Introduction To Resonant. Circuits. Resonance in series & parallel RLC circuits

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1 Introduction To esonant Circuits esonance in series & parallel C circuits Basic Electrical Engineering (EE-0)

2 esonance In Electric Circuits Any passive electric circuit ill resonate if it has an inductor and capacitor. esonance is characterized by the input voltage and current being in phase. The driving point impedance (or admittance) is completely real hen this condition exists. In this presentation e ill consider (a) series resonance, and (b) parallel resonance. Basic Electrical Engineering (EE-0)

3 Series esonance Consider the series C circuit shon belo. V = V M 0 V + _ I C The input impedance is given by: Z j( ) C The magnitude of the circuit current is; I I V m ( ) C Basic Electrical Engineering (EE-0) 3

4 Series esonance esonance occurs hen, C At resonance e designate as o and rite; o C This is an important equation to remember. It applies to both series And parallel resonant circuits. Basic Electrical Engineering (EE-0) 4

5 Series esonance The magnitude of the current response for the series resonance circuit is as shon belo. I V m V m Half poer point o Bandidth: BW = BW = Basic Electrical Engineering (EE-0) 5

6 Series esonance The peak poer delivered to the circuit is; P V m The so-called half-poer is given hen I V m. We find the frequencies, and, at hich this half-poer occurs by using; ( ) C Basic Electrical Engineering (EE-0) 6

7 Series esonance After some insightful algebra one ill find to frequencies at hich the previous equation is satisfied, they are: an d C C The to half-poer frequencies are related to the resonant frequency by o Basic Electrical Engineering (EE-0) 7

8 Series esonance The bandidth of the series resonant circuit is given by; BW b We define the Q (quality factor) of the circuit as; o Q C C Using Q, e can rite the bandidth as; o BW o Q These are all important relationships. Basic Electrical Engineering (EE-0) 8

9 Series esonance An Observation: If Q > 0, one can safely use the approximation; BW BW o and o These are useful approximations. Basic Electrical Engineering (EE-0) 9

10 Series esonance An Observation: By using Q = o / in the equations for and e have; and Q Q o Q Q o Basic Electrical Engineering (EE-0) 0

11 Amplitude Series esonance Program Output Q = 0, 4, (rad/sec) Basic Electrical Engineering (EE-0)

12 Parallel esonance Background Consider the circuits shon belo: V I C I V jc j V I C V I j jc Basic Electrical Engineering (EE-0)

13 Series esonance Duality I V jc j V I j jc We notice the above equations are the same provided: I C V If e make the inner-change, then one equation becomes the same as as the the other. For such case, e e say say the the one one circuit is is the dual of of the the other. other. Basic Electrical Engineering (EE-0) 3

14 Parallel esonance Background What What this this means means is is is that that for for for all all the all the the equations equations e e have have e have derived derived for for the the parallel parallel resonant resonant circuit, circuit, circuit, e e can can e use use can use for for the the series series resonant resonant circuit circuit provided provided e e make make e make the the substitutions: replaced be replaced by C C replaced by Basic Electrical Engineering (EE-0) 4

15 Basic Electrical Engineering (EE-0) 5 Parallel esonance Parallel esonance Series esonance Q O C O C O C Q o BW BW ) ( C BW BW, C, C C C,,, Q Q o, Q Q o

16 esonance Example : Determine the resonant frequency for the circuit belo. Z I N j( ) jc j jc ( C ( C) j) jc At resonance, the phase angle of Z must be equal to zero. Basic Electrical Engineering (EE-0) 6

17 esonance Analysis ( C ( C) j) jc For zero phase; This gives; C ( C) ( C C C or o ( C C ) Basic Electrical Engineering (EE-0) 7

18 Extension of Series esonance Peak Voltages and esonance: V + _ + V _ V S + _ I C + _ V C We kno the folloing: When = o =, V S and I are in phase, the driving point impedance C is purely real and equal to. A plot of I shos that it is maximum at = o. We kno the standard equations for series resonance applies: Q, BW, etc. Basic Electrical Engineering (EE-0) 8

19 eflection: Extension of Series esonance A question that arises is hat is the nature of V, V, and V C? A little reflection shos that V is a peak value at o. But e are not sure about the other to voltages. We kno that at resonance they are equal and they have a magnitude of QxV S. Irin shos that the frequency at hich the voltage across the capacitor is a maximum is given by; o Q max The above being true, e might ask, hat is the frequency at hich the voltage across the inductor is a maximum? We anser this question by simulation Basic Electrical Engineering (EE-0) