Comparison of Series and Parallel Resonance Circuits

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COMPARISON OF SERIES AND

PARALLEL RESONANCE

CIRCUITS

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ELECTRICAL RESONANCE

Electrical resonance occurs in an electric circuit at a particular resonance frequency where the imaginary parts of circuit element impedances or admittances cancel each

other. In some circuits this happens when the impedance between the input and output of the circuit is almost zeroand the transfer function is close to one.

Resonant circuits exhibit ringing and can generatehigher voltages and currents than are fed into them. They arewidely used in wireless (radio) transmission for bothtransmission and reception.


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RLC Circuit Introduction

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An RLC circuit (or LCR circuit) is an electrical circuitconsisting of a resistor, an inductor, and a capacitor,connected in series or in parallel. The RLC part of the name

is due to those letters being the usual electrical symbols for resistance, inductance and capacitance respectively. Thecircuit forms a harmonic oscillator for current and willresonate in a similar way as an LC circuit will. The main

difference that the presence of the resistor makes is that anyoscillation induced in the circuit will die away over time if itis not kept going by a source.

This effect of the resistor is called damping. The

presence of the resistance also reduces the peak resonantfrequency somewhat. Some resistance is unavoidable in realcircuits, even if a resistor is not specifically included as acomponent. A pure LC circuit is an ideal which really onlyexists in theory.


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Resonance In Electric Circuits

Any passive electric circuit will resonate if it has an inductor

and capacitor.

Resonance is characterized by the input voltage and current

being in phase. The driving point impedance (or admittance)

is completely real when this condition exists.

In this presentation we will consider

(a) series resonance, and

(b) parallel resonance.


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SERIES RESONANCE

CONDITION

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Series Resonance

Consider the series RLC circuit shown below.

R L

C+

_ IV

V = VM 0

The input impedance is given by:

1

( ) Z R j wL

wC

The magnitude of the circuit current is;

2 2

| |1

( )

mV I I

R wL

wC


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Series Resonance

Resonance occurs when,1

wLwC

At resonance we designate w as wo and write;

This is an important equation to remember. It applies

to both series and parallel resonant circuits.

1 1,

2

o o f

LC LC


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Series Resonance

The magnitude of the current response for the series

resonance circuit is as shown below.

mV

R

2

mV

R

w

|I|

wow1 w2

Bandwidth:

BW = wBW = w2 – w1

Half power point


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Series Resonance

The peak power delivered to the circuit is;2

mV

P R

The so-called half-power is given when2

mV

I R

.

We find the frequencies, w1 and w2, at which this half-power

occurs by using;

2 212 ( ) R R wLwC


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Series Resonance

After some insightful algebra one will find two frequencies at which

the previous equation is satisfied, they are:

2

1

1

2 2

R Rw

L L LC

and 2

2

1

2 2

R Rw

L L LC

The two half-power frequencies are related to the resonant frequency by

1 2ow w w


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Series Resonance

The bandwidth of the series resonant circuit is given by;

2 1b

R BW w w w

L

We define the Q (quality factor) of the circuit as;

1 1o

o

w L LQ R w RC R C

Using Q, we can write the bandwidth as;

ow BW Q

These are all important relationships.


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jwL jwC

RV I

11

jwC jwL R I V

1

We notice the above equations are the same provided:

V I

R R

1

C L

If we make the inner-change,then one equation becomes

the same as the other.

For such case, we say the onecircuit is the dual of the other.

Series Resonance

Duality

If we make the inner-change,then one equation becomes

the same as the other.

For such case, we say the one

circuit is the dual of the other.


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The features of series resonance:

The impedance is purely resistive, Z = R;

• The supply voltage Vs and the current I are in

phase (cosq = 1)

• The magnitude of the transfer function H (ω) =

Z(ω) is minimum;

• The inductor voltage and capacitor voltage can

be much more than the source voltage.

QU L TY F CTOR OF ER E RE ON NCE


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QUALITY FACTOR OF SERIES RESONANCE

1 2

1

,

2 2

o

o

o

o o

LQ

R RC

R B

L Q

B B

oQ

B

Peak EnergyStored 2

Energy Dissipated in one Period at Resonance

1o

o

Q

LQ R RC


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PARALLEL RESONANCE

CONDITION


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The properties of the parallel RLC circuit can be obtained from theduality relationship of electrical circuits and considering that the

parallel RLC is the dual impedance of a series RLC. Considering this

it becomes clear that the differential equations describing this circuitare identical to the general form of those describing a series RLC.

For the parallel circuit, the attenuation α is given by

and the damping factor is consequently


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This is the inverse of the expression for ζ in the series circuit.Likewise, the other scaled parameters, fractional bandwidthand Q are also the inverse of each other. This means that a

wide band, low Q circuit in one topology will become anarrow band, high Q circuit in the other topology whenconstructed from components with identical values.

The Q and fractional bandwidth of the parallel circuitare given by


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Comparison of Series and Parallel Resonance Circuits

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