its positive peak and the current at zero. This completes one full cycle of the energy exchange between the capacitor and inductor: (Figure below)
Capacitor fully charged: voltage at (+) peak, inductor fully discharged: zero current.
This oscillation will continue with steadily decreasing amplitude due to power losses from stray resistances in the circuit, until the process stops altogether. Overall, this behavior is akin to that of a pendulum: as the pendulum mass swings back and forth, there is a transformation of energy taking place from kinetic (motion) to potential (height), in a similar fashion to the way energy is transferred in the capacitor/inductor circuit back and forth in the alternating forms of current (kinetic motion of electrons) and voltage (potential electric energy).
At the peak height of each swing of a pendulum, the mass briefly stops and switches directions. It is at this point that potential energy (height) is at a maximum and kinetic energy (motion) is at zero. As the mass swings back the other way, it passes quickly through a point where the string is pointed straight down. At this point, potential energy (height) is at zero and kinetic energy (motion) is at maximum. Like the circuit, a pendulum’s backandforth oscillation will continue with a steadily dampened amplitude, the result of air friction (resistance) dissipating energy. Also like the circuit, the pendulum’s position and velocity measurements trace two sine waves (90 degrees out of phase) over time: (Figure below)
Pendelum transfers energy between kinetic and potential energy as it swings low to high.
In physics, this kind of natural sinewave oscillation for a mechanical system is called Simple Harmonic Motion (often abbreviated as "SHM"). The same underlying principles govern both the oscillation of a capacitor/inductor circuit and the action of a pendulum, hence the similarity in effect. It is an interesting property of any pendulum that its periodic time is governed by the length of the string holding the mass, and not the weight of the mass itself. That is why a pendulum will keep swinging at the same frequency as the oscillations decrease in amplitude. The oscillation rate is independent of the amount of energy stored in it.
The same is true for the capacitor/inductor circuit. The rate of oscillation is strictly dependent on the sizes of the capacitor and inductor, not on the amount of voltage (or current) at each respective peak in the waves. The ability for such a circuit to store energy in the form of oscillating voltage and current has earned it the name tank circuit. Its property of maintaining a single, natural frequency regardless of how much or little energy is actually being stored in it gives it special significance in electric circuit design.
However, this tendency to oscillate, or resonate, at a particular frequency is not limited to circuits exclusively designed for that purpose. In fact, nearly any AC circuit with a combination of capacitance and inductance (commonly called an "LC circuit") will tend to manifest unusual effects when the AC power source frequency approaches that natural frequency. This is true regardless of the circuit’s intended purpose.
If the power supply frequency for a circuit exactly matches the natural frequency of the circuit’s LC combination, the circuit is said to be in a state of resonance. The unusual effects will reach maximum in this condition of resonance. For this reason, we need to be able to predict what the resonant frequency will be for various combinations of L and C, and be aware of what the effects of resonance are.

REVIEW:

A capacitor and inductor directly connected together form something called a tank circuit, which oscillates (or resonates) at one particular frequency. At that frequency, energy is alternately shuffled between the capacitor and the inductor in the form of alternating voltage and current 90 degrees out of phase with each other.

