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Course no. 9
Technical University of Cluj-Napoca
Author: Prof. Radu Ciupa
The Theory of Electric Circuits
About the Course
• Aims– To provide students with an introduction to electric circuits
• Objectives
– To introduce the concept of resonance in electric circuits
• Resonance in electric circuits:
- Resonance in a series circuit - Resonance in a parallel circuit- Resonance in real circuits- Resonance in inductively coupled circuits
Electrostatics
Topics of the course
Chapter 4
RESONANCE IN ELECTRIC CIRCUITS
Friday, April 21, 2023 4
Chapter 4. Resonance in electric circuits.
Friday, April 21, 2023 5
- at resonance : Q = 0 (meaning X = 0 or B = 0)
- the phase shift between I and U is 0 (sin φ = 0)
Remarks: a) X = 0 corresponds to the series resonance,
b) B = 0 corresponds to the parallel resonance
c) at resonance the current has an extreme
Chapter 4. Resonance in electric circuits.
Friday, April 21, 2023 6
4.1 RESONANCE IN A SERIES CIRCUIT.
-The resonance condition:
Q = 0=>X = 0
or
- the angular resonant frequency:
)1
(C
LjRZ
22 )1
(C
LR
U
Z
UI
RC
Larctg
RX
arctg
1
01
CLX
C
L
1
LC
10
Chapter 4. Resonance in electric circuits.
Friday, April 21, 2023 7
The vector diagram (phasorial representation):Remarks:
a)R does not influence the resonance.
b) U = UR = RI
is minimum (X = 0),
the current is maximum:
c) UL = UC : voltage resonance!
R = X+R = Z 22
maxIR
UI
d) if → overvoltages.
, because →
the characteristic impedance of a series resonant circuit
UUU CL
> or 1 >
=
= 000 RLR
L
RI
LI
U
UL LC
1 0 R
C
LL
LC > =
1
= =
1 =
00 C
L
CL
Chapter 4. Resonance in electric circuits.
Friday, April 21, 2023 8
Q = R
= RI
I =
UU =
UU CL
- the quality factor (or Q-factor)
It shows how many times the voltage across the inductor or across the capacitance of a series resonant circuit (at resonance) is greater than the applied voltage.
- the damping factorR
Qd =
1 =
RC
1 - L
=
C
1 - L = X - X = X
C1
- L + R
U = I
CL
arctan
2
2
- for the R, L, C series circuit:
Chapter 4. Resonance in electric circuits.
Friday, April 21, 2023 9
Chapter 4. Resonance in electric circuits.
Friday, April 21, 2023 10
C1
- L + R
U
C = I
C = U C
1
12
2
0 = U C
2
-2 2
0
d = c
C1
-L+R
UL = I = U L
2
2
0 = U L
d = L 20
-2
2
R
U = I 0 L 0c
Note: Ucmax = ULmax
Chapter 4. Resonance in electric circuits.
Friday, April 21, 2023 11
4.2 RESONANCE IN A PARALLEL CIRCUIT.
-The resonance condition:
Q = 0 => B = 0
- the angular resonant frequency:
Remark: In practice (that is having real L and C), the resonant frequencies
are different for the series and parallel connections.
01
CL
B C
L
1
LC
10
CjU+Lj
U+
R
U = I
I+I+I = I CLR
C-L
j-R
U = I
1
1
jB)-(GU = I
Chapter 4. Resonance in electric circuits.
Friday, April 21, 2023 12
The vector diagram (phasorial representation):
Remarks:
a)I = IR = GU = U/R
is minimum (B
= 0),
the current is minimum:
b) IL = IC : current resonance!
minIGUI
c) if → overcurrents.
, because
the characteristic admittance of a parallel resonant circuit
III CL
LC
1 0
G=B+G=Y 22
R < C
= L R
> L
= C
1or
1
10 0
= L
C =
L = C
0 0
1
Chapter 4. Resonance in electric circuits.
Friday, April 21, 2023 13
R = G
=Q
G
= Q
= d1
- the quality factor
- the damping factor
I+I+I = I
CU = I
L
U = I
R
U = I
CLR
C
L
R
)I-I(+I = I
C-L
j-R
U = I
CLR22
1
1
Chapter 4. Resonance in electric circuits.
Friday, April 21, 2023 14
4.3 RESONANCE IN REAL CIRCUITS.
Cj+R = Z Lj+R = Z
1 and 2211
Cj
-R = Y
Lj+R = Y
1and
1
2
21
1
C+R
C-L+R
Lj -
L+R
R + L+R
R = Y+Y = Y e
2222
2221
2222
2
2221
121 1
1
1
0 = 1
1
2222
2221
C+R
C-L+R
L = Be
In a parallel circuit, the resonance
occurs when Be = 0
Chapter 4. Resonance in electric circuits.
Friday, April 21, 2023 15
R-R-
LC =
R-CL
R-CL
LC =
22
2
21
2
22
21
0
1
1
Remarks:
a)if there is resonance;
b)if there is no resonance;
c)if , → the same as for a series resonant circuit
d)if , → the resonance can occur at any frequency
in this case:
and or and 2121 <R <R >R >R
R>>R R<<R 2121 or
21 RRLC
= 1
0
21 RR0
00
=
1
2
21
21
c-Lj+R
Cj
-RL)j+(R =
Z+Z
ZZ = Z e
Chapter 4. Resonance in electric circuits.
Friday, April 21, 2023 16
4.4 RESONANCE IN INDUCTIVELY COUPLED CIRCUITS.
122
22
211
111
+
1 - + = 0
+
1 - + =
IMjIC
LjR
IMjIC
LjRU
It is possible to obtain resonance in the primary circuit, in the secondary circuit or simultaneously in both circuits.
+ + = 0
= =
1
222
1
211e1
1
1
MjI
IjXR
I
IMjjXRZ
I
U
C-L = X ,
C
1-L = X
222
111
1
where
jX+R
Mj- =
I
I
221
2
Chapter 4. Resonance in electric circuits.
Friday, April 21, 2023 17
X+R
XM-Xj +X+R
MR+R = jX+R
M+jX+R = Z e 22
22
222
122
22
222
1
22
22
111
It results:
The resonance occur when Xe = 0 :
Remarks: a)the resistance R1 does not influence the resonance, while R2 influences the resonance;
b)the realization of the resonance of inductively coupled circuits is important especially in radio frequencies circuits, we can approximate
X+R
XM = X 22
22
222
1
22 XR
Chapter 4. Resonance in electric circuits.
Friday, April 21, 2023 18
M = XX 2221
M = C
L C
-L 22
22
11
1
1
0 1 0201202
201
224 = +)+(-)k-(
where: is the coefficient of coupling,
is the resonance frequency of the primary circuit
is the resonance frequency of the secondary circuit
LL
M = k
21
CL =
11
01
1
CL =
22
02
1
Chapter 4. Resonance in electric circuits.
Friday, April 21, 2023 19
)k-(
)k-(-)+(+ =
2
202
201
22202
201
202
201
00 12
14 '' ,'
d d-d = k rc 4
4
2 (critical coupling)