A circuit RL is an electrical circuit containing a resistance and a reel in series. It is said that the reel is opposed transitorily to the establishment of the current in the circuit.

The differential equation which governs the circuit is the following one:

U = L (di ⁄ dt) + R_{t}*i

Circuit RL series

U = L (di ⁄ dt) + R

Circuit RL series

- With:
- U : terminal voltage of the assembly, out of V
- I : the intensity of the electric current in has
- L : the inductance of the reel out of H
- R
_{t}: total resistance of the circuit out of Ω

Transitory mode

The general solution, associated with the initial condition

i_{bobine} (t = 0) = 0, is:

i_{bobine} = U/R_{t} (1 - e^{-t ⁄ T})

i

i

- ibobine: the intensity of the electric current crossing the assembly, in has
- L : the inductance of the reel out of H
- R
_{t}: the total resistance of the circuit in Ω - U : the tension of the generator, out of V
- T : time in S
- τ : the time-constant of the circuit, in S

It is the time-constant τ which characterizes the duration of the transitory mode. Thus, the closed-circuit current is established to 1% near at the end of a duration of 5t.

When the current becomes permanent, the equation is simplified in U=Ri because Ldi ⁄ dt=0.

Permanent sinusoidal mode

In permanent sinusoidal mode, the circuit can be characterized by a complex impedance Z Worth Z = R_{t} + jωL.

Into electrokinetic, a circuit RLC is a linear circuit containing an electrical resistance, a reel (inductance) and a condenser (capacity).

There exist two types of circuits RLC series or parallel, according to the interconnection of the three types of components. The behavior of a circuit RLC is generally described by a differential equation of the second order where circuits RL or circuits RC behave like first order circuits.

Using a generator of signals, it is possible to inject into the circuit of the oscillations and to observe in certain cases a resonance, characterized by an increase in the current when the selected entry signal corresponds to the own pulsation of the circuit, calculable starting from the differential equation which governs it.

Circuit RLC in series

Circuit subjected to a level of tension

If a circuit RLC series is subjected to a level of tension E, the law of the meshs imposes the relation :

E = u_{C} + u_{L} + u_{R} = u_{C} + L * (di ⁄ dt) + R_{t}i

By introducing the relation characteristic of the condenser :

i_{C} = i = C * (du_{C} ⁄ dt)

one obtains the differential equation of the second order :

LC * (d²u_{c} ⁄ dt²) + R_{t}C * (du_{c} ⁄ dt) + u_{C} = E

Circuit subjected to a level of tension

If a circuit RLC series is subjected to a level of tension E, the law of the meshs imposes the relation :

E = u

By introducing the relation characteristic of the condenser :

i

one obtains the differential equation of the second order :

LC * (d²u

- E : the electromotive force of the generator, in volts
- u
_{C}: the terminal voltage of the condenser, in volts - L : the inductance of the reel, in henrys
- I : intensity of the electric current in the circuit, in amps
- Q : the electric charge of the condenser, in coulombs
- C : electric capacity of the condenser, in farads
- R
_{t}: the total resistance of the circuit, in ohms - T : time in seconds

In the case of a mode without losses, That is to say for R_{t} = 0

one obtains a solution putting itself in the form :

u_{c} = E + A cos [(2πt ⁄ T_{0}) + φ]

T_{0} = 2π√LC

one obtains a solution putting itself in the form :

u

T

- With :
- T0 the period of oscillation, in seconds
- With and φ two constants to be determined thanks to the initial conditions of the circuit.

What gives us :

ƒ_{0} = 1 ⁄ 2π√LC

where ƒ is the Eigen frequency of the circuit, in hertz.

ƒ

where ƒ is the Eigen frequency of the circuit, in hertz.