When the power supply frequency for an AC circuit exactly matches that circuit’s natural oscillation frequency as set by the L and C components, a condition of resonance will have been reached.
1.2. Simple Parallel (Tank Circuit) Resonance
A condition of resonance will be experienced in a tank circuit (Figure below) when the reactances of the capacitor and inductor are equal to each other. Because inductive reactance increases with increasing frequency and capacitive reactance decreases with increasing frequency, there will only be one frequency where these two reactances will be equal.
Simple parallel resonant circuit (tank circuit).
In the above circuit, we have a 10 µF capacitor and a 100 mH inductor. Since we know the equations for determining the reactance of each at a given frequency, and we’re looking for that point where the two reactances are equal to each other, we can set the two reactance formulae equal to each other and solve for frequency algebraically:
So there we have it: a formula to tell us the resonant frequency of a tank circuit, given the values of inductance (L) in Henrys and capacitance © in Farads. Plugging in the values of L and C in our example circuit, we arrive at a resonant frequency of 159.155 Hz.
What happens at resonance is quite interesting. With capacitive and inductive reactances equal to each other, the total impedance increases to infinity, meaning that the tank circuit draws no current from the AC power source! We can calculate the individual impedances of the 10 µF capacitor and the 100 mH inductor and work through the parallel impedance formula to demonstrate this mathematically:
As you might have guessed, I chose these component values to give resonance impedances that were easy to work with (100 Ω even). Now, we use the parallel impedance formula to see what happens to total Z:
We can’t divide any number by zero and arrive at a meaningful result, but we can say that the result approaches a value of infinity as the two parallel impedances get closer to each other. What this means in practical terms is that, the total impedance of a tank circuit is infinite (behaving as an open circuit) at resonance. We can plot the consequences of this over a wide power supply frequency range with a short SPICE simulation: (Figure below)
Resonant circuit sutitable for SPICE simulation.
freq i(v1) 3.162E04 1.000E03 3.162E03 1.0E02                                  1.000E+02 9.632E03 . . . . * 1.053E+02 8.506E03 . . . . * . 1.105E+02 7.455E03 . . . . * . 1.158E+02 6.470E03 . . . . * . 1.211E+02 5.542E03 . . . . * . 1.263E+02 4.663E03 . . . . * . 1.316E+02 3.828E03 . . . .* . 1.368E+02 3.033E03 . . . *. . 1.421E+02 2.271E03 . . . * . . 1.474E+02 1.540E03 . . . * . . 1.526E+02 8.373E04 . . * . . . 1.579E+02 1.590E04 . * . . . . 1.632E+02 4.969E04 . . * . . . 1.684E+02 1.132E03 . . . * . . 1.737E+02 1.749E03 . . . * . . 1.789E+02 2.350E03 . . . * . . 1.842E+02 2.934E03 . . . *. . 1.895E+02 3.505E03 . . . .* . 1.947E+02 4.063E03 . . . . * . 2.000E+02 4.609E03 . . . . * .                                 
tank circuit frequency sweep v1 1 0 ac 1 sin c1 1 0 10u * rbogus is necessary to eliminate a direct loop * between v1 and l1, which SPICE can't handle rbogus 1 2 1e12 l1 2 0 100m .ac lin 20 100 200 .plot ac i(v1) .end
The 1 picoohm (1 pΩ) resistor is placed in this SPICE analysis to overcome a limitation of SPICE: namely, that it cannot analyze a circuit containing a direct inductorvoltage source loop. (Figure below) A very low resistance value was chosen so as to have minimal effect on circuit behavior.
This SPICE simulation plots circuit current over a frequency range of 100 to 200 Hz in twenty even steps (100 and 200 Hz inclusive). Current magnitude on the graph increases from left to right, while frequency increases from top to bottom. The current in this circuit takes a sharp dip around the analysis point of 157.9 Hz, which is the closest analysis point to our predicted resonance frequency of 159.155 Hz. It is at this point that total current from the power source falls to zero.
The plot above is produced from the above spice circuit file ( *.cir), the command (.plot) in the last line producing the text plot on any printer or terminal. A better looking plot is produced by the "nutmeg" graphical postprocessor, part of the spice package. The above spice ( *.cir) does not require the plot (.plot) command, though it does no harm. The following commands produce the plot below: (Figure below)
spice b r resonant.raw resonant.cir ( b batch mode, r raw file, input is resonant.cir) nutmeg resonant.raw
From the nutmeg prompt:
>setplot ac1 (setplot {enter} for list of plots) >display (for list of signals) >plot mag(v1#branch) (magnitude of complex current vector v1#branch)
Nutmeg produces plot of current I(v1) for parallel resonant circuit.
Incidentally, the graph output produced by this SPICE computer analysis is more generally known as a Bode plot. Such graphs plot amplitude or phase shift on one axis and frequency on the other. The steepness of a Bode plot curve characterizes a circuit’s "frequency response," or how sensitive it is to changes in frequency.

REVIEW:

Resonance occurs when capacitive and inductive reactances are equal to each other.

For a tank circuit with no resistance ®, resonant frequency can be calculated with the following formula:


The total impedance of a parallel LC circuit approaches infinity as the power supply frequency approaches resonance.

