RESONANCE
GROUP NO 19
GROUP MEMBERS:
UMM-E-FARWA CS-024
SYEDA HIBA AHMAD CS-045
SIDRA SHAMS CS-071
GROUP PRESENTATION
Electrical Resonance
Resonance is a condition established by the application of a particular
frequency (the resonant frequency) to a series or parallel R-L-C circuit
when the imaginary parts of circuit element i.e. the impedances (total
resistance offered by the circuit) or admittances (opposite to impedance)
cancel each other.
Electrically resonating circuits exhibit ringing and generate higher
currents and voltages than are fed into them. Some associated terms
are defined as follows.
LC Circuit
From the above definition we can see that for electrical resonance in a
circuit, an inductor and a capacitor is required. Hence, the ideal circuit
for resonance is the LC circuit, in which we assume the resistance is
zero. However, all real circuits have some resistance in them, and thus a
more accurate explanation for practical electrical resonance is the RLC
circuit.
RLC Circuit
An RLC circuit consists of a resistor, inductor and capacitor connected in
either series or parallel. The phenomenon is more easily described when
series and parallel circuits are considered individually.
Impedance
Electrical impedance is the measure of the opposition that a circuit
presents to the passage of a current when a voltage is applied. In
quantitative terms, it is the complex ratio of the voltage to the current in
an alternating current (AC) circuit.
Impedance extends the concept of resistance to AC circuits, and
possesses both magnitude and phase, unlike resistance, which has only
magnitude. When a circuit is driven with direct current (DC), there is no
distinction between impedance and resistance
The impedance caused by these two effects is collectively referred to
as reactance and forms the imaginary part of complex impedance
whereas resistance forms the real part. The symbol for impedance is Z.
ZT =R+ XL +X c
Hence,
XL = Xc at resonant frequency.
The transfer of power to the system is maximum at this particular
frequency and for frequencies above and Impedance extends the
concept of resistance to AC circuits, and possesses both magnitude
and phase, unlike resistance, which has only magnitude. When a circuit
is driven with direct current (DC), there is no distinction between
impedance and resistance below, the power transfer drops off to
significantly lower levels.
In some circuits this happens when the impedance between the input
and output of the circuit is almost zero and the transfer function is close
to one.
Admittance (Y)
The admittance (Y) is a measure of how easily a circuit or device will
allow a current to flow. It is defined as the inverse of the impedance (Z).
Its SI unit is siemens (symbol S) or mho, and the symbol ℧.
Y =1/ZT
Susceptance (B)
In electrical engineering, susceptance (B) is the imaginary part of
admittance. In SI units, susceptance is measured in Siemens.
Conductance (G)
The real part of admittance is conductance. In SI units, conductance is
measured in siemens.
Resonance with capacitors and inductors:
 Resonance of a circuit involving capacitors and inductors occurs
because the collapsing magnetic field of the inductor generates an
electric current in its windings that charges the capacitor, and then the
discharging capacitor provides an electric current that builds the
magnetic field in the inductor, and the process is repeated continually. In
some cases, resonance occurs when the inductive reactance and the
capacitive reactance of the circuit are of equal magnitude, causing
electrical energy to oscillate between the magnetic field of the inductor
and the electric field of the capacitor.

A resistive element will always be present in a resonant circuit due to
coil resistance and it is also required to obtained the responsive curve.

At resonance, the series impedance of the two elements is at a minimum
and the parallel impedance is at maximum.
Series RLC Circuit
Resonance in a RLC circuit in which the
circuit elements (inductor, capacitor,
resistance) are connected in series occurs
when the inductive and capacitive
reactances are equal in magnitude but 180
degrees out of phase.
Resistance
The resistance in such a circuit is the sum of the internal resistance of
the source, the internal resistance of the inductor and any added
resistance to control the curve.
R = Rs + Rl + Rd
Impedance
The total impedance of the network is given by
ZT = R + j(XL – Xc)
During resonance,
XL = X c
Hence, XL – Xc = 0, and Zt = R, which is the minimum amount of
impedance possible in a circuit.
Resonant Frequency
To calculate the resonant frequency at which this occurs:
XL = X c
As XL = ωL,
and Xc = 1/ωC, the equation becomes
ωL = 1/ωC,
and as ω = 2πfr,
Current and Voltage During Resonance
The current through the circuit at
resonance is:
I = E/R ∠ 0 degrees,
where E = applied voltage and R =
resistance. It is maximum, as the
impedence is at minimum.