Circuit subjected to a sinusoidal tension

The complex transformation applied to the various tensions makes it possible to write the law of the meshs in the form :

U_{G} = U_{C} + U_{L} + U_{R}

maybe, by introducing the complex impedances

U_{G} = -(j ⁄ C_{ω}) I + jLωI + R_{t}I = [R_{t} + j(LCω² -1 ⁄ Cω)] I

The angular frequency of resonance in intensity of such a circuit ω_{0} is given by :

ω_{0} = 1 ⁄ √LC

For this frequency the relation above becomes :

U_{G} = U_{R} = R_{t}I

and one a :

U_{L} = -U_{C} = j ⁄ R_{t} √(L ⁄ C) U_{G}

U

maybe, by introducing the complex impedances

U

The angular frequency of resonance in intensity of such a circuit ω

ω

For this frequency the relation above becomes :

U

and one a :

U

Circuit RLC in parallel

Parallel circuit RLC, known as antiresonant circuit

i_{r} = u ⁄ R

di_{l} ⁄ dt = u ⁄ L

i_{c} = dq ⁄ dt = C (du ⁄ dt)

because q = Cu

i = i_{r} + i_{l} + i_{c}

di ⁄ dt = C (d²u ⁄ dt²) + (1du ⁄ Rdt) + u ⁄ L

Parallel circuit RLC, known as antiresonant circuit

i

di

i

because q = Cu

i = i

di ⁄ dt = C (d²u ⁄ dt²) + (1du ⁄ Rdt) + u ⁄ L

Caution : the branch C is in short-circuit : one cannot connect has, B directly at the boundaries of a generator E, it is necessary to add a resistance to him.

- The two initial conditions are :
- i
_{l0}guard its value before the powering because inductance is opposed to the variation of the current - q
_{0}guard its value before the powering u_{0}= q_{0}⁄ C

Circuit subjected to a sinusoidal tension

The complex transformation applied to the various intensities gives

I = I_{r} + I_{l} + I_{c}

maybe, by introducing the complex impedances

I = (1 ⁄ R) U + (1 ⁄ jLω) U + jCωU

that is to say

I = (1 ⁄ R) + j [Cω - (1 ⁄ Lω)] U

The angular frequency of resonance in intensity of such a circuit ω_{0} is given by :

ω = 1 ⁄ √LC

For this frequency the relation above becomes

I = I_{r} = (1 ⁄ R) U

and one has

I_{c} = -I_{l} = j√(C ⁄ L) U

The complex transformation applied to the various intensities gives

I = I

maybe, by introducing the complex impedances

I = (1 ⁄ R) U + (1 ⁄ jLω) U + jCωU

that is to say

I = (1 ⁄ R) + j [Cω - (1 ⁄ Lω)] U

The angular frequency of resonance in intensity of such a circuit ω

ω = 1 ⁄ √LC

For this frequency the relation above becomes

I = I

and one has

I

Use of circuits RLC

Circuits RLC are generally used to produce filters of frequency, or transformers of impedance.

Thus, parallel circuit RLC is commonly called antiresonant circuit because it reduces to zero certain often undesirable frequencies for the apparatus in which it is integrated, allowing for example to eliminate the parasites in a receiver.

A circuit RC is an electrical circuit, composed of a resistance and a condenser assembled in series or parallel. In their configuration series, circuits RC make it possible to produce low-pass or high-pass electronic filters. The time-constant t of a circuit RC is given by the product of the value of these two elements which compose the circuit.

Circuit series

Circuit RC series

Transfer transfer functions

That is to say Z_{C}(ω) impedance of the condenser

Z_{C}(ω) = 1 ⁄ jCω

The terminal voltage of resistance or the condenser

can be calculated by regarding the assembly as a tension divider uncharged

V_{C}(ω) = [Z_{C}(ω) ⁄ Z_{C}(ω) + R] V_{in}(ω) = [1 ⁄ 1 + jRCω] V_{in}(ω)

V_{R}(ω) = [R ⁄ Z_{C}(ω) + R] V_{in}(ω) = [jRCω ⁄ 1 + jRCω] V_{in}(ω)

