Formula Sheets: Resonance | Network Theory (Electric Circuits) - Electrical Engineering (EE) PDF Download

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Net w ork Theory: Resonance F orm ula Sheet for
Electrical GA TE
Resonance Basics
• Definition : Resonance o ccurs in an RLC circuit when the inductiv e and capaci-
tiv e reactances cancel eac h other, resulting in maxim um curren t (series) or maxim um
imp edance (parallel).
• Resonan t F requency :
?
0
=
1
v
LC
(rads
-1
), f
0
=
1
2p
v
LC
(Hz)
where L is inductance (H ), C is capacitance (F ).
Series Resonance
• Imp edance : Z = R+j
(
?L-
1
?C
)
.
• A t Resonance : ?L =
1
?C
, so Z = R (minim um imp edance).
• Curren t at Resonance : I
max
=
V
R
, where V is the source v oltage (RMS or p eak).
• V oltage A cross Comp onen ts :
V
L
= I
max
·?
0
L, V
C
= I
max
·
1
?
0
C
• Qualit y F actor (Q ) :
Q =
?
0
L
R
=
1
R
v
L
C
=
1
?
0
RC
• Bandwidth : BW =
?
0
Q
=
R
L
(rads
-1
) .
• Half-P o w er F requencies :
?
1
= ?
0
-
BW
2
, ?
2
= ?
0
+
BW
2
P arallel Resonance
• A dmittance : Y =
1
R
+j
(
?C-
1
?L
)
.
• A t Resonance : ?C =
1
?L
, so Y =
1
R
(minim um admittance, maxim um imp edance).
• Imp edance at Resonance : Z
max
= R (ideal case) or Z
max
˜
L
RC
(practical case).
• Qualit y F actor (Q ) :
Q = R
v
C
L
= R?
0
C =
R
?
0
L
• Bandwidth : BW =
?
0
Q
=
1
RC
(rads
-1
). Curren t at Resonance : Minimumcurrent,asimpedanceismaximum.
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Net w ork Theory: Resonance F orm ula Sheet for
Electrical GA TE
Resonance Basics
• Definition : Resonance o ccurs in an RLC circuit when the inductiv e and capaci-
tiv e reactances cancel eac h other, resulting in maxim um curren t (series) or maxim um
imp edance (parallel).
• Resonan t F requency :
?
0
=
1
v
LC
(rads
-1
), f
0
=
1
2p
v
LC
(Hz)
where L is inductance (H ), C is capacitance (F ).
Series Resonance
• Imp edance : Z = R+j
(
?L-
1
?C
)
.
• A t Resonance : ?L =
1
?C
, so Z = R (minim um imp edance).
• Curren t at Resonance : I
max
=
V
R
, where V is the source v oltage (RMS or p eak).
• V oltage A cross Comp onen ts :
V
L
= I
max
·?
0
L, V
C
= I
max
·
1
?
0
C
• Qualit y F actor (Q ) :
Q =
?
0
L
R
=
1
R
v
L
C
=
1
?
0
RC
• Bandwidth : BW =
?
0
Q
=
R
L
(rads
-1
) .
• Half-P o w er F requencies :
?
1
= ?
0
-
BW
2
, ?
2
= ?
0
+
BW
2
P arallel Resonance
• A dmittance : Y =
1
R
+j
(
?C-
1
?L
)
.
• A t Resonance : ?C =
1
?L
, so Y =
1
R
(minim um admittance, maxim um imp edance).
• Imp edance at Resonance : Z
max
= R (ideal case) or Z
max
˜
L
RC
(practical case).
• Qualit y F actor (Q ) :
Q = R
v
C
L
= R?
0
C =
R
?
0
L
• Bandwidth : BW =
?
0
Q
=
1
RC
(rads
-1
). Curren t at Resonance : Minimumcurrent,asimpedanceismaximum.
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P o w er i n Resonan t Circuits
• • P o w er at Resonance (Series) : P = I
2
rms
R , where I
rms
=
V rms
R
.
• P o w er F actor : cos? = 1 (unit y at resonance).
• Reactiv e P o w er : Q = 0 (since X
L
= X
C
at resonance).
Key Notes
• Use SI units: V oltage (V ), Curren t (A ), Resistance (? ), Inductance (H ), Capacitance
(F ), F requency (Hz or rads
-1
).
• Series resonance: Minim um imp edance, maxim um curren t.
• P arallel resonance: Maxim um imp edance, minim um curren t.
• Q indicates sharpness of resonance; higher Q means narro w er bandwidth.
• F or GA TE, c hec k circuit configuration (series/parallel) and use RMS v alues for p o w er.
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