Linear analog electronic filters 

Simple filters

edit 
An LC circuit is a resonant circuit or tuned circuit that consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electric current can alternate between them 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 applications 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. For a model incorporating resistance see RLC circuit.
Contents 
An LC circuit can store electrical energy vibrating at its resonant frequency. A capacitor stores energy in the electric field between its plates, depending on the voltage across it, and an inductor stores energy in its magnetic field, 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. However, the current will continue, because inductors resist changes in current, and energy will be extracted from the magnetic field to keep it flowing. 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). 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 oscillations are very fast, typically hundreds to billions of times per second.
By Kirchhoff's voltage law, we know that the voltage across the capacitor, V_{C}, must equal the voltage across the inductor, :
Likewise, by Kirchhoff's current law, the current through the capacitor plus the current through the inductor must equal zero:
From the constitutive relations for the circuit elements, we also know that
and
After rearranging and substituting, we obtain the second order differential equation
We now define the parameter ω as follows:
With this definition, we 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 and by considering the initial conditions.
Since the exponential is complex, the solution represents a sinusoidal alternating current.
If the initial conditions are such that , then we can use Euler's formula to obtain a real sinusoid with amplitude 2A and angular frequency
Thus, the resulting solution becomes:
The initial conditions that would satisfy this result are:
and
The resonance effect occurs when inductive and capacitive reactances are equal in absolute value. (Notice that the LC circuit does not, by itself, resonate. The word resonance refers to a class of phenomena in which a small driving perturbation gives rise to a large effect in the system. The LC circuit must be driven, for example by an AC power supply, for resonance to occur (below).) 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 determines their "Q" and so selectivity. For a series resonant circuit, 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.
Here R, L, and C are in series in an ac circuit. Inductive reactance magnitude () increases as frequency increases while capacitive reactance magnitude () decreases with increase in frequency. At a particular frequency these two reactances are equal in magnitude but opposite in sign. The frequency at which this happens is the resonant frequency () for the given circuit.
Hence, at :
Converting angular frequency into hertz we get
Here f is the resonant frequency. Then rearranging,
In a series AC circuit, X_{C} leads by 90 degrees while X_{L} lags by 90. Therefore, they cancel each other out. The only opposition to a current is coil resistance. Hence in series resonance the current is maximum at resonant frequency.
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 and substituting we have
Writing this expression under a common denominator gives
Note that the numerator implies if ω^{2}LC = 1 the total impedance Z will be zero and otherwise nonzero. Therefore the series connected circuit, when connected to a circuit in series, will act as a bandpass filter having zero impedance at the resonant frequency of the LC circuits.
Here a coil (L) and capacitor (C) are connected in parallel with an AC power supply. 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 Z_{L} and Z_{C} and simplification, gives
Note that but for all other values of ω^{2}LC the impedance is finite (and therefore less than infinity). Hence the parallel connected circuit will act as bandstop filter having infinite impedance at the resonant frequency of the LC
LC circuits behave as electronic resonators, which are a key component in many applications:
Linear analog electronic filters 