Circuit RC series

Transfer transfer functions

That is to say Z

Z

The terminal voltage of resistance or the condenser

can be calculated by regarding the assembly as a tension divider uncharged

V

V

One will note H_{C} the transfer transfer function obtained by regarding the terminal voltage of the condenser as output voltage and H_{R} if one uses that at the boundaries of resistance. H_{C} and H_{R} are obtained respectively thanks to the expressions of V_{C} and V_{R}

H_{C}(ω) = V_{C}(ω) ⁄ V_{in}(ω) = 1 ⁄ 1 + jRCω

H_{R}(ω) = V_{R}(ω) ⁄ V_{in}(ω) = jRCω ⁄ 1 + jRCω

H

For a dipole, one can write the transfer transfer function in the form H(ω) = Ge^{jφ}, where G is the profit of the dipole and φ its phase.

Thus

H_{C}(ω) = G_{C}e^{jφc}

with

G_{C} = 1 ⁄ √[1 + (ωRC)²]

and

φ_{C} = arctan (-ωRC)

In the same way for H_{R}

H_{R}(ω) = G_{R}e^{jφR}

with

G_{R} = ωRC ⁄ √[1 + (ωRC)²]

and

φ_{R} = arctan (1 ⁄ ωRC)

H

with

G

and

φ

In the same way for H

H

with

G

and

φ

Frequential analysis

A frequential analysis of the assembly makes it possible to determine which frequencies the filter rejects or accepts. For the low frequencies, H_{C} has a module close to one and a phase close to zero. The more the frequency increases, the more its module decreases for tending towards zero and its phase of -π ⁄ 2. Conversely, H_{R} has a module close to zero with the low frequencies and a phase close to -π ⁄ 2 and when the frequency increases, its module tend towards one and its phase towards zero.

When ω = 0:

G_{C} → 1 and φ_{C} → 0

G_{R} → 0 et φ_{R} → 90° = π ⁄ 2

When ω → ∞

G_{C} → 0 et φ_{C} → -90° = -π ⁄ 2

G_{R} → and φ_{R} → 0

G

G

When ω → ∞

G

G

Thus, when the exit of the filter is taken on the condenser the behavior is of the type filters low-pass : the high frequencies are attenuated and the low frequencies pass. If the exit is taken on resistance, the reverse occurs and the circuit behaves like a high-pass filter.

The cut-off frequency ƒ_{c} of the circuit which defines the limit in 3 dB

between the attenuated frequencies and those which are not it is equal to

ƒ_{c} = 1 ⁄ 2πRC (in Hz)

between the attenuated frequencies and those which are not it is equal to

ƒ

Temporal analysis

for reasons of simplicity, the temporal analysis will be carried out by using the transform of Laplace p

by supposing that the circuit is subjected to a level of tension of amplitude V in entry

V_{in} = 0 for t = 0 and V_{in} = V otherwise

V_{in} (P) = V ⁄ p

V_{C} (p) = H_{C} (p) V_{in} (p) = (1 ⁄ 1 + pRC) * V ⁄ p

V_{R} (p) = H_{R} (p) V_{in} (p) = (pRC ⁄ 1 + pRC) * V ⁄ p

for reasons of simplicity, the temporal analysis will be carried out by using the transform of Laplace p

by supposing that the circuit is subjected to a level of tension of amplitude V in entry

V

V

V

V

The transform of opposite Laplace of these expressions gives

V_{C} (t) = V (1 - e^{-t ⁄ RC})

V_{R} (t) = Ve^{-t ⁄ RC}

in this case, the condenser takes care and the tension on its terminals tends towards V

while that at the boundaries of resistance tends towards 0

graphic determination of τ by the observation of V_{C} (t)

V

V

in this case, the condenser takes care and the tension on its terminals tends towards V

while that at the boundaries of resistance tends towards 0

graphic determination of τ by the observation of V

Circuit RC has a time-constant, generally noted τ = RC, representing the time which the tension takes to carry out 63% (1 - e^{-1}) of the variation necessary to pass from its initial value to its end value.