A Bode plot is a graph plotting waveform amplitude or phase on one axis and frequency on the other.
1.3. Simple Series Resonance
A similar effect happens in series inductive/capacitive circuits. (Figure below) When a state of resonance is reached (capacitive and inductive reactances equal), the two impedances cancel each other out and the total impedance drops to zero!
Simple series resonant circuit.
With the total series impedance equal to 0 Ω at the resonant frequency of 159.155 Hz, the result is a short circuit across the AC power source at resonance. In the circuit drawn above, this would not be good. I’ll add a small resistor (Figure below) in series along with the capacitor and the inductor to keep the maximum circuit current somewhat limited, and perform another SPICE analysis over the same range of frequencies: (Figure below)
Series resonant circuit suitable for SPICE.
series lc circuit v1 1 0 ac 1 sin r1 1 2 1 c1 2 3 10u l1 3 0 100m .ac lin 20 100 200 .plot ac i(v1) .end
Series resonant circuit plot of current I(v1).
As before, circuit current amplitude increases from bottom to top, while frequency increases from left to right. (Figure above) The peak is still seen to be at the plotted frequency point of 157.9 Hz, the closest analyzed point to our predicted resonance point of 159.155 Hz. This would suggest that our resonant frequency formula holds as true for simple series LC circuits as it does for simple parallel LC circuits, which is the case:
A word of caution is in order with series LC resonant circuits: because of the high currents which may be present in a series LC circuit at resonance, it is possible to produce dangerously high voltage drops across the capacitor and the inductor, as each component possesses significant impedance. We can edit the SPICE netlist in the above example to include a plot of voltage across the capacitor and inductor to demonstrate what happens: (Figure below)
series lc circuit v1 1 0 ac 1 sin r1 1 2 1 c1 2 3 10u l1 3 0 100m .ac lin 20 100 200 .plot ac i(v1) v(2,3) v(3) .end
Plot of Vc=V(2,3) 70 V peak, V_{L}=v(3) 70 V peak, I=I(V1#branch) 0.532 A peak
According to SPICE, voltage across the capacitor and inductor reach a peak somewhere around 70 volts! This is quite impressive for a power supply that only generates 1 volt. Needless to say, caution is in order when experimenting with circuits such as this. This SPICE voltage is lower than the expected value due to the small (20) number of steps in the AC analysis statement (.ac lin 20 100 200). What is the expected value?
Given: fr = 159.155 Hz, L = 100mH, R = 1 XL = 2πfL = 2π(159.155)(100mH)=j100Ω XC = 1/(2πfC) = 1/(2π(159.155)(10µF)) = j100Ω Z = 1 +j100 j100 = 1 Ω I = V/Z = (1 V)/(1 Ω) = 1 A VL = IZ = (1 A)(j100) = j100 V VC = IZ = (1 A)(j100) = j100 V VR = IR = (1 A)(1)= 1 V Vtotal = VL + VC + VR Vtotal = j100 j100 +1 = 1 V
The expected values for capacitor and inductor voltage are 100 V. This voltage will stress these components to that level and they must be rated accordingly. However, these voltages are out of phase and cancel yielding a total voltage across all three components of only 1 V, the applied voltage. The ratio of the capacitor (or inductor) voltage to the applied voltage is the "Q" factor.
Q = VL/VR = VC/VR

REVIEW:

The total impedance of a series LC circuit approaches zero as the power supply frequency approaches resonance.

The same formula for determining resonant frequency in a simple tank circuit applies to simple series circuits as well.

Extremely high voltages can be formed across the individual components of series LC circuits at resonance, due to high current flows and substantial individual component impedances.
1.4. Applications of Resonance
So far, the phenomenon of resonance appears to be a useless curiosity, or at most a nuisance to be avoided (especially if series resonance makes for a shortcircuit across our AC voltage source!). However, this is not the case. Resonance is a very valuable property of reactive AC circuits, employed in a variety of applications.
One use for resonance is to establish a condition of stable frequency in circuits designed to produce AC signals. Usually, a parallel (tank) circuit is used for this purpose, with the capacitor and inductor directly connected together, exchanging energy between each other. Just as a pendulum can be used to stabilize the frequency of a clock mechanism’s oscillations, so can a tank circuit be used to stabilize the electrical frequency of an AC oscillator circuit. As was noted before, the frequency set by the tank circuit is solely dependent upon the values of L and C, and not on the magnitudes of voltage or current present in the oscillations: (Figure below)
Resonant circuit serves as stable frequency source.
Another use for resonance is in applications where the effects of greatly increased or decreased impedance at a particular frequency is desired. A resonant circuit can be used to "block" (present high impedance toward) a frequency or range of frequencies, thus acting as a sort of frequency "filter" to strain certain frequencies out of a mix of others. In fact, these particular circuits are called filters, and their design constitutes a discipline of study all by itself: (Figure below)
Resonant circuit serves as filter.
In essence, this is how analog radio receiver tuner circuits work to filter, or select, one station frequency out of the mix of different radio station frequency signals intercepted by the antenna.

REVIEW:

Resonance can be employed to maintain AC circuit oscillations at a constant frequency, just as a pendulum can be used to maintain constant oscillation speed in a timekeeping mechanism.

