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*Introduction:*

The LC circuit can store electrical energy vibrating at resonant frequency. The capacitor stores energy in electric field between its plates, depending on voltage across it, and inductor stores energy in its magnetic field, depending on current through it.

This oscillator comprises of capacitor and coil connected in parallel. Assume capacitor is charged by the battery. Once capacitor is charged, one plate of capacitor has more electrons than other plate, therefore it is charged. Now, when it is discharged through wire, electrons return to positive plate, therefore making capacitor's plates neutral, or discharged. Though, this action works differently when discharge capacitor through a coil. When current is applied through the coil, the magnetic field is produced around the coil. This magnetic field produces the voltage across coil which opposes direction of electron flow. Due to this, capacitor doesn't discharge right away. Smaller the coil, faster capacitor discharges. Now interesting part happens. Once the capacitor is completely discharged through coil, magnetic field begins to collapse around coil. Voltage induced from collapsing magnetic field recharges capacitor oppositely. Then capacitor starts to discharge through coil again, generating magnetic field. This procedure continues until capacitor is completely discharged because of resistance.

Technically this basic LC circuit produces sine wave which loses voltage in every cycle. To overcome this, extra voltage is applied to keep oscillator from losing voltage. Though, to keep this oscillator going well, switching method is utilized. The vacuum tube (or a solid-state equivalent likes a FET) is utilized to keep LC circuit oscillating. Benefit of using vacuum tube is that they can oscillate at specified frequencies like a thousand cycles per second.

*Time Domain Solution:*

By Kirchhoff's voltage law, voltage across capacitor, VC, should equal voltage across inductor, V_{L}:

V_{C} = V_{L}

From constitutive relations for circuit elements, we also know that

V_{L}(t) = L(d_{iL}/dt) and i_{c}(t) = C(dV_{C}/dt)

Rearranging and substituting provides second order differential equation

d^{2}_{i}(t)/dt^{2} + (1/LC)i(t) = 0

Parameter ω, radian frequency, can be stated as:

ω = √1/LC

If initial conditions are such that A= B, then we can utilized Euler's formula to get the real sinusoid with amplitude 2A and angular frequency

Therefore, resulting solution becomes:

i(t) = 2Acos(ωt)

Initial conditions which would satisfy result are:

i(t = 0) = 2A and di/dt(t = 0) = 0

*Resonance Effect:*

Resonance effect takes place when inductive and capacitive reactances are equal in absolute value. Frequency at which this equality holds for particular circuit is known as resonant frequency. Resonant frequency of LC circuit is

ω = √1/LC

Where L is inductance in Henries, and C is capacitance in Farads. Angular frequency has units of radians per second. Equivalent frequency in units of hertz is

f = ω/2π = 1/2π√LC

LC circuits are frequently utilized as filters; the L/C ratio is one of factors which find out their Q and so selectivity. For the series resonant circuit with the given resistance, the higher the inductance and the lower the capacitance, the narrower the filter bandwidth. For the parallel resonant circuit opposite applies. Positive feedback around tuned circuit (regeneration) can also increase selectivity.

**Series LC Circuit:**

Resonance:

Here L and C are in series in the ac circuit. Inductive reactance magnitude (X_{L}) increases as frequency increases while capacitive reactance magnitude (X_{C}) decreases with increase in frequency. At particular frequency these two reactances are equal in magnitude but opposite in sign. Frequency at which this occurs is resonant frequency (f_{r}) for given circuit.

Hence at f_{r}

X_{L} = -X_{C}

ωL = 1/ωC

Converting angular frequency in hertz we get

2πfL = 1/2πfC

Here f is resonant frequency. Then rearranging,

f = 1/2π√LC

In the series AC circuit, X_{C} leads by 90 degrees while X_{L} lags by 90. Thus, they cancel each other out. Only opposition to current is coil resistance. Therefore in series resonance current is maximum at resonant frequency. At f_{r}, current is maximum. Circuit impedance is minimum. In this state circuit is known as an acceptor circuit.

Below f_{r}, X_{L} << (-X_{C}). Therefore circuit is capacitive

Above f_{r}, X_{L}>>(-X_{C}). Therefore circuit is inductive

Impedance:

Consider impedance of series LC circuit. Total impedance is given by sum of inductive and capacitive impedances:

Z = Z_{L} + Z_{C}

By writing inductive impedance as Z_{L} = jωL and capacitive impedances as Z_{C} = 1/jωC and substituting we have

Z = jωL + 1/jωC therefore Z = (ω^{2}LC - 1)j/ωC

Numerator implies that if ω^{2}LC = 1 total impedance Z will be zero and otherwise non-zero. Thus series LC circuit, when connected in series with the load, will act as band-pass filter having zero impedance at resonant frequency of LC circuit.

*Parallel LC Circuit:*

Resonance:

Here coil (L) and capacitor (C) are joined in parallel with AC power supply. Let R be internal resistance of coil. When X_{L} equals X_{C}, reactive branch currents are equal and opposite. Therefore they cancel out each other to give minimum current in main line. As total current is minimum, in this state total impedance is maximum. Resonant frequency given by:

f = 1/2π√LC

Therefore I=V/Z, as per Ohm's law. At f_{r}, line current is minimum. Total impedance is maximum. In this state cct is known as rejector circuit.

Below f_{r}, circuit is inductive. Above f_{r}, circuit is capacitive.

Impedance:

The same analysis may be applied to parallel LC circuit. Total impedance is then given by:

Z = Z_{L}Z_{C}/(Z_{L} + Z_{C}) and after substitution of Z_{L} and Z_{C} and simplification, provides

Z = -jωL/ω^{2}LC - 1

Note that limω^{2}LC→1 Z = ∞

But for all other values of ω^{2}LC impedance is finite (and thus less than infinity). Therefore parallel LC circuit connected in series with load will act as band-stop filter having infinite impedance at resonant frequency of LC circuit.