It is also possible to derive these expressions from the differential equations describing the circuit

V_{in} - V_{C} ⁄ R = C * (dV_{C} ⁄ dt)

V_{R} = V_{in} - V_{C}

The solutions are exactly the same ones as those obtained by the transform of Laplace.

V

V

The solutions are exactly the same ones as those obtained by the transform of Laplace.

Integrator

high frequency,that is to say if ω » 1 ⁄ RC, the condenser do not have time to take care and the tension on its terminals remains weak

thus

V_{R} ≈ V_{in}

and the intensity in the circuit is thus worth

I ≈ V_{in} ⁄ R

Like

V_{C} = 1 ⁄ C ∫_{0}^{t} Idt

one obtains

V_{C} ≈ 1 ⁄ RC ∫_{0}^{t} V_{in}dt

high frequency,that is to say if ω » 1 ⁄ RC, the condenser do not have time to take care and the tension on its terminals remains weak

thus

V

and the intensity in the circuit is thus worth

I ≈ V

Like

V

one obtains

V

The terminal voltage of the condenser thus integrates the tension of entry and the circuit behaves like an integrating assembly, that is to say like a low-pass filter.

Shunting device

low frequency, that is to say if

ω « 1 ⁄ RC

the condenser has time to take care almost completely

Then

I ≈ V_{in} ⁄ (1 ⁄ jωC)

V_{in} ≈ 1 ⁄ jωC ≈ V_{C}

Now

V_{R} = IR = C * (dV_{C} ⁄ dt) * R

V_{R} ≈ RC * (dV_{in} ⁄ dt)

low frequency, that is to say if

ω « 1 ⁄ RC

the condenser has time to take care almost completely

Then

I ≈ V

V

Now

V

V

The terminal voltage of resistance thus derives the tension from entry and the circuit behaves like an assembly shunting device, that is to say like a high-pass filter.

Intensity

The intensity of the current is the same one in all the circuit

since it is about a circuit series

I (ω) = V_{in} (ω) ⁄ R + Z_{C} = jCω ⁄ 1 + jRCω * V_{in} (ω)

The intensity of the current is the same one in all the circuit

since it is about a circuit series

I (ω) = V

Impulse response

The impulse response is the transform of opposite Laplace of the corresponding transfer transfer function and represents the response of the circuit to an impulse.

For the condenser

H_{C} (t) = 1 ⁄ RC * e^{-t ⁄ RC} u (t) = 1 ⁄ τ * e^{-t ⁄ τ} u (t)

where U (t) is the function of Heaviside and τ = RC is the time-constant.

for resistance

H_{R} (t) = - 1 ⁄ RC * e^{-t ⁄ RC} u (t) = -1 ⁄ τ * e^{-t ⁄ τ u (t)}

H

where U (t) is the function of Heaviside and τ = RC is the time-constant.

for resistance

H

Parallel circuit

Parallel circuit RC is generally of an interest less than circuit RC series : the output voltage being equal to the tension of entry, it can be used like only filters supplied with a power source.

The intensities in the two dipoles are

I_{R} = V_{in} ⁄ R

I_{C} = jωCV_{in}

I

I

The current in the condenser is out of phase of 90° compared to the current of entry and resistance.

Subjected to a level of tension, the condenser takes care quickly and can be regarded as an open circuit, the circuit behaving consequently like a simple resistance.

A circuit LLC is an electrical circuit containing a reel (L) and a condenser. Thus the electric phenomenon of resonance is obtained.

This type of circuit is used in the filters, the tuners and the mixers of frequencies. Consequently, its use is widespread in the transmissions without wire in broadcasting, as much for the emission that the reception.