Resonance can be exploited for its impedance properties: either dramatically increasing or decreasing impedance for certain frequencies. Circuits designed to screen certain frequencies out of a mix of different frequencies are called filters.
1.5. Resonance in SeriesParallel Circuits
In simple reactive circuits with little or no resistance, the effects of radically altered impedance will manifest at the resonance frequency predicted by the equation given earlier. In a parallel (tank) LC circuit, this means infinite impedance at resonance. In a series LC circuit, it means zero impedance at resonance:
However, as soon as significant levels of resistance are introduced into most LC circuits, this simple calculation for resonance becomes invalid. We’ll take a look at several LC circuits with added resistance, using the same values for capacitance and inductance as before: 10 µF and 100 mH, respectively. According to our simple equation, the resonant frequency should be 159.155 Hz. Watch, though, where current reaches maximum or minimum in the following SPICE analyses:
Parallel LC circuit with resistance in series with L.
resonant circuit v1 1 0 ac 1 sin c1 1 0 10u r1 1 2 100 l1 2 0 100m .ac lin 20 100 200 .plot ac i(v1) .end
Resistance in series with L produces minimum current at 136.8 Hz instead of calculated 159.2 Hz
Minimum current at 136.8 Hz instead of 159.2 Hz!
Parallel LC with resistance in serieis with C.
Here, an extra resistor (R_{bogus}) (Figure below)is necessary to prevent SPICE from encountering trouble in analysis. SPICE can’t handle an inductor connected directly in parallel with any voltage source or any other inductor, so the addition of a series resistor is necessary to "break up" the voltage source/inductor loop that would otherwise be formed. This resistor is chosen to be a very low value for minimum impact on the circuit’s behavior.
resonant circuit v1 1 0 ac 1 sin r1 1 2 100 c1 2 0 10u rbogus 1 3 1e12 l1 3 0 100m .ac lin 20 100 400 .plot ac i(v1) .end
Minimum current at roughly 180 Hz instead of 159.2 Hz!
+ (((22009.png)))
Resistance in series with C shifts minimum current from calculated 159.2 Hz to roughly 180 Hz.
+
Switching our attention to series LC circuits, (Figure below) we experiment with placing significant resistances in parallel with either L or C. In the following series circuit examples, a 1 Ω resistor (R_{1}) is placed in series with the inductor and capacitor to limit total current at resonance. The "extra" resistance inserted to influence resonant frequency effects is the 100 Ω resistor, R_{2}. The results are shown in (Figure below).
Series LC resonant circuit with resistance in parallel with L.
resonant circuit v1 1 0 ac 1 sin r1 1 2 1 c1 2 3 10u l1 3 0 100m r2 3 0 100 .ac lin 20 100 400 .plot ac i(v1) .end
Maximum current at roughly 178.9 Hz instead of 159.2 Hz!
Series resonant circuit with resistance in parallel with L shifts maximum current from 159.2 Hz to roughly 180 Hz.
And finally, a series LC circuit with the significant resistance in parallel with the capacitor. (Figure below) The shifted resonance is shown in (Figure below)
+ (((02105.png)))
Series LC resonant circuit with rsistance in parallel with C.
resonant circuit v1 1 0 ac 1 sin r1 1 2 1 c1 2 3 10u r2 2 3 100 l1 3 0 100m .ac lin 20 100 200 .plot ac i(v1) .end
Maximum current at 136.8 Hz instead of 159.2 Hz!
Resistance in parallel with C in series resonant circuit shifts curreent maximum from calculated 159.2 Hz to about 136.8 Hz.
The tendency for added resistance to skew the point at which impedance reaches a maximum or minimum in an LC circuit is called antiresonance. The astute observer will notice a pattern between the four SPICE examples given above, in terms of how resistance affects the resonant peak of a circuit:

Parallel ("tank") LC circuit:

R in series with L: resonant frequency shifted down

R in series with C: resonant frequency shifted up
+

Series LC circuit:

R in parallel with L: resonant frequency shifted up

R in parallel with C: resonant frequency shifted down
Again, this illustrates the complementary nature of capacitors and inductors: how resistance in series with one creates an antiresonance effect equivalent to resistance in parallel with the other. If you look even closer to the four SPICE examples given, you’ll see that the frequencies are shifted by the same amount, and that the shape of the complementary graphs are mirrorimages of each other!
Antiresonance is an effect that resonant circuit designers must be aware of. The equations for determining antiresonance "shift" are complex, and will not be covered in this brief lesson. It should suffice the beginning student of electronics to understand that the effect exists, and what its general tendencies are.
Added resistance in an LC circuit is no academic matter. While it is possible to manufacture capacitors with negligible unwanted resistances, inductors are typically plagued with substantial amounts of resistance due to the long lengths of wire used in their construction. What is more, the resistance of wire tends to increase as frequency goes up, due to a strange phenomenon known as the skin effect where AC current tends to be excluded from travel through the very center of a wire, thereby reducing the wire’s effective crosssectional area. Thus, inductors not only have resistance, but changing, frequencydependent resistance at that.
As if the resistance of an inductor’s wire weren’t enough to cause problems, we also have to contend with the "core losses" of ironcore inductors, which manifest themselves as added resistance in the circuit. Since iron is a conductor of electricity as well as a conductor of magnetic flux, changing flux produced by alternating current through the coil will tend to induce electric currents in the core itself (eddy currents). This effect can be thought of as though the iron core of the transformer were a sort of secondary transformer coil powering a resistive load: the lessthanperfect conductivity of the iron metal. This effects can be minimized with laminated cores, good core design and highgrade materials, but never completely eliminated.
One notable exception to the rule of circuit resistance causing a resonant frequency shift is the case of series resistorinductorcapacitor ("RLC") circuits. So long as all components are connected in series with each other, the resonant frequency of the circuit will be unaffected by the resistance. (Figure below) The resulting plot is shown in (Figure below).
Series LC with resistance in series.
series rlc circuit v1 1 0 ac 1 sin r1 1 2 100 c1 2 3 10u l1 3 0 100m .ac lin 20 100 200 .plot ac i(v1) .end
Maximum current at 159.2 Hz once again!
Resistance in series resonant circuit leaves current maximum at calculated 159.2 Hz, broadening the curve.
Note that the peak of the current graph (Figure below) has not changed from the earlier series LC circuit (the one with the 1 Ω token resistance in it), even though the resistance is now 100 times greater. The only thing that has changed is the "sharpness" of the curve. Obviously, this circuit does not resonate as strongly as one with less series resistance (it is said to be "less selective"), but at least it has the same natural frequency!
It is noteworthy that antiresonance has the effect of dampening the oscillations of freerunning LC circuits such as tank circuits. In the beginning of this chapter we saw how a capacitor and inductor connected directly together would act something like a pendulum, exchanging voltage and current peaks just like a pendulum exchanges kinetic and potential energy. In a perfect tank circuit (no resistance), this oscillation would continue forever, just as a frictionless pendulum would continue to swing at its resonant frequency forever. But frictionless machines are difficult to find in the real world, and so are lossless tank circuits. Energy lost through resistance (or inductor core losses or radiated electromagnetic waves or . . .) in a tank circuit will cause the oscillations to decay in amplitude until they are no more. If enough energy losses are present in a tank circuit, it will fail to resonate at all.
Antiresonance’s dampening effect is more than just a curiosity: it can be used quite effectively to eliminate unwanted oscillations in circuits containing stray inductances and/or capacitances, as almost all circuits do. Take note of the following L/R time delay circuit: (Figure below)
L/R time delay circuit
The idea of this circuit is simple: to "charge" the inductor when the switch is closed. The rate of inductor charging will be set by the ratio L/R, which is the time constant of the circuit in seconds. However, if you were to build such a circuit, you might find unexpected oscillations (AC) of voltage across the inductor when the switch is closed. (Figure below) Why is this? There’s no capacitor in the circuit, so how can we have resonant oscillation with just an inductor, resistor, and battery?
Inductor ringing due to resonance with stray capacitance.
All inductors contain a certain amount of stray capacitance due to turntoturn and turntocore insulation gaps. Also, the placement of circuit conductors may create stray capacitance. While clean circuit layout is important in eliminating much of this stray capacitance, there will always be some that you cannot eliminate. If this causes resonant problems (unwanted AC oscillations), added resistance may be a way to combat it. If resistor R is large enough, it will cause a condition of antiresonance, dissipating enough energy to prohibit the inductance and stray capacitance from sustaining oscillations for very long.
Interestingly enough, the principle of employing resistance to eliminate unwanted resonance is one frequently used in the design of mechanical systems, where any moving object with mass is a potential resonator. A very common application of this is the use of shock absorbers in automobiles. Without shock absorbers, cars would bounce wildly at their resonant frequency after hitting any bump in the road. The shock absorber’s job is to introduce a strong antiresonant effect by dissipating energy hydraulically (in the same way that a resistor dissipates energy electrically).
+

REVIEW:

Added resistance to an LC circuit can cause a condition known as antiresonance, where the peak impedance effects happen at frequencies other than that which gives equal capacitive and inductive reactances.

Resistance inherent in realworld inductors can contribute greatly to conditions of antiresonance. One source of such resistance is the skin effect, caused by the exclusion of AC current from the center of conductors. Another source is that of core losses in ironcore inductors.

In a simple series LC circuit containing resistance (an "RLC" circuit), resistance does not produce antiresonance. Resonance still occurs when capacitive and inductive reactances are equal.
1.6. Q and Bandwidth of a Resonant Circuit
The Q, quality factor, of a resonant circuit is a measure of the "goodness" or quality of a resonant circuit. A higher value for this figure of merit corresponds to a more narrow bandwith, which is desirable in many applications. More formally, Q is the ratio of power stored to power dissipated in the circuit reactance and resistance, respectively:
Q = Pstored/Pdissipated = I2X/I2R Q = X/R where: X = Capacitive or Inductive reactance at resonance R = Series resistance.
This formula is applicable to series resonant circuits, and also parallel resonant circuits if the resistance is in series with the inductor. This is the case in practical applications, as we are mostly concerned with the resistance of the inductor limiting the Q. Note: Some text may show X and R interchanged in the "Q" formula for a parallel resonant circuit. This is correct for a large value of R in parallel with C and L. Our formula is correct for a small R in series with L.
A practical application of "Q" is that voltage across L or C in a series resonant circuit is Q times total applied voltage. In a parallel resonant circuit, current through L or C is Q times the total applied current.
1.6.1. Series resonant circuits
A series resonant circuit looks like a resistance at the resonant frequency. (Figure below) Since the definition of resonance is X_{L}=X_{C}, the reactive components cancel, leaving only the resistance to contribute to the impedance. The impedance is also at a minimum at resonance. (Figure below) Below the resonant frequency, the series resonant circuit looks capacitive since the impedance of the capacitor increases to a value greater than the decreasing inducitve reactance, leaving a net capacitive value. Above resonance, the inductive reactance increases, capacitive reactance decreases, leaving a net inductive component.
At resonance the series resonant circuit appears purely resistive. Below resonance it looks capacitive. Above resonance it appears inductive.
Current is maximum at resonance, impedance at a minumum. Current is set by the value of the resistance. Above or below resonance, impedance increases.
Impedance is at a minumum at resonance in a series resonant circuit.
The resonant current peak may be changed by varying the series resistor, which changes the Q. (Figure below) This also affects the broadness of the curve. A low resistance, high Q circuit has a narrow bandwidth, as compared to a high resistance, low Q circuit. Bandwidth in terms of Q and resonant frequency:
BW = fc/Q Where fc = resonant frquency Q = quality factor
A high Q resonant circuit has a narrow bandwidth as compared to a low Q
Bandwidth is measured between the 0.707 current amplitude points. The 0.707 current points correspond to the half power points since P = I^{2}R, (0.707)^{2} = (0.5). (Figure below)
Bandwidth, Δf is measured between the 70.7% amplitude points of series resonant circuit.
BW = Δf = fhfl = fc/Q Where fh = high band edge, fl = low band edge fl = fc  Δf/2 fh = fc + Δf/2 Where fc = center frequency (resonant frequency)
+
In Figure above, the 100% current point is 50 mA. The 70.7% level is 0.707(50 mA)=35.4 mA. The upper and lower band edges read from the curve are 291 Hz for f_{l} and 355 Hz for f_{h}. The bandwidth is 64 Hz, and the half power points are ± 32 Hz of the center resonant frequency:
BW = Δf = fhfl = 355291 = 64 fl = fc  Δf/2 = 32332 = 291 fh = fc + Δf/2 = 323+32 = 355
Since BW = f_{c}/Q:
Q = fc/BW = (323 Hz)/(64 Hz) = 5
1.6.2. Parallel Resonant Circuits
A parallel resonant circuit is resistive at the resonant frequency. (Figure below) At resonance X_{L}=X_{C}, the reactive components cancel. The impedance is maximum at resonance. (Figure below) Below the resonant frequency, the parallel resonant circuit looks inductive since the impedance of the inductor is lower, drawing the larger proportion of current. Above resonance, the capacitive reactance decreases, drawing the larger current, thus, taking on a capacitive characteristic.
A parallel resonant circuit is resistive at resonance, inductive below resonance, capacitive above resonance.
Impedance is maximum at resonance in a parallel resonant circuit, but decreases above or below resonance. Voltage is at a peak at resonance since voltage is proportional to impedance (E=IZ). (Figure below)
Parallel resonant circuit: Impedance peaks at resonance.
A low Q due to a high resistance in series with the inductor produces a low peak on a broad response curve for a parallel resonant circuit. (Figure below) conversely, a high Q is due to a low resistance in series with the inductor. This produces a higher peak in the narrower response curve. The high Q is achieved by winding the inductor with larger diameter (smaller gague), lower resistance wire.
Parallel resonant response varies with Q.
The bandwidth of the parallel resonant response curve is measured between the half power points. This corresponds to the 70.7% voltage points since power is proportional to E^{2}. ((0.707)^{2}=0.50) Since voltage is proportional to impedance, we may use the impedance curve. (Figure below)
Bandwidth, Δf is measured between the 70.7% impedance points of a parallel resonant circuit.
In Figure above, the 100% impedance point is 500 Ω. The 70.7% level is 0.707(500)=354 Ω. The upper and lower band edges read from the curve are 281 Hz for f_{l} and 343 Hz for f_{h}. The bandwidth is 62 Hz, and the half power points are ± 31 Hz of the center resonant frequency:
BW = Δf = fhfl = 343281 = 62 fl = fc  Δf/2 = 31231 = 281 fh = fc + Δf/2 = 312+31 = 343
Q = fc/BW = (312 Hz)/(62 Hz) = 5
2. Navy (todo)
2.1. 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.