As we know that E = V and R =
Rs + Rl + Rd, the equation
becomes:
Or simply:
While the input voltage and current are in phase at resonance, they are
180 degrees out of phase in the capacitor and inductor.
VL = (I ∠ 0 degrees)( XL ∠ 90 degrees) = I XL ∠ 90 degrees
VC = (I ∠ 0 degrees)( Xc ∠ - 90 degrees) = I Xc ∠ - 90 degrees
However, their magnitudes are equal, that is: VL = VC
Power at Resonance
The average power at the resistor is:
Pr = I2R
The reactive power at the capacitor is:
Pc = I2Xc
And the reactive power at the inductor is:
PL = I2XL
Reactive power is the kind of power in which the average value is 0, and
the change in energy is also equal to zero.
Using the following substitution:
Pavg becomes:
Or simply:
Quality Factor
For a series resonant circuit, this is defined as the ratio of the reactive
power
to the average power at resonance. It is represented by Q, and
for series circuits, by Qseries or Qs.
It shows how much energy has been stored as compared to dissipated.
The less the dissipation, the greater the Qs factor and the more intense
and constricted the region of resonance. An LC circuit – the ideal circuit
from before – has a quality factor Qs of infinity.
Mathematically, for series resonant circuits:
Qs = reactive power / average power
Qs = I2XL/ I2R
Qs = XL / R
Qs = ωsL / R
Furthermore, as ωs = 2πfs
and
Qs can also be written as:
, in terms of circuit parameters.
Circuits with low Qs are highly damped and exhibit many losses, while
those with high Qs are underdamped and do not have many losses. Qs is
directly proportional to selectivity, as it depends inversely on bandwidth.
Selectivity and Bandwidth of SRC
RLC resonant circuits respond mainly to a few select frequencies –
which gives resonating circuits the ability to “filter” or remove frequencies
not close to its resonating frequency.
Selectivity is the precision of response of resonant circuits to their
resonating frequencies – that is, the higher the selectivity, the fewer
frequencies across which the circuit shows a response.
The quality factor is directly proportional to selectivity. The higher the
selectivity, the higher the Qs.
The frequencies f1 and f2 at which current I falls to 0.707 (1/√2) of its
maximum value are called half-power frequencies, where
is the maximum value of current.
The bandwidth (f2 − f1) is called the half-power bandwidth or simply the
bandwidth of the circuit.
Thus, the definition of selectivity becomes the ratio of the resonant
frequency to the half-power bandwidth, and is:
Mathematical proof of the current falling to 1/√2 its original value is as
follows.
We know that the current in the series RLC circuit is given by,
Let ω2 (for f2) be such a frequency that
Then at frequency ω2, I is:
And its magnitude is:
Thus, ω2 radians/sec. (or f2 Hertz) gives the upper half-power frequency.
Similarly, let ω1 be such a frequency that,
Then the current at frequency ω1 is given by,
With a magnitude:
Furtehrmore, the power at these frequencies is as follows:
The Parallel Resonance Circuit
Parallel resonant circuits are 3-element networks that contain two
reactive components making it a second-order circuit, both are
influenced by variations in the supply frequency and both have a
frequency point where their two reactive components cancel each other
out influencing the characteristics of the circuit. Both series and parallel
circuits have a resonant frequency point.
Consider the parallel RLC circuit below.
Parallel RLC Circuit
For parallel RLC circuits:
A parallel circuit containing a resistance, R, an inductance I and a
capacitance, C will produce a parallel resonance (also called antiresonance) circuit when the resultant current through the parallel
combination is in phase with the supply voltage. At resonance there will
be a large circulating current between the inductor and the capacitor due
to the energy of the oscillations.
A parallel resonant circuit stores the circuit energy in the magnetic
field of the inductor and the electric field of the capacitor. This energy is
constantly being transferred back and forth between the inductor and the
capacitor which results in zero current and energy being drawn from the
supply. This is because the corresponding instantaneous values of IL
and IC will always be equal and opposite and therefore the current drawn
from the supply is the vector addition of these two currents and the
current flowing in IR.
