### Circuit RL

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) + Rt*i

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
Rt: total resistance of the circuit out of Ω
Transitory mode
The general solution, associated with the initial condition
ibobine (t = 0) = 0, is:
ibobine = U/Rt (1 - e-t ⁄ T)
ibobine: the intensity of the electric current crossing the assembly, in has
L : the inductance of the reel out of H
Rt : 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 = Rt + jωL.

### Circuit RLC

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 = uC + uL + uR = uC + L * (di ⁄ dt) + Rti
By introducing the relation characteristic of the condenser :
iC = i = C * (duC ⁄ dt)
one obtains the differential equation of the second order :
LC * (d²uc ⁄ dt²) + RtC * (duc ⁄ dt) + uC = E
E : the electromotive force of the generator, in volts
uC : 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
Rt : 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 Rt = 0
one obtains a solution putting itself in the form :
uc = E + A cos [(2πt ⁄ T0) + φ]
T0 = 2π√LC
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 :
ƒ0 = 1 ⁄ 2π√LC
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 :
UG = UC + UL + UR
maybe, by introducing the complex impedances
UG = -(j ⁄ Cω) I + jLωI + RtI = [Rt + j(LCω² -1 ⁄ Cω)] I
The angular frequency of resonance in intensity of such a circuit ω0 is given by :
ω0 = 1 ⁄ √LC
For this frequency the relation above becomes :
UG = UR = RtI
and one a :
UL = -UC = j ⁄ Rt √(L ⁄ C) UG
Circuit RLC in parallel

Parallel circuit RLC, known as antiresonant circuit
ir = u ⁄ R
dil ⁄ dt = u ⁄ L
ic = dq ⁄ dt = C (du ⁄ dt)
because q = Cu
i = ir + il + ic
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 :
il0 guard its value before the powering because inductance is opposed to the variation of the current
q0 guard its value before the powering u0 = q0 ⁄ C
Circuit subjected to a sinusoidal tension
The complex transformation applied to the various intensities gives
I = Ir + Il + Ic
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 ω0 is given by :
ω = 1 ⁄ √LC
For this frequency the relation above becomes
I = Ir = (1 ⁄ R) U
and one has
Ic = -Il = j√(C ⁄ L) U
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.

### Circuit RC

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 ZC(ω) impedance of the condenser
ZC(ω) = 1 ⁄ jCω
The terminal voltage of resistance or the condenser
can be calculated by regarding the assembly as a tension divider uncharged
VC(ω) = [ZC(ω) ⁄ ZC(ω) + R] Vin(ω) = [1 ⁄ 1 + jRCω] Vin(ω)
VR(ω) = [R ⁄ ZC(ω) + R] Vin(ω) = [jRCω ⁄ 1 + jRCω] Vin(ω)
One will note HC the transfer transfer function obtained by regarding the terminal voltage of the condenser as output voltage and HR if one uses that at the boundaries of resistance. HC and HR are obtained respectively thanks to the expressions of VC and VR
HC(ω) = VC(ω) ⁄ Vin(ω) = 1 ⁄ 1 + jRCω
HR(ω) = VR(ω) ⁄ Vin(ω) = jRCω ⁄ 1 + jRCω
For a dipole, one can write the transfer transfer function in the form H(ω) = Ge, where G is the profit of the dipole and φ its phase.
Thus
HC(ω) = GCejφc
with
GC = 1 ⁄ √[1 + (ωRC)²]
and
φC = arctan (-ωRC)
In the same way for HR
HR(ω) = GRejφR
with
GR = ωRC ⁄ √[1 + (ωRC)²]
and
φR = arctan (1 ⁄ ωRC)
Frequential analysis
A frequential analysis of the assembly makes it possible to determine which frequencies the filter rejects or accepts. For the low frequencies, HC 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, HR 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:
GC → 1 and φC → 0
GR → 0 et φR → 90° = π ⁄ 2
When ω → ∞
GC → 0 et φC → -90° = -π ⁄ 2
GR → and φR → 0
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 ƒc of the circuit which defines the limit in 3 dB
between the attenuated frequencies and those which are not it is equal to
ƒc = 1 ⁄ 2πRC (in Hz)
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
Vin = 0 for t = 0 and Vin = V otherwise
Vin (P) = V ⁄ p
VC (p) = HC (p) Vin (p) = (1 ⁄ 1 + pRC) * V ⁄ p
VR (p) = HR (p) Vin (p) = (pRC ⁄ 1 + pRC) * V ⁄ p
The transform of opposite Laplace of these expressions gives
VC (t) = V (1 - e-t ⁄ RC)
VR (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 VC (t)
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
Vin - VC ⁄ R = C * (dVC ⁄ dt)
VR = Vin - VC
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
VR ≈ Vin
and the intensity in the circuit is thus worth
I ≈ Vin ⁄ R
Like
VC = 1 ⁄ C ∫0t Idt
one obtains
VC ≈ 1 ⁄ RC ∫0t Vindt
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 ≈ Vin ⁄ (1 ⁄ jωC)
Vin ≈ 1 ⁄ jωC ≈ VC
Now
VR = IR = C * (dVC ⁄ dt) * R
VR ≈ RC * (dVin ⁄ dt)
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) = Vin (ω) ⁄ R + ZC = jCω ⁄ 1 + jRCω * Vin (ω)
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
HC (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
HR (t) = - 1 ⁄ RC * e-t ⁄ RC u (t) = -1 ⁄ τ * e-t ⁄ τ u (t)
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
IR = Vin ⁄ R
IC = jωCVin
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.

### Circuit LLC

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
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
ω0 = √ 1 ⁄ LC
What gives us the Eigen frequency
or of resonance of a circuit LLC in hertz
ƒ0 = ω0 ⁄ 2π = 1 ⁄ 2π√LC
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 = ZL + ZC
With
ZL = jωL
impedance of the reel and
ZC = 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
ω0 = √1 ⁄ LC
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 = ZLZC ⁄ ZL + ZC
After substitution of ZL and ZC
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
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 Rs series, or to a parallel resistance Rp large.
One can calculate Q according to resistances
Q = Z ⁄ Rs = Rp ⁄ Z
where Z is the module of the impedances of the reel or the condenser.

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