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1 /26 Curs 11 Electrotehnică
Electrotechnics
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11.1. Series resonance (voltage resonance)
11.2. Parallel resonance (currents resonance)
11.3. Mixt resonance (series-parallel resonance,
resonance in real circuits)
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▪ in electrical circuits having inductors and capacitors, due to the fact that
their reactances cancel each other, there can be cases in which the total
equivalent reactance of the whole circuit is zero, although the circuit
has reactive components;
▪ The phase angle φ is zero;
▪ The reactive power (Q=U·I·sinφ) consumed in the circuit is null;
▪ These type of circuits are called resonant circuits.
Resonance condition
Resonance = operating regime of an electrical circuit, in which the phase
shift between the voltage U and current intensity I at the terminals of the
circuit is cancelled.
Resonance condition:
or
0, 0, Q=0X B= =At the resonance:
= = = 0, 0, 0, Q=0X B
0 =
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11.1. Series resonance
(voltage resonance)
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R L CU U U U= + +
R C
I
L
U
❑ The series circuit R, L, C powered by a sinusoidal voltage is considered.
▪Ohm’s law in complex:
1
R
L
C
U R I
U j L I
U Ij C
=
=
=
= + + 1
U R I j L I Ij C
= + −
1U I R j L
C
= + − 1
U R I j L I j IC
1Z=R j L
C
+ −
= U Z I
− =1
L XC
Z R jX= +
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❑ The circuit is resonant if it meets the resonance condition:
= = =0, 0, 0X Q
=1
LC
This relation shows that in the circuit the resonance can be realized by
the variation of the following parameters:
▪ through the variation of the frequency; angular frequency and resonance
frequency have the expressions:
= =1
, f2
1r r
LC LC
▪ through the variation of the inductivity; the resonance inductivity has
the expression:
2
1rL
C=
− =1
0LC =2 1LC
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▪ through the variation of the capacitance; the resonance capacitance
has the expression:
2
1rC
L=
=
+ −
1
UI
R j LC
❑ Determination of the current expression from the circuit:
At resonance:= =min rZ Z R
= U
IR
=
+ −
2
2 1
UI
R LC
= + −
1U I R j L
C
− =1
0LC =Z R
U UI = =
Z R= maxrI I
=U
IZ
=U
IZ
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❑ The phasorial diagram of the circuit at resonance is:
- The current is considered as phase origin
0
C
1U j I
C= −
U
I
RU R I=
LU j L I=
R C
I
L
U
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▪At resonance, the inductor and capacitor voltages are equal and of
opposite signs
▪The voltage at the circuit terminals is equal with the voltage on the
resistor:
LU j L I=
1CU j I
C= −
1L
C
=
= −LU CU
=R L= +UU U + RCU UAs it can be observed from the voltage phase diagram, there may be
situations when, operating in resonant regime where the voltage at the
inductor terminals (equal to that at the capacitor terminals) exceeds even the
voltage at the circuit terminals U.
LU CU U=
In this case it is said to cause surges in the circuit (voltages higher than the
supply voltage) and for this reason the series resonance is called voltage
resonance.
=LU CU
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The assessment of the posibilities and the values of the overvoltages which
can appear in such circuits is usually made with the help of the value called
characteristic impedance, noted with ρ, which has the dimension of an
impedance and is defined with the relation:
=L
C
The ratio:=
Rq
is called quality factor of the series circuit R, L, C and it is the ratio
between the characteristic impedance and the resistence.
r
1L
r
RC
= L
CR
❑ There can be a surge only in circuits in which:
=1
rLC
Taking into
account that:
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R 1d
q= =
The inverse of the quality factor:
is called amortization factor of the series R, L, C circuit.
The current is passing through a maximum at resonance (ω=ωr), when
its value is:
r
UI
R=
Conclusions – Series resonance
At series resonance the voltage at the inductor terminals is equal with the
voltage at the capacitor terminals and satisfies the relationship:
L C
UU U
d= =
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maxrI I=
0 =
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11.2. Parallel resonance
(current resonance)
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R CLI I I I= + +
❑ The parallel circuit formed from ideal elements R, L, C, powered
with a sinusoidal voltage is considered.
