Parallel Resonance and Parallel RLC Resonant Circuit

 

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Parallel Resonance Circuit The Parallel Resonance Circuit In many ways a parallel resonance circuit is exactly the same as the series resonance circuit we looked at in the previous tutorial. Both are 3-element networks that contain two reactive components making them 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 characteri stics of the circuit. Both circuits have a resonant frequenc frequency y point. The difference this time however, is that a parallel resonance circuit is influenced by the currents flowing through each parallel branch within the parallel LC tank circuit. A tank circuit is circuit is a parallel combination of L and C that is used in filter networks to either select or reject AC frequencies. Consider the parallel RLC circuit below.

Parallel RLC Circuit

  Let us define what we already know about parallel RLC circuits.

A parallel circuit containing a resistance, R, an inductance, L and a capacitance, C will produce a parallel resonance (also resonance (also called anti-resonance) 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, then parallel circuits produce current resonance. 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 corr corresponding esponding instan instantaneous taneous values o off IL and IC will always be equal and oppo opposite site and therefo therefore re the current drawn from the supply is the vector vector addition of these two currents currents and the current flowi flowing ng 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 from the previ previous ous series re resonance sonance tutori tutorial al that resonance takes place when VL = -VC and this situation occurs occurs when the two reactances reactances are equal, equal, XL = XC. The admittan admittance ce of a paralle parallell circuit circuit is giv given en as:

  Resonance Reso nance oc occurs curs wh when en XL = XC and the imagina imaginary ry parts of Y be become come ze zero. ro. The Then: n:

  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 capacitor are connected series. Also at resonance the parallel LC tank circuit acts like an open circuit with the circuit current beingor determined by the resistor,inRparallel only. Soor 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. Also at resonance, as the impedance of the circuit circuit is now that of resistance only, the to total tal circuit current, I will be “in-phase” “in-phase” with the supply voltage, voltage, VS. 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 flows thro through ugh the ccircui ircuitt at resonance resonance,, if both L a and nd C rema remain in con constant. stant. Th Then en the impedan impedance ce of the ci circuit rcuit at reson resonance ance Z = RMAX is calle called d 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 admittance must  must be at its minimum and one of the characteristics of a parallel resonance circuit 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 parallel resonance ccircuit ircuit the imag imaginary inary part of admi admittance, ttance, i.e. the susce susceptance, ptance, B is zero because BL = BC as shown.

Susceptance at Resonance

 From above, the inductive susceptance , B is inversely prop proportional ortional to the frequ frequency ency as represented by the hype hyperbolic rbolic curve. The capacitive  L susceptance , BC is directly proport proportional ional to the frequency and is therefore repr represented esented 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 were 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 the resonant frequency, ƒr the current drawn from the supply must be “in-phase” with the applied voltage as effectively there is only the resistance present in the parallel 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 current flowing through through the circuit must also be at its minimum minimum as the inductive inductive and capacitive branch curr currents ents are equal ( IL = IC ) and are are 180o out o off phas phase. e. 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 currents IL and IL are equal and cancelling g giving iving a net reactive current current equal to zero zero.. Then at resonance the above e equation quation 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 graph sho shows ws us that th the e respo response nse sta starts rts at its ma maximu ximum m value value,, reach reaches es its mi minimu nimum m value a att the reso resonance nance ffrequ requency ency wh when en 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, current, even at resonance but as they are equal and at oppo opposition sition ( 180o out-of-phase ) they effectiv effectively ely 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  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. reversed. Which is why a parallel resonance circuit is also called an Anti-resonance Anti-resonance circuit.  circuit.

Bandwidth & Selectivity of a Parallel Resonance Circuit The bandwidth of a parallel resonance circuit is defined in exactly the same way as for the series resonance circuit. The upper and lower cut-off  frequencies given as: ƒupper and ƒlower respectively denote the half-power frequencies where the power dissipated in the circuit is half of the full power dissipated at the resonant 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 maxim um resonant resonant vvalue alue,, ( 0.707 x I )2 R. As with the series circuit, if the resonant frequency remains constant, an increase in the quality factor, Q will cause a decrease in the bandwidth and likewise, a decrease in the quality quality factor will cause an increase increase in the bandwidth as defined by: BW = ƒr /Q or BW = ƒ2 - ƒ2. Also changing the ratio between the inductor, L and the capacitor, C, or the value of the resistance, R the bandwidth and therefore the frequency response of the circuit will be changed for a fixed resonant frequency. This technique is used extensively in tuning circuits for radio and television transmitters and receivers. The selectivity or Q-factor Q-factor for  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 in a parallel circuit it gives the current magnification.

Bandwidth of a Parallel Resonance Circuit

Parallel Resonance Example No1 A parallel resonance network consisting of a resistor of 60 Ω, a capacitor of 120uF and an inductor of 200mH is connected across a sinusoidal supply voltage which has a constant output of 100 volts at all frequencies. Calculate, the resonant frequency, frequency, the quality factor and the bandwidth of the circuit, the circuit current at resonance and current magnification.

Resonant Frequency, ƒr

Inductive Reactance at Resonance, X L

 

Quality factor, Q

Bandwidth, BW

The upper and lower -3dB frequency points, ƒH and ƒL

Circuit Current at Resonance, I T At resonance the dynamic impedance of the circuit is equal to R

Current Magnification, Imag

Note that the current drawn from the supply at resonance (the resistive current) is only 1.67 amps, while the current flowing around the LC tank circuit is larger at 2.45 amps. We can check this value by calculating the current flowing through the inductor (or capacitor) at resonance.

