Electrotechnics - users.utcluj.rousers.utcluj.ro/~claudiah/Electrotechnics/Cursuri... · Mixt resonance (series-parallel resonance, resonance in real circuits) 3 /26 Curs 11 Electrotehnic

1 /26 Curs 11 Electrotehnică

Electrotechnics


11.1. Series resonance (voltage resonance)

11.2. Parallel resonance (currents resonance)

11.3. Mixt resonance (series-parallel resonance,

resonance in real circuits)


▪ 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 =


11.1. Series resonance

(voltage resonance)


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= +


❑ 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


▪ 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


❑ 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


▪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


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:


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= =


maxrI I=

0 =


11.2. Parallel resonance

(current resonance)


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


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


▪ 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


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


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


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

= =


minrI I=

0 =


11.3. Mixt resonance

(series-parallel resonance,

rezonance in real circuits)


❑ 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


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

+ − − =


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 =


(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


The end

(Another step towards

the Final !!!)

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Electrotechnics - users.utcluj.rousers.utcluj.ro/~claudiah/Electrotechnics/Cursuri... · Mixt resonance (series-parallel resonance, resonance in real circuits) 3 /26 Curs 11 Electrotehnic