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