Parallel Resonance Tutorial Summary We have seen that Parallel Resonance circuits Resonance circuits are similar to series resonance circuits. Resonance occurs in a parallel RLC circuit when the total circuit current is “in-phase” with the supply voltage as the two reactive components cancel each other out. At resonance the admittance of the circuit is at its minimum and is equal to the conductance of the circuit. Also at resonance the current drawn from the supply is also at its minimum and is determined by the value of the parallel resistance. The equation used to calculate the resonant frequency point is the same for the previous series circuit. However, while the use of either pure or impure components in the series RLC circuit does not affect the calculation of the resonance frequency, but in a parallel RLC circuit it does. In this tutorial about parallel resonance, we have assumed that the components are purely inductive and purely capacitive with negligible resistance. However in reality the coil will contain some resistance. Then the equation for calculating the parallel resonant frequency of a circuit is therefore modified to account for the additional resistance.

Resonant Frequency using Impure Components

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19 Responses to “Parallel Resonance Circuit” Older comments (http://www.electr (http://www.electronics-tutorials onics-tutorials.ws/accircuits/parall .ws/accircuits/parallel-resonance.htm el-resonance.html/comment-pag l/comment-page-1#comments) e-1#comments)

xyz can any one explain me how current through inductor and capacitor becomes high when the total current of the circuit is minimum???? Reply (/accircuits/para (/accircuits/parallel-resonance.h llel-resonance.html?replytocom= tml?replytocom=5820#res 5820#respond) pond)

February 7th, 2015 (http://www.e (http://www.electronics-tuto lectronics-tutorials.ws/accircuits/parallel-re rials.ws/accircuits/parallel-resonance.html/co sonance.html/comment-page mment-page-2#comment -2#comment-5820) -5820)

Wayne Storr (http://www.elect (http://www.electronics-tutorials.w ronics-tutorials.ws) s) Current magnification occurs in parallel RLC circuits (as opposed to voltage magnification in series RLC circuits) due to the resonance condition between the inductor and capacitor. We have seen above that at the resonance frequency, the magnitudes of the inductive and capacitive reactances are equal and draw a current as a result. However their phases are exact opposites so cancel out around the LC tank circuit leaving just the resistance to draw current from the supply, since at resonance, I = V/G. At resonance, current magnification magnification Q = resistance/reactance = R/X. Above, R = 60 ohms and X = 40.8 ohms, therefore Q = 1.47 (no units). The resistor current is 100 volts/60 ohms = 1.67, then the inductor or capacitor current (they are the same) will be Q x IR = 1.47 x 1.67 = 2.45A as shown. Reply (/accircuits/pa (/accircuits/parallel-resonanc rallel-resonance.html?replytoco e.html?replytocom=5843#re m=5843#respond) spond)

February 9th, 2015 (http://www.e (http://www.electronics-tuto lectronics-tutorials.ws/accircuits/parallel-re rials.ws/accircuits/parallel-resonance.html/co sonance.html/comment-page mment-page-2#comment -2#comment-5843) -5843)

Benjamin If the same reactive components in a parallel RLC circuit were reconfigured in a series configuration, would: A) impedance increase, and B) resonance stay the same? Thank you for your time! Reply (/accircuits/para (/accircuits/parallel-resonance.h llel-resonance.html?replytocom= tml?replytocom=5595#res 5595#respond) pond)

 January 18th, 2015 2015 (http://www.electronics-t (http://www.electronics-tutorials.ws/accircuits/p utorials.ws/accircuits/parallel-resonance.ht arallel-resonance.html/commentml/comment-page-2#comm page-2#comment-5595) ent-5595)

Wayne Storr (http://www.elect (http://www.electronics-tutorials.w ronics-tutorials.ws) s)

 

In a series circuit, as the frequency increases upwards from zero, the total impedance of the circuit decreases to its minimum value at resonance because at the resonant frequency the reactances XL and Xc cancel each other out leaving only the resistance across the supply. Then at resonance Z = R. In a parallel circuit, as the frequency increases the total admittance increases to its maximum at the resonance frequency because again at the resonant frequency the reactances XL and Xc cancel each other out leaving only the resistance across the supply. Then Y = 1/R. In both series and parallel circuits, if the same values for L and C are used then the point of resonance will be exactly the same frequency as resistance is unaffected by frequency change. Reply (/accircuits/pa (/accircuits/parallel-resonanc rallel-resonance.html?replytoco e.html?replytocom=5602#re m=5602#respond) spond)

 January 18th, 2015 2015 (http://www.electronics-t (http://www.electronics-tutorials.ws/accircuits/p utorials.ws/accircuits/parallel-resonance.ht arallel-resonance.html/commentml/comment-page-2#comm page-2#comment-5602) ent-5602)

Benjamin Hello,

I am having an unfruitful conversation on-line with my instructor regarding series and parallel configurations, and whether impedance increases if (resonance @ 100 Khz) the same reactive components were put into a series configuration as opposed to a parallel configuration. I have already taken the test, and I have been beating my head against the wall for a couple of days now. I maintain that impedance would increase if the same reactive components were put into series configuration as opposed to parallel. I also maintain that resonance would would not change. I am just working to learn this stuff, and would welcome some advice. I may be dead wrong. If I am though, I would like to know. Reply (/accircuits/para (/accircuits/parallel-resonance.h llel-resonance.html?replytocom= tml?replytocom=5594#res 5594#respond) pond)

 January 18th, 2015 2015 (http://www.electronics-t (http://www.electronics-tutorials.ws/accircuits/p utorials.ws/accircuits/parallel-resonance.ht arallel-resonance.html/commentml/comment-page-2#comm page-2#comment-5594) ent-5594)

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