Circuit LLC series and parallel

Electric resonance

Operation

Electric resonance

Operation

The electric phenomenon of resonance occurs in an electrical circuit at a frequency of resonance given where the imaginary parts of the impedance and admittance of the elements of circuit are cancelled. In certain circuits, electric resonance takes place when impedance between the entry and the exit of the circuit is nearly zero and the transfer transfer function is about the unit. The resonant circuits comprise repercussions and can generate higher voltages and currents that those which they receive, which makes them useful for the transmission without wire.

In a circuit made up of condensers and reels, the magnetic field in a reel induces an electric current in rollings up of this reel to charge a condenser. When it discharges, the condenser produced an electric current which reinforces the magnetic field in the reel. This process is repeated continuously, in a way comparable with the process of swinging of a mechanical clock. In certain cases, resonance takes place when the reactances of reel and condenser are equal magnitudes, so that electrical energy oscillates between the magnetic field of the reel and the electric field of the condenser.

Frequency of resonance

The own pulsation or of resonance of a circuit LLC is

ω_{0} = √ 1 ⁄ LC

What gives us the Eigen frequency

or of resonance of a circuit LLC in hertz

ƒ_{0} = ω_{0} ⁄ 2π = 1 ⁄ 2π√LC

The own pulsation or of resonance of a circuit LLC is

ω

What gives us the Eigen frequency

or of resonance of a circuit LLC in hertz

ƒ

Impedance

A circuit LLC series

the impedance of a circuit series is given by

the sum of the impedances of each one of its components

Z = Z_{L} + Z_{C}

With

Z_{L} = jωL

impedance of the reel and

Z_{C} = 1 ⁄ jωC

impedance of the condenser

Z = jωL + 1 ⁄ jωC

what gives us once reduced to the same denominator

Z = (ω²LC - 1)j ⁄ ωC

it will be noticed that the impedance is null with the pulsation of resonance

ω_{0} = √1 ⁄ LC

A circuit LLC series

the impedance of a circuit series is given by

the sum of the impedances of each one of its components

Z = Z

With

Z

impedance of the reel and

Z

impedance of the condenser

Z = jωL + 1 ⁄ jωC

what gives us once reduced to the same denominator

Z = (ω²LC - 1)j ⁄ ωC

it will be noticed that the impedance is null with the pulsation of resonance

ω

The circuit thus behaves like a filter band pass or a filter band suppressor, according to how it is laid out in the network considered.

A parallel circuit LLC

the impedance of the circuit is given by the formula

Z = Z_{L}Z_{C} ⁄ Z_{L} + Z_{C}

After substitution of Z_{L} and Z_{C}

by their literal formulas, one obtains

Z = L ⁄ C ⁄ (ω²LC - 1)i ⁄ ωC

Who is simplified in

Z = -Lωi ⁄ ω²LC - 1

the impedance of the circuit is given by the formula

Z = Z

After substitution of Z

by their literal formulas, one obtains

Z = L ⁄ C ⁄ (ω²LC - 1)i ⁄ ωC

Who is simplified in

Z = -Lωi ⁄ ω²LC - 1

Parallel circuit LLC thus has an infinite impedance to the frequency of resonance. According to the way in which one lays out it in a network, it will be able to act as a filter passes band or like a filter band crosses.

Selectivity

Band-width with -3dB

Band-width with -3dB

Circuits LLC are often used as filters. If a circuit LLC is used as filter passes band, one generally defines his band-width in -3dB around his frequency of resonance. One will have then the over-tension coefficient Q equal to the relationship between the frequency of resonance and the band-width. Q = ƒ0 ⁄ BP

The higher the over-tension coefficient is, the more the circuit is selective.

Q high corresponds to a resistance weak R_{s} series, or to a parallel resistance R_{p} large.

One can calculate Q according to resistances

Q = Z ⁄ R_{s} = R_{p} ⁄ Z

where Z is the module of the impedances of the reel or the condenser.

Q = Z ⁄ R

where Z is the module of the impedances of the reel or the condenser.

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