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Linear analog electronic filters 

Simple filters

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Linear analog electronic filters 

Simple filters

edit 
An inductorcapacitor circuit (LC circuit) is an electric circuit composed of inductors and capacitors. A second order LC circuit is composed of one inductor and one capacitor and is the simplest type of LC circuit.
A second order LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. The circuit can act as an electrical resonator, an electrical analogue of a tuning fork, storing energy oscillating at the circuit's resonant frequency.
LC circuits are used either for generating signals at a particular frequency, or picking out a signal at a particular frequency from a more complex signal. They are key components in many electronic devices, particularly radio equipment, used in circuits such as oscillators, filters, tuners and frequency mixers.
An LC circuit is an idealized model since it assumes there is no dissipation of energy due to resistance. Any practical implementation of an LC circuit will always include loss resulting from small but nonzero resistance within the components and connecting wires. The purpose of an LC circuit is usually to oscillate with minimal damping, so the resistance is made as low as possible. While no practical circuit is without losses, it is nonetheless instructive to study this ideal form of the circuit to gain understanding and physical intuition. For a circuit model incorporating resistance, see RLC circuit.
An LC circuit can store electrical energy oscillating at its resonant frequency. See the animation at right. A capacitor stores energy in the electric field (E) between its plates, depending on the voltage across it, and an inductor stores energy in its magnetic field (B), depending on the current through it.
If a charged capacitor is connected across an inductor, charge will start to flow through the inductor, building up a magnetic field around it and reducing the voltage on the capacitor. Eventually all the charge on the capacitor will be gone and the voltage across it will reach zero. However, the current will continue, because inductors resist changes in current. The energy to keep it flowing is extracted from the magnetic field, which will begin to decline. The current will begin to charge the capacitor with a voltage of opposite polarity to its original charge. When the magnetic field is completely dissipated the current will stop and the charge will again be stored in the capacitor, with the opposite polarity as before. Then the cycle will begin again, with the current flowing in the opposite direction through the inductor.
The charge flows back and forth between the plates of the capacitor, through the inductor. The energy oscillates back and forth between the capacitor and the inductor until (if not replenished by power from an external circuit) internal resistance makes the oscillations die out. Its action, known mathematically as a harmonic oscillator, is similar to a pendulum swinging back and forth, or water sloshing back and forth in a tank. For this reason the circuit is also called a tank circuit. The oscillation frequency is determined by the capacitance and inductance values. In typical tuned circuits in electronic equipment the oscillations are very fast, thousands to billions of times per second.
The resonance effect occurs when inductive and capacitive reactances are equal in magnitude. The frequency at which this equality holds for the particular circuit is called the resonant frequency. The resonant frequency of the LC circuit is
where L is the inductance in henries, and C is the capacitance in farads. The angular frequency has units of radians per second.
The equivalent frequency in units of hertz is
LC circuits are often used as filters; the L/C ratio is one of the factors that determines their "Q" and so selectivity. For a series resonant circuit with a given resistance, the higher the inductance and the lower the capacitance, the narrower the filter bandwidth. For a parallel resonant circuit the opposite applies. Positive feedback around the tuned circuit ("regeneration") can also increase selectivity (see Q multiplier and Regenerative circuit).
Stagger tuning can provide an acceptably wide audio bandwidth, yet good selectivity.
The resonance effect of the LC circuit has many important applications in signal processing and communications systems.
LC circuits behave as electronic resonators, which are a key component in many applications:
By Kirchhoff's voltage law, the voltage across the capacitor, V_{C}, plus the voltage across the inductor, V_{L} must equal zero:
Likewise, by Kirchhoff's current law, the current through the capacitor equals the current through the inductor:
From the constitutive relations for the circuit elements, we also know that
and
Rearranging and substituting gives the second order differential equation
The parameter ω_{0}, the resonant angular frequency, is defined as:
Using this can simplify the differential equation
The associated polynomial is
Thus,
or
Thus, the complete solution to the differential equation is
and can be solved for A and B by considering the initial conditions.
Since the exponential is complex, the solution represents a sinusoidal alternating current.
Since the electric current i is a physical quantity, it must be realvalued. As a result, it can be shown that the constants A and B must be complex conjugates:
Now, let
Therefore,
Next, we can use Euler's formula to obtain a real sinusoid with amplitude I _{0}, angular frequency ω_{0} = (LC)^{−1/2}, and phase angle .
Thus, the resulting solution becomes:
and
The initial conditions that would satisfy this result are:
and
In the series configuration of the LC circuit, the inductor L and capacitor C are connected in series, as shown here. The total voltage v across the open terminals is simply the sum of the voltage across the inductor and the voltage across the capacitor. The current i flowing into the positive terminal of the circuit is equal to the current flowing through both the capacitor and the inductor.
Inductive reactance magnitude () increases as frequency increases while capacitive reactance magnitude () decreases with the increase in frequency. At one particular frequency these two reactances are equal in magnitude but opposite in sign. The frequency at which this happens is called the resonant frequency () for the given circuit.
Hence, at resonance:
Solving for , we have
which is defined as the resonant angular frequency of the circuit.
Converting angular frequency (in radians per second) into frequency (in hertz), we have
In a series configuration, X_{C} and X_{L} cancel each other out. In real, rather than idealised components the current is opposed, mostly by the resistance of the coil windings. Thus, the current supplied to a series resonant circuit is a maximum at resonance.
In the series configuration, resonance occurs when the complex electrical impedance of the circuit approaches zero.
First consider the impedance of the series LC circuit. The total impedance is given by the sum of the inductive and capacitive impedances:
By writing the inductive impedance as Z_{L} = jωL and capacitive impedance as Z_{C} = (jωC)^{−1} and substituting we have
Writing this expression under a common denominator gives
Finally, defining the natural angular frequency as
the impedance becomes
The numerator implies that in the limit as the total impedance Z will be zero and otherwise nonzero. Therefore the series LC circuit, when connected in series with a load, will act as a bandpass filter having zero impedance at the resonant frequency of the LC circuit.
In the parallel configuration, the inductor L and capacitor C are connected in parallel, as shown here. The voltage v across the open terminals is equal to both the voltage across the inductor and the voltage across the capacitor. The total current i flowing into the positive terminal of the circuit is equal to the sum of the current flowing through the inductor and the current flowing through the capacitor.
Let R be the internal resistance of the coil. When X_{L} equals X_{C}, the reactive branch currents are equal and opposite. Hence they cancel out each other to give minimum current in the main line. Since total current is minimum, in this state the total impedance is maximum.
Resonant frequency given by: .
Note that any reactive branch current is not minimum at resonance, but each is given separately by dividing source voltage (V) by reactance (Z). Hence I=V/Z, as per Ohm's law.
The same analysis may be applied to the parallel LC circuit. The total impedance is then given by:
and after substitution of and and simplification, gives
which further simplifies to
where
Note that
but for all other values of the impedance is finite. The parallel LC circuit connected in series with a load will act as bandstop filter having infinite impedance at the resonant frequency of the LC circuit. The parallel LC circuit connected in parallel with a load will act as bandpass filter.
The first evidence that a capacitor and inductor could produce electrical oscillations was discovered in 1826 by French scientist Felix Savary.^{[1]}^{[2]} He found that when a Leyden jar was discharged through a wire wound around an iron needle, sometimes the needle was left magnetized in one direction and sometimes in the opposite direction. He correctly deduced that this was caused by a damped oscillating discharge current in the wire, which reversed the magnetization of the needle back and forth until it was too small to have an effect, leaving the needle magnetized in a random direction. American physicist Joseph Henry repeated Savary's experiment in 1842 and came to the same conclusion, apparently independently.^{[3]}^{[4]} British scientist William Thomson (Lord Kelvin) in 1853 showed mathematically that the discharge of a Leyden jar through an inductance should be oscillatory, and derived its resonant frequency.^{[1]}^{[3]}^{[4]} British radio researcher Oliver Lodge, by discharging a large battery of Leyden jars through a long wire, created a tuned circuit with its resonant frequency in the audio range, which produced a musical tone from the spark when it was discharged.^{[3]} In 1857, German physicist Berend Wilhelm Feddersen photographed the spark produced by a resonant Leyden jar circuit in a rotating mirror, providing visible evidence of the oscillations.^{[1]}^{[3]}^{[4]} In 1868, Scottish physicist James Clerk Maxwell calculated the effect of applying an alternating current to a circuit with inductance and capacitance, showing that the response is maximum at the resonant frequency.^{[1]} The first example of an electrical resonance curve was published in 1887 by German physicist Heinrich Hertz in his pioneering paper on the discovery of radio waves, showing the length of spark obtainable from his sparkgap LC resonator detectors as a function of frequency.^{[1]}
One of the first demonstrations of resonance between tuned circuits was Lodge's "syntonic jars" experiment around 1889.^{[1]}^{[3]} He placed two resonant circuits next to each other, each consisting of a Leyden jar connected to an adjustable oneturn coil with a spark gap. When a high voltage from an induction coil was applied to one tuned circuit, creating sparks and thus oscillating currents, sparks were excited in the other tuned circuit only when the circuits were adjusted to resonance. Lodge and some English scientists preferred the term "syntony" for this effect, but the term "resonance" eventually stuck.^{[1]} The first practical use for LC circuits was in the 1890s in sparkgap radio transmitters to allow the receiver and transmitter to be tuned to the same frequency. The first patent for a radio system that allowed tuning was filed by Lodge in 1897, although the first practical systems were invented in 1900 by Italian radio pioneer Guglielmo Marconi.^{[1]}
The Wikibook Circuit Idea has a page on the topic of: How do We Create Sinusoidal Oscillations? 
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