In the solution of AC parallel resonance circuits we know that the supply
voltage is common for all branches, so this can be taken as our
reference vector. Each parallel branch must be treated separately as
with series circuits so that the total supply current taken by the parallel
circuit is the vector addition of the individual branch currents. Then there
are two methods available to us in the analysis of parallel resonance
circuits. We can calculate the current in each branch and then add
together or calculate the admittance of each branch to find the total
current.
We know from the previous series resonance tutorial that resonance
takes place when VL = - VC and that this situation occurs when the two
reactances are equal, that is, XL = XC. The admittance of a parallel
circuit is given as:
Resonance occurs when XL = XC and the imaginary parts of Y become
zero. Then:
Notice that at resonance the parallel circuit produces the same equation
as for the series resonance circuit. Therefore, it makes no difference if
the inductor or capacitor are connected in parallel or series. Also at
resonance the parallel LC tank circuit acts like an open circuit with the
circuit current being determined by the resistor, R only. So the total
impedance of a parallel resonance circuit at resonance becomes just the
value of the resistance in the circuit and Z = R as shown.
At resonance, the impedance of the parallel circuit is at its maximum
value and equal to the resistance of the circuit and we can change the
circuit's frequency response by changing the value of this resistance.
Changing the value of R affects the amount of current that flows through
the circuit at resonance, if both L and C remain constant. Then the
impedance of the circuit at resonance Z = RMAX is called the "dynamic
impedance" of the circuit.
Impedance in a Parallel Resonance Circuit
Note that if the parallel circuits impedance is at its maximum at
resonance then consequently, the circuits admittance must be at its
minimum and one of the characteristics of a parallel resonance circuit is
that admittance is very low limiting the circuits current. Unlike the series
resonance circuit, the resistor in a parallel resonance circuit has a
damping effect on the circuits bandwidth making the circuit less
selective.
Also, since the circuit current is constant for any value of impedance, Z,
the voltage across a parallel resonance circuit will have the same shape
as the total impedance and for a parallel circuit the voltage waveform is
generally taken from across the capacitor.
We now know that at the resonant frequency, ƒr the admittance of the
circuit is at its minimum and is equal to the conductance, G given by 1/R
because in a parallel resonance circuit the imaginary part of admittance,
i.e. the susceptance, B is zero because BL = BC as shown.
Susceptance at Resonance
From above, the inductive susceptance, BL is inversely proportional to
the frequency as represented by the hyperbolic curve. The capacitive
susceptance, BC is directly proportional to the frequency and is
therefore represented by a straight line. The final curve shows the plot of
total susceptance of the parallel resonance circuit versus the frequency
and is the difference between the two susceptance's.
Then we can see that at the resonant frequency point where it crosses
the horizontal axis the total circuit susceptance is zero. Below the
resonant frequency point, the inductive susceptance dominates the
circuit producing a "lagging" power factor, whereas above the resonant
frequency point the capacitive susceptance dominates producing a
"leading" power factor. So at resonant frequency, the circuits current
must be "in-phase" with the applied voltage as there effectively there is
only the resistance in the circuit so the power factor becomes one or
unity, ( θ = 0o ).
Current in a Parallel Resonance Circuit
As the total susceptance is zero at the resonant frequency, the
admittance is at its minimum and is equal to the conductance, G.
Therefore at resonance the current flowing through the circuit must also
be at its minimum as the inductive and capacitive branch currents are
equal ( IL = IC ) and are 180o out of phase.
We remember that the total current flowing in a parallel RLC circuit is
equal to the vector sum of the individual branch currents and for a given
frequency is calculated as:
At resonance, currents IL and IL are equal and cancelling giving a net
reactive current equal to zero. Then at resonance the above equation
becomes.
Since the current flowing through a parallel resonance circuit is the
product of voltage divided by impedance, at resonance the impedance,
Z is at its maximum value, ( =R ). Therefore, the circuit current at this
frequency will be at its minimum value of V/R and the graph of current
against frequency for a parallel resonance circuit is given as.
Parallel Circuit Current at Resonance
The frequency response curve of a parallel resonance circuit shows that
the magnitude of the current is a function of frequency and plotting this
onto a graph shows us that the response starts at its maximum value,
reaches its minimum value at the resonance frequency when IMIN = IR
and then increases again to maximum as ƒ becomes infinite. The result
of this is that the magnitude of the current flowing through the inductor, L
and the capacitor, C tank circuit can become many times larger than the
supply current, even at resonance but as they are equal and at
opposition ( 180o out-of-phase ) they effectively cancel each other out.
