لخّصلي

RLC circuits are circuits that contain an ohmic resistor R, an inductor L and
a capacitor C connected in series or parallel and connected to an alternating current
source as shown in the following figures.
Series connection Parallel connection
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When alternating current passes through the ohmic resistance, it encounters
resistance as a result of the collision of the current's electrons with the conductor's
atoms and molecules, and energy is exhausted in it in the form of heat. The
alternating voltage and alternating current are in phase, that is, they have the same
phase angle
When alternating current passes through an induction coil (without
resistance), the coil’s self-induction generates a reverse induced electromotive
force in the coil that resists the original current, and this resistance is called the
“inductive reactance of the coil” and is symbolized by the symbol (XL). It can be
defined as “the impedance or resistance encountered by an alternating current when
passing through the coil due to its self-induction” and its unit is ohm. Factors on
which the inductive reactance of a coil (XL) depends:
 frequency (f)
 Coefficient of self-inductance of the coil (L)
𝑿𝑳 = 𝟐𝝅 𝒇 𝑳 ----------------------------(1)
It is clear from this equation that the relationship between
the inductive reactance and the frequency is a direct
relationship, as the inductive reactance of the coil
increases with increasing the frequency.

When alternating current passes through the capacitor, it winds up a type of
resistance called the “capacitive reactance of the capacitor” because it arises from
the capacitance of the capacitor and is denoted by the symbol “XC”. It can be
defined as "the impedance or resistance encountered by alternating current when
passing through a capacitor its capacitance" and its unit is ohm. The capacitive
reactance of the capacitor (XC) depends on:
 frequency (f)
 capacitance of the capacitor (C)
𝑿𝑪 =
𝟏
𝟐𝝅 𝒇 𝑪
-----------------------------(2)
It is clear from this equation that the relationship between
capacitive reactance and frequency is an inverse
relationship, as the capacitive reactance of a capacitor
decreases with increasing the frequency.
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AC voltage (V) and impedance (Z) in RLC circuit
When we connect an alternating voltage source with a resistor, an inductor and
a capacitor in series, as shown in the figure below, the electric current will pass and
the following voltage differences VR, VL, and VC will be generated on the resistor,
coil and capacitor, respectively. Using the vector method, the total potential
difference V is:
𝑽 = √𝑽𝑹
𝟐 + (𝑽𝑳 − 𝑽𝑪
)
𝟐
And by compensating for the voltage
differences:
V=IZ VR= IR VL = IXL VC = IXC
we get
𝐈𝐙 = 𝐈 √𝐑𝟐 + (𝐗𝐋 − 𝐗𝐂
)
𝟐
Therefore, the total impedance (Z) of the RLC circuit in the case of series
connection is:
𝐙 = √𝐑𝟐 + (𝐗𝐋 − 𝐗𝐂
)
𝟐 -----------------------------(3)
Resonance in RLC circuit
The state of resonance occurs when the inductive reactance of the coil is
equal to the capacitive reactance of the capacitor, i.e. the effect of each one
cancels out. In this case, the circuit resistance is as small as possible and the
current in the circuit becomes as large as possible, and the current is in phase with
the potential difference:
𝑿𝑳 = 𝑿𝑪
𝟐𝝅 𝒇 𝑳 =
𝟏
𝟐𝝅 𝒇 𝑪
𝒇 =
𝟏
𝟐𝝅 √𝑳 𝑪
---------------------------------(4)
This frequency is called the frequency of the resonant circuit. and it depends on:
 Coefficient of self-inductance of the coil (L)
 Capacitance of the capacitor (C)
And the total circuit impedance (Z) in the case of series connection is:
𝐙 = √𝐑𝟐 + (𝐗𝐋 − 𝐗𝐂
)
𝟐 = 𝐑 -----------(5)
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Some Applications of Resonance Circuits
Resonance circuits are used in receivers such as radio and television, as each
radio or television station has a specific frequency. In the receiver, we receive the
frequency that passes in the resonance circuit, and whose resistance is the least
possible, and the rest of the frequencies do not pass because the impedance of the
receiving circuit is large and the capacitance of the capacitor changes (By managing
the condenser plates to change the area) it is possible to move between stations