Simple filters

edit 
An LC circuit is a resonant circuit or tuned circuit that consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electric current can alternate between them 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 applications 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. For a model incorporating resistance see RLC circuit.
Contents 
An LC circuit can store electrical energy vibrating at its resonant frequency. A capacitor stores energy in the electric field between its plates, depending on the voltage across it, and an inductor stores energy in its magnetic field, 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, and 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 used. In typical tuned circuits in electronic equipment the oscillations are very fast, thousands to millions of times per second.
By Kirchhoff's voltage law, the voltage across the capacitor, V_{C}, must equal the voltage across the inductor, V_{L}:
Likewise, by Kirchhoff's current law, the current through the capacitor plus the current through the inductor must equal zero:
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, the radian frequency, can be defined as: ω = (LC)^{−1/2}. Using this can simplify the differential equation
The associated polynomial is s^{2} +ω^{2} = 0, 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.
If the initial conditions are such that A = B, then we can use Euler's formula to obtain a real sinusoid with amplitude 2A and angular frequency ω = (LC)^{−1/2}.
Thus, the resulting solution becomes:
The initial conditions that would satisfy this result are:
and
The resonance effect occurs when inductive and capacitive reactances are equal in absolute value. (Notice that the LC circuit does not, by itself, resonate. The word resonance refers to a class of phenomena in which a small driving perturbation gives rise to a large effect in the system. The LC circuit must be driven, for example by an AC power supply, for resonance to occur (below).) 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 $\backslash omega\backslash ,$ 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.
Here L and C are connected in series to an AC power supply. Inductive reactance magnitude ($X\_L\backslash ,$) increases as frequency increases while capacitive reactance magnitude ($X\_C\backslash ,$) decreases with the increase in frequency. At a particular frequency these two reactances are equal in magnitude but opposite in sign. The frequency at which this happens is the resonant frequency ($f\_r\backslash ,$) for the given circuit.
Hence, at $f\_r\backslash ,$ :
Converting angular frequency into hertz we get
Here f is the resonant frequency. Then rearranging,
In a series AC circuit, X_{C} leads by 90 degrees while X_{L} lags by 90. Therefore, they cancel each other out. The only opposition to a current is coil resistance. Hence in series resonance the current is maximum at resonant frequency.
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
The numerator implies that if ω^{2}LC = 1 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.
Here a coil (L) and capacitor (C) are connected in parallel with an AC power supply. 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: $f\; =\; \{1\; \backslash over\; \{2\; \backslash pi\; \backslash sqrt\{LC\}\}\}$.
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 $Z\_\{L\}$ and $Z\_\{C\}$ and simplification, gives
Note that
but for all other values of $\backslash omega^\{2\}\; L\; C$ the impedance is finite (and therefore less than infinity). Hence 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.
This article may require cleanup to meet Wikipedia's quality standards. Please improve this article if you can. The talk page may contain suggestions. (March 2009) 
LC circuits behave as electronic resonators, which are a key component in many applications:
An LC circuit consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electrical current can alternate between them at an angular frequency of
LC circuits are key components in many applications 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. For a model incorporating resistance see RLC circuit.
Contents 
The resonance effect occurs when inductive and capacitive reactances are equal. See: Reactance. [Notice that the LC circuit does not, by itself, resonate. The word resonance refers to a class of phenomena in which a small driving perturbation gives rise to a large effect in the system. The LC circuit must be driven, for example by an AC power supply, for resonance to occur (below).] The frequency at which this equality holds for the particular circuit is called the resonant frequency. The resonant frequency of the LC circuit (in radians per second) is
The equivalent frequency in units of hertz is
Here R, L, and C are in series in an ac circuit. Inductive reactance (X_{L}) increases as frequency increases while capacitive reactance (X_{C}) decreases with increase in frequency. At a particular frequency these two reactances are equal in magnitude but opposite in phase. The frequency at which this happens is the resonant frequency (f_{r}) for the given circuit.
Hence, at f_{r} :
X_{L} = X_{C}
Converting angular frequency into hertz we get
Here f is the resonant frequency. Then rearranging,
In a series ac circuit, X_{C} leads by 90 degrees while X_{L} lags by 90. Therefore, they both cancel each other out. The only opposition to a current is coil resistance. Hence in series resonance the current is maximum at resonant frequency.
Here a coil (L) and capacitor (C) are connected in parallel with an ac power supply. 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.
By Kirchhoff's voltage law, we know that the voltage across the capacitor, V_{C} must equal the voltage across the inductor, V_{L}:
Likewise, by Kirchhoff's current law, the current through the capacitor plus the current through the inductor must equal zero:
From the constitutive relations for the circuit elements, we also know that
and
After rearranging and substituting, we obtain the second order differential equation
We now define the parameter ω as follows:
With this definition, we can simplify the differential equation:
The associated polynomial is s^{2} + ω^{2} = 0, 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.
If the initial conditions are such that A = B, then we can use Euler's formula to obtain a real sinusoid with amplitude 2A and angular frequency .
Thus, the resulting solution becomes:
The initial conditions that would satisfy this result are:
and
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 and substituting we have
Writing this expression under a common denominator gives
Note that the numerator implies if ω^{2}LC = 1 the total impedance Z will be zero and otherwise nonzero. Therefore the series connected circuit, when connected to a circuit in parallel, will act as a bandpass filter having zero impedance at the resonant frequency of the LC circuit.
The same analysis may be applied to the parallel LC circuit. The total impedance is then given by:
and after substitution of Z_{L} and Z_{C} we have
which simplifies to
Note that but for all other values of ω^{2}LC the impedance is finite (and therefore less than infinity). Hence the parallel connected circuit will act as bandstop filter having infinite impedance at the resonant frequency of the LC circuit.
LC circuits are often used as filters; the L/C ratio determines their selectivity. For a series resonant circuit, the higher the inductance and the lower the capacitance, the narrower the filter bandwidth. For a parallel resonant circuit the opposite applies.
LC circuits behave as electronic resonators, which are a key component in many applications:
LC circuit is an electronic circuit made up of an inductor and a capacitor.
Lc

LC circuit's resonant frequency is equal to: $\backslash omega=\backslash sqrt\{1\backslash over\; LC\}$
The angular frequency ω has units of radians per second.
LC circuits are used for creating signals at a particular frequency, or picking out a signal at a particular frequency from a more complex signal. An ideal LC circuit does not have resistance.
At LC circuit energy saves in the capacitor's electric field.
$\{U\}=\{q^2\backslash over2C\}$
U is energy and q is electric charge. At LC circuit energy also save in the inductor's magnetic field.
$\{U\}=\{Li^2\backslash over2\}$
U is energy and i is electric current that flows in inductor.
Let's analyze an LC circuit's vibration. Vibrating LC circuit's total energy is U.
$U=\{q^2\backslash over2C\}+\{Li^2\backslash over2\}$
Because circuit's resistance is 0, there is no energy that transmits to heat energy, and U is maintained regularity. $\{dU\backslash over\; dt\}\; =\; 0$
So LC circuit's vibration is shown like that $\{Ld^2q\backslash over\; dt^2\}+\{q\backslash over\; C\}=0$
First consider the Electrical impedance of the series LC circuit. The total impedance is given by the sum of the inductive and capacitive impedances
$Z\; =\; Z\_\{L\}\; +\; Z\_\{C\}$
By writing the inductive impedance as $Z\_\{L\}\; =\; j\; \backslash omega\; L$ and capacitive impedance as $Z\_\{C\}\; =\; \backslash frac\{1\}\{j\{\backslash omega\; C\}\}$
Resultingly the series connected circuit, when connected to a circuit in series, will act as a bandpass filter having zero impedance at the resonant frequency of the LC circuits.
The same analysis may be applied to the parallel LC circuit. The total impedance is then given by
$Z=\backslash frac\{Z\_\{L\}Z\_\{C\}\}\{Z\_\{L\}+Z\_\{C\}\}$
and after substitution of $Z\_\{L\}$ and $Z\_\{C\}$ we have
$Z=\backslash frac\{\backslash frac\{L\}\{C\}\}\{\backslash frac\{(\backslash omega^\{2\}LC1)j\}\{\backslash omega\; C\}\}$
which simplifies to
$Z=\backslash frac\{L\backslash omega\; j\}\{\backslash omega^\{2\}LC1\}$ .
Resultingly the parallel connected circuit will act as bandstop filter having infinite impedance at the resonant frequency of the LC circuit.