As a parallel resonance circuit only functions on resonant frequency, this
type of circuit is also known as an Rejector Circuit because at
resonance, the impedance of the circuit is at its maximum thereby
suppressing or rejecting the current whose frequency is equal to its
resonant frequency. The effect of resonance in a parallel circuit is also
called "current resonance".
The calculations and graphs used above for defining a parallel
resonance circuit are similar to those we used for a series circuit.
However, the characteristics and graphs drawn for a parallel circuit are
exactly opposite to that of series circuits with the parallel circuits
maximum and minimum impedance, current and magnification being
reversed. Which is why a parallel resonance circuit is also called an
Anti-resonance circuit.
Selectivity and Bandwidth of a PRC
The bandwidth of a parallel resonance circuit is defined in exactly the
same way as for the series resonance circuit. The upper and lower cutoff frequencies given as: ƒupper and ƒlower respectively denote the halfpower frequencies where the power dissipated in the circuit is half of the
full power dissipated at the resonant frequency 0.5( I2 R ) which gives us
the same -3dB points at a current value that is equal to 70.7% of its
maximum resonant value, ( 0.707 x I )2 R.
The selectivity or Q-factor for a parallel resonance circuit is generally
defined as the ratio of the circulating branch currents to the supply
current and is given as:
Note that the Q-factor of a parallel resonance circuit is the inverse of the
expression for the Q-factor of the series circuit. Also in series resonance
circuits the Q-factor gives the voltage magnification of the circuit,
whereas a parallel circuit it gives the current magnification.
Bandwidth of a Parallel Resonance Circuit
Applications
 They are widely used in wireless (radio) transmission for both
transmission and reception.
 Resonance is used for tuning and filtering, because it occurs at a
particular frequency for given values of inductance and capacitance.
 Resonance can be employed to maintain AC circuit oscillations at a
constant frequency, just as a pendulum can be used to maintain
constant oscillation speed in a timekeeping mechanism.
 It can be detrimental to the operation of communications circuits by
causing unwanted sustained and transient oscillations that may cause
noise, signal distortion, and damage to circuit elements.
 Resonance can be exploited for its impedance properties: either
dramatically increasing or decreasing impedance for certain frequencies.
Circuits designed to screen certain frequencies out of a mix of different
frequencies are called filters.
References
 Serway & Beichner Ch 33
 http://encyclopedia.thefreedictionary.com/electrical+Resonance/
 http://hyperphysics.phyastr.gsu.edu/%E2%80%8Chbase/electric/serres.html
 http://armymunitions.tpub.com/mm0308/mm03080035.htm/
 http://www.electronics-tutorials.ws/accircuits/parallel-resonance.html/
http://www.electronics-tutorials.ws/accircuits/series-resonance.html/
 http://wps.prenhall.com/chet_boylestad_introduct_12/137/35100/898583
0.cw/index.html
 http://hyperphysics.phyastr.gsu.edu/%E2%80%8Chbase/sound/reson.html
 http://pages.uoregon.edu/dparks/206/resonance/
 http://www.utc.edu/Faculty/Tatiana-Allen/elecres.html
 http://www.powerelectricalblog.com/2007/03/electrical-resonance-andresonant.html
 http://www.its.bldrdoc.gov/fs-1037/dir-031/_4576.htm
 http://www.intuitor.com/resonance/circuits.html
 http://www.animations.physics.unsw.edu.au/jw/AC.html
 http://www.radio-electronics.com/info/formulae/q-quality-factor/basicstutorial.php
 http://www.globalspec.com/reference/9525/348308/chapter-12-6selectivity-and-bandwidth
From Wikipedia, the free encyclopedia
 http://en.wikipedia.org/wiki/Electrical_impedance
 http://en.wikipedia.org/wiki/Admittance/
 http://en.wikipedia.org/wiki/Resonance
 http://en.wikipedia.org/wiki/Electrical_resonance
Check this for MCQ’s and fill in the blanks:
 http://wps.prenhall.com/chet_boylestad_introduct_12/137/35100/898583
0.cw/index.html