2.1.1. 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.
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:
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 1.
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 [fignavy_mod9_00004] and [fignavy_mod9_00006] are joined in Figure 2 to show the relative values of X_{L} and X_{L} at resonance, below resonance, and above resonance.
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, a.c. 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.
2.2. 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. 
2.2.1. 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 3. Note that the resonant frequency of each of these components is 205 kHz, as figured in the problem.
2.2.1.1. How the Ideal SeriesLC Circuit Responds to the Resonant Frequency (205 kHz)
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 X_{C} 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.
Now, apply Ohm’s law for a.c. 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.
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.
2.2.1.2. How the Ideal SeriesLC Circuit Respond to a Frequency Below Resonance (100 kHz)
Given:
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:
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.
2.2.1.3. How the Ideal SeriesLC Circuit Responds to a Frequency Above Resonance (300 kHz)
Given:
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:
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.
2.2.1.4. 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.
2.2.1.5. 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 4.
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?
2.2.1.6. 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 a.c. 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 the following tank circuit figures, the waveform is of the charging and discharging CAPACITOR VOLTAGE. In Figure 5, the switch has been moved to position C. The d.c. voltage is applied across the capacitor, and the capacitor charges to the potential of the battery.
In Figure 6, 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 7 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.
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, Figure 8.
In Figure 9, 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.
In Figure 10, the capacitor now discharges back through the coil. This discharge current causes the magnetic field to build up again around the coil.
In Figure 11, the capacitor is completely discharged. The magnetic field is again at maximum.
In Figure 12, 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.
In Figure 13, 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.
The number of times per second that these events in Figure 5 through Figure 13 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 (I_{2}R 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 14. The charging and discharging action will continue until all of the energy has been radiated or dissipated as heat.
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 I_{2}R losses is still another reason for having low resistance.
A "perfect" tuned circuit would produce the continuous sine wave shown in Figure 15. Its frequency would be that of the circuit.
Because we don’t have perfection, another way of causing a circuit to oscillate indefinitely would be to apply a continuous a.c. 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 5 through Figure 13 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?
2.2.2. 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. X_{L} still equals X_{C} at resonance. Because the inductor and capacitor are in parallel, however, the circuit has the basic characteristics of an a.c. parallel circuit. The parallel hookup causes frequency selection to be accomplished in a different manner. It gives the circuit different characteristics. The first of these characteristics is the ability to store energy.
2.2.2.1. The Characteristics of a Typical ParallelResonant Circuit
Look at Figure 16. In this circuit, as in other parallel circuits, the voltage is the same across the inductor and capacitor. The currents through the components vary inversely with their reactances in accordance with Ohm’s law. The total current drawn by the circuit is the vector sum of the two individual component currents. Finally, these two currents, I_{L} and I_{C}, are 180 degrees out of phase because the effects of L and C are opposite. There is not a single fact new to you in the above. It is all based on what you have learned previously about parallel a.c. circuits that contain L and C.
Now, at resonance, X_{L} is still equal to X_{C}. Therefore, I_{L} must equal I_{C}. Remember, the voltage is the same; the reactances are equal; therefore, according to Ohm’s law, the currents must be equal. But, don’t forget, even though the currents are equal, they are still opposites. That is, if the current is flowing "up" in the capacitor, it is flowing "down" in the coil, and vice versa. In effect, while the one component draws current, the other returns it to the source. The net effect of this "give and take action" is that zero current is drawn from the source at resonance. The two currents yield a total current of zero amperes because they are exactly equal and opposite at resonance.
A circuit that is completed and has a voltage applied, but has zero current, must have an INFINITE IMPEDANCE (apply Ohm’s law — any voltage divided by zero yields infinity).
By now you know that we have just ignored our old friend resistance from previous discussions. In an actual circuit, at resonance, the currents will not quite counteract each other because each component will have different resistance. This resistance is kept extremely low, but it is still there. The result is that a relatively small current flows from the source at resonance instead of zero current. Therefore, a basic characteristic of a practical parallelLC circuit is that, at resonance, the circuit has MAXIMUM impedance which results in MINIMUM current from the source. This current is often called line current. This is shown by the peak of the waveform for impedance and the valley for the line current, both occurring at f_{r} the frequency of resonance in Figure 16.
