RLC Circuits
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ELI the ICE man
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Series RLC Circuit
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Impedance in a series RLC
circuit
1
XC =
2πfC
X L = 2πfL
Z= R +X
2
2
Z = R + (X L − X C )
2
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2
Series RLC circuit
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Current in a series RLC circuit
I = I R = I L = IC
VS
I=
Z
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Voltage in a series RLC circuit
VR = I × R
VL = I × X L
VC = I × X C
VS = V + (VL − VC )
2
R
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2
Voltage in a series RLC circuit
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Phase angle in a series RLC
circuit
VL − VC
θ = arctan
VR
X L − XC
θ = arctan
R
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Power in a series RLC circuit
PR = I × R
2
PA = VS × I
R PR
PF = cosθ = =
Z PA
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Parallel RLC circuit
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Voltage in a parallel RLC circuit
VR = VL =VC =VS
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Current in a parallel RLC circuit
V
IR =
R
V
IL =
XL
V
IC =
XC
I T = I + I = I + (I L − I C )
2
R
2
X
2
R
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2
Phase angle in a parallel RLC
circuit
I L − IC
θ = arctan
IR
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Example RLC parallel circuit
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Impedance in a parallel RLC
circuit
VS
Z=
IT
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Power in a parallel RLC circuit
PR = I × R
2
R
PA = VS × I T
PR
PF = cosθ =
PA
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Resonance
Resonance – A circuit condition that occurs when the
inductive reactance (XL) is equal to the capacitive
reactance (XC).
F0 = resonant frequency in hertz
f0 =
1
2π LC
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Series resonance of an RLC
circuit
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Quality factor for a series RLC
resonance circuit
X L VL VC X C
=
=
=
Q=
R VR VR
R
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Bandwidth of series resonant
RLC circuit
Bandwidth – The width of the group or band of
frequencies between the half-power points.
Frequency Response Curve – A graph
indicating a circuit’s response to different
frequencies.
Cutoff frequency – The frequency at which the
gain of the circuit falls below 0.707 of the
maximum current or half power (-3dB).
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Bandwidth of series resonant
RLC circuit (continued)
f0
BW =
Q f0
Q f o = Quality
Factor
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at
f0
Series resonant circuit
bandwidth
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Bandwidth of series resonant
RLC circuit (continued)
Half Power Point – A point at which power is
50%. This half power point corresponds to
70.7% of the total current.
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Parallel resonance
Parallel Resonance Circuit – A circuit having
an inductor and capacitor in parallel with one
another, offering a high impedance at the
frequency of resonance.
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Parallel resonant circuit
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Flywheel action and
tank circuit
Flywheel Action – The sustaining effect of
oscillation in an LC circuit due to the
charging and discharging of the capacitor and
the expansion and contraction of the
magnetic field around the inductor.
Tank Circuit – A circuit made up of a coil
and a capacitor that is capable of storing
electric energy.
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Quality factor in a parallel
resonance
VC VL
Q=
=
VS VS
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Quality factor in a tank circuit at
resonance
I Tank Z Tank
Q=
=
IS
XL
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Bandwidth in a parallel circuit at
resonance
f0
BW =
Q f0
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Selectivity
Tuned Circuit – A circuit that can have its
components’ values varied so that the circuit
responds to one selected frequency, yet
heavily attenuates all other frequencies.
Selectivity – The characteristic of a circuit to
discriminate between the wanted signal and
the unwanted signal.
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Applications of RLC circuits
Low-pass filter
High-pass filter
Bandpass filter
Band-stop filter
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Low-pass filter
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High-pass Filter
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Bandpass filter
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Band-stop filter
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Complex numbers
Complex Numbers – Numbers composed of a real
number and an imaginary number
Real Numbers – Numbers that have no imaginary part
Imaginary number – A complex number whose
imaginary part is not zero
j Operator – A prefix used to indicate an imaginary
number
Complex Plane – A plane whose points are identified
by means of complex
numbers
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Complex plane
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Complex numbers examples
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Coordinates
Rectangular Coordinates – A Cartesian coordinate
system whose straight-line axes or coordinate planes
are perpendicular .
x + jy
Polar Coordinates – Either of two numbers that
locate a point in a plane by its distance from a fixed
point or a line and the angle this line makes with a
fixed line.
r<
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Representing phasors. (a)
Rectangular notation. (b) Polar
notation
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Polar to rectangular conversion
r<θ
x + jy
x = r cos θ
y = r sin θ
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Rectangular to polar conversion
r ⟨θ
x + jy
r= x +y
2
2
y
θ = arctan
x
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Learn to use your calculator to perform
P
R
and
R
P
Conversions.
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Addition of complex numbers
(A + jB) + (C + jD)
= (A + C) + j(B + D)
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Subtraction of complex numbers
(A + jB) − (C + jD)
= (A − C) + j(B − D)
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Multiplication of complex
numbers
(A < θ1) × (B < θ 2)
= A × B < (θ 1 + θ 2)
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Division of complex numbers
A<θ1
B<θ2
= A/B < (θ 1 − θ 2)
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Complex number in ac circuits
On the complex plane:
R is on the Real axis,
XL is on the +j axis, and
XC is on the –j axis.
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Impedance in a series ac circuit
In rectangular form:
ZT = R + j(XL-XC)
In polar form:
ZT = R + ( X L − X C )
2
2
 X L − XC 
⟨ arctan

R


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Current
VS
I=
ZT
Where VS and ZT are in polar form
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End of RLC Circuits
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