▪ First Kirchhoff theorem:
=
=
=
R
C
L
UI
R
UI
j L
I j C U
= + + U U
I j C UR j L
= − −
1 1I U j C
R L
= − + U U
I j j C UR L
−
1 Y=G-j C
L
= I U Y
− =1
C BL
Y G jB= −=1
RG
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The circuit is resonant if it meets the resonance condition:
− =1
0CL
1 1C L
L C
= =
This relation shows that in the circuit the resonance can appear by
varying the following parameters:
▪ by varying the frequency, angular frequency and resonance frequency
having the expressions: 1 1, f
2r r
LC LC
= =
▪by varying the inductivity, the resonance inductivity having the
expression:
2
1rL
C=
The parallel resonance condition is identical with the series resonance
condition.
=
=
B 0,
0
=2 1LC
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▪ by varying the capacitance, the resonance capacitance having the
expression:
2
1rC
L=
= − −
1 1I U j C
R L
The determination of the current from the circuit:
At resonance:
= = =minY1
r Y GR
U
IR
=
= + −
2
2
1 1I U C
LR
=I UY
=I UY
10C
L
− = =
1Y
R
UI =UY=
R
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The phase diagram of the circuit at resonance:
- The voltage is considered as phase origin
0
L
UI
j L=
U
I
R
UI
R=
cI j C U=
R C
I
LU
IR IL IC
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At resonance the total curent is equal with the one on the branch
containing the resistance:
L
UI
j L=
CI j CU=
=1
CL
== − L L CI II CI
= R L C R= II I I+ + I
As it can also be observed from the currents phase diagram, at resonance
the current passing through the inductor (equal with the one passing
through the capacitor), in some situations, can be even higher than the
total circuit current I.C LI I I=
So there is the possibility of overcurrents occurence in the parallel R, L, C
circuit, and for this reason the parallel resonance is called currents
resonance.
r
1L=r R
C
The circuits in which:
= r
1 1
LrC
R
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The assessment of this posibility is usually made with the help of a value
called characteristic admitance, noted with , defined with the relation:
1= = =r
rL
CC
L
The quality factor of the parallel R, L, C circuit:
1d
q=
The inverse of the quality factor:
is called the amortisation factor of the parallel R, L, C circuit.
qG
R
= =
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minrI I=
0 =
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11.3. Mixt resonance
(series-parallel resonance,
rezonance in real circuits)
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❑ The circuit formed by parallel connecting two series circuits R, L
and R, C is considered.
1 1= +Z R j L
▪ The complex impedances of the two
branches are:
2 2
1= +Z R
j C
▪The equivalent impedance of the circuit
will be:
( )
( )
1 2
1 2
1 21 2
1
1
+ + = =
+ + + −
e
R j L Rj CZ Z
ZZ Z
R R j LC
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After simple transformation, the form below is obtained:
( ) ( )
2 22 2 2 2 21 11 2 2 1 2 22 2 2
2 22 2
1 2 1 2
1 1
+ + + + − −
= +
+ + − + + −
e
R RL LR R R R L R LR
C CC CZ j
R R L R R LC C
In the considered circuit there will be resonance when the resonance
condition will be met, namely when the equivalent reactance ( the
coefficient of j, the imaginary part of Ze ) is cancelled, that is:
2 22 12 2
0RL L
LRC CC
+ − − =
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Solving this equation, the resonance angular frequency is obtained:
21
22
1r
LR
CLLC RC
−
=
−
This expression shows that in the considered circuitul, the resonance
angular frequency also depends on the reresistences R1 and R2 from the
circuit, unlike the resonances from the seried and parallel R, L, C
circuits, where the resonance angular frequency was only based on the L
and C parameters.
Observations
❑When the resistences R1 and R2 are equal (or when they are zero) the
resonance angular frequency is equal to the ideal resonance angular
frequency: 1r
LC =
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(or inverse), results an imaginary resonance angular frequency, thus the
circuit will not be resonant for any angular frequency.
results a non determination of the resonance angular frequency, thus the
circuit is resonant for any angular frequency. This circuit is called
completely aperiodic and is used in electrical measurements.
2 21 2= =
LR R
C
❑ For:
❑ In the particular case:
and
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The end
(Another step towards
the Final !!!)
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