**Applications of Resonance Effect:**

- Most common application is tuning. For instance, when we tune the radio to particular station, LC circuits are set at resonance for that specific carrier frequency.
- The series resonant circuit gives voltage magnification.
- The parallel resonant circuit gives current magnification.
- The parallel resonant circuit can be utilized as load impedance in output circuits of RF amplifiers. Because of high impedance, gain of amplifier is maximum at resonant frequency
- Both parallel and series resonant circuits are utilized in induction heating.

*LC OR Oscillatory Circuit:*

Assume capacitor has been fully-charged from dc source. As S is open, it can't discharge through L. When S is closed electrons move from plate A to plate B through coil L. This electron flow decreases strength of electric field and therefore amount of energy stored in it. As electronic current begins flowing, self-induced emf in coil opposes current flow. Therefore, rate of discharge of electrons is rather slowed down. Due to flow of current, magnetic field is set up that stores energy given out by electric field.

As plate A loses its electrons by discharge, electron current has the tendency to die down and will really reduce to zero when all excess electrons on A are driven over to plate B so that both plates are reduced to same potential. At that time, there is no electric field but magnetic field has maximum value.

Though, due to self-induction (or electrical inertia) of coil, more electrons are transferred to plate B than are essential to make up electron deficiency there. It means that now plate B has more electrons than A. Therefore, capacitor becomes charged again though in opposite direction as. Magnetic field L collapses and energy given out by it is stored in electric field of capacitor. After this, capacitor begins discharging in opposite direction so that, now, electrons move from plate B to plate A. Electric field begins collapsing while magnetic field begins building up again though in opposite direction. The condition when capacitor becomes completely discharged once again. Though, these discharging electrons overshoot and again excess amount of electrons flow to plate A, thereby charging capacitor once more. This sequence of charging and discharging continues. The back and forth motion of electrons between two plates of capacitor comprises oscillatory current. It may be also noted that during this procedure, electric energy of capacitor is converted in magnetic energy of coil and vice versa. These oscillations of capacitor discharge are damped as energy is dissipated away slowly so that their amplitude becomes zero after a short time. There are two reasons for loss of the energy:

- Some energy is lost in form of heat produced in resistance of coil and connecting wires
- some energy is lost in form of electromagnetic (EM) waves which are radiated out from circuit through which the oscillatory current is passing.

**Frequency of LC Circuit or Oscillatory Current:**

Frequency of time-period of oscillatory current depends on two factors:

(a) Capacitance of Capacitor Larger the capacitor, greater the time required for reversal of discharge current i.e. lower its frequency.

(b) Self-inductance of Coil Larger the self-inductance, greater the internal effect and therefore longer the time needed by current to stop flowing during discharge of capacitor. Frequency of oscillatory discharge current is given by

f = 1/2π√LC = 159/√LC kHz

Where L = self-inductance in µH and C = capacitance in µF

Damped oscillations so produced are not good for radio transmission purpose due to their limited range and excessive distortion. For good radio transmission, we require un damped oscillations that can be produced if some additional energy is supplied in correct phase and correct direction to LC circuit for making up I^{2}R losses continually occurring in circuit.

*Frequency Stability of an Oscillator:*

The ability of the oscillator to maintain the constant frequency of oscillation is known as its frequency stability. The factors affect frequency stability are given below:

Operating Point of the Active Device: The Q-point of active device (i.e. transistor) is so chosen as to confine circuit operation on linear portion of characteristic. Operation on non-linear portion varies parameters of transistor that, in turn, affects frequency stability of oscillator.

Inter-element Capacitances: Any changes in inter-element capacitances of the transistor particularly the collector- to emitter capacitance cause changes in oscillator output frequency, therefore affecting frequency stability. Effect of changes in inter-element capacitances can be neutralized by adding the swamping capacitor across offending elements-added capacitance being made part of the tank circuit.

Temperature Variations: Variations in temperature cause changes in transistor parameters and also change values of resistors, capacitors and inductors utilized in circuit. As such changes occur slowly, they cause slow change (known as drift) in oscillator output frequency.

*Essentials of a Feedback LC Oscillator:*

The essential components of a feedback LC oscillator are

A resonator that consists of LC circuit. It is also called as frequency-determining network (FDN)or tank circuit. The amplifier whose function is to amplify oscillations produced by resonator.

A positive feedback network (PFN) whose function is to transfer part of output energy to resonant LC circuit in proper phase. Amount of energy fed back is sufficient to meet losses in LC circuit.

Essential condition for maintaining oscillations and for finding value of frequency is

βA = 1 + j0 or βA∠Φ = 1∠0

It means that

The feedback factor or loop gain |βA| = 1

Net phase shift around loop is 0° (or the integral multiple of 360°). In other words, feedback must be positive. The above conditions form Barkhausen criterion for maintaining the steady level of oscillation at specific frequency. Majority of oscillators utilized in radio receivers and transmitters use tuned circuits with positive feedback. Variations in oscillator circuits are due to different way by which feedback is applied. Few basic circuits are:

- Armstrong or Tickler or Tuned-base Oscillator: It uses inductive feedback from collector to tuned LC circuit in base of transistor.
- Tuned Collector Oscillator-it also uses inductive coupling but LC tuned circuit is in collector circuit.
- Hartley Oscillator-Here feedback is supplied inductively.
- Colpitts Oscillator-Here feedback is supplied capacitively.
- Clapp Oscillator-it is slight modification of Colpitts oscillator.

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