There is little difference between the circuit pulsed by the battery in Figure 5 through Figure 13 that oscillated at its resonant (or natural) frequency, and the circuit we have just discussed. The equal and opposite currents in the two components are the same as the currents that charged and discharged the capacitor through the coil.
For a given source voltage, the current oscillating between the reactive parts will be stronger at the resonant frequency of the circuit than at any other frequency. At frequencies below resonance, capacitive current will decrease; above the resonant frequency, inductive current will decrease. Therefore, the oscillating current (or circulating current, as it is sometimes called), being the lesser of the two reactive currents, will be maximum at resonance.
If you remember, the basic resonant circuit produced a "damped" wave. A steady amplitude wave was produced by giving the circuit energy that would keep it going. To do this, the energy had to be at the same frequency as the resonant frequency of the circuit.
So, if the resonant frequency is "timed" right, then all other frequencies are "out of time" and produce waves that tend to buck each other. Such frequencies cannot produce strong oscillating currents.
In our typical parallelresonant (LC) circuit, the line current is minimum (because the impedance is maximum). At the same time, the internal oscillating current in the tank is maximum. Oscillating current may be several hundred times as great as line current at resonance.
In any case, this circuit reacts differently to the resonant frequency than it does to all other frequencies. This makes it an effective frequency selector.
2.2.2.2. Summary of Resonance
Both series and parallelLC circuits discriminate between the resonant frequency and all other frequencies by balancing an inductive reactance against an equal capacitive reactance.
In series, these reactances create a very low impedance. In parallel, they create a very high impedance. These characteristics govern how and where designers use resonant circuits. A low impedance requirement would require a seriesresonant circuit. A highimpedance requirement would require the designer to use a parallelresonant circuit.
2.2.2.3. Tuning a Band of Frequencies
Our resonant circuits so far have been tuned to a single frequency  the resonant frequency. This is fine if only one frequency is required. However, there are hundreds of stations on many different frequencies.
Therefore, if we go back to our original application, that of tuning to different radio stations, our resonant circuits are not practical. The reason is because a tuner for each frequency would be required and this is not practical.
What is a practical solution to this problem? The answer is simple. Make either the capacitor or the inductor variable. Remember, changing either L or C changes the resonant frequency. Now you know what has been happening all of these years when you "pushed" the button or "turned" the dial. You have been changing the L or C in the tuned circuits by the amount necessary to adjust the tuner to resonate at the desired frequency. No matter how complex a unit, if it has LC tuners, the tuners obey these basic laws.
Q9. What is the term for the number of times per second that tank circuit energy is either stored in the inductor or capacitor? Q10. In a parallelresonant circuit, what is the relationship between impedance and current? Q11. When is line current minimum in a parallelLC circuit?
2.3. Bandwidth
If circuit Q is low, the gain of the circuit at resonance is relatively small. The circuit does not discriminate sharply (reject the unwanted frequencies) between the resonant frequency and the frequencies on either side of resonance, as shown by the curve in Figure 17. The range of frequencies included between the two frequencies (426.4 kHz and 483.6 kHz in this example) at which the current drops to 70 percent of its maximum value at resonance is called the BANDWIDTH of the circuit.
It is often necessary to state the band of frequencies that a circuit will pass. The following standard has been set up: the limiting frequencies are those at either side of resonance at which the curve falls to a point of .707 (approximately 70 percent) of the maximum value. This point is called the HALFPOWER point. Note that in Figure 17 and in Figure 18, the seriesresonant circuit has two halfpower points, one above and one below the resonant frequency point. The two points are designated upper frequency cutoff (f_{co}) and lower frequency cutoff (f_{co}) or simply f1 and f2. The range of frequencies between these two points comprises the bandwidth. Figure 17 and Figure 18 illustrate the bandwidths for low and highQ resonant circuits. The bandwidth may be determined by use of the following formulas:
For example, by applying the formula we can determine the bandwidth for the curve shown in Figure 17.
If the Q of the circuit represented by the curve in Figure 18 is 45.5, what would be the bandwidth?
If Q equals 7.95 for the lowQ circuit as in Figure 17, we can check our original calculation of the bandwidth.
The Q of the circuit can be determined by transposing the formula for bandwidth to:
To find the Q of the circuit using the information found in the last example problem:
Q12. What is the relationship of the coil to the resistance of a circuit with high "Q"? Q13. What is the band of frequencies called that is included between the two points at which current falls to 70 percent of its maximum value in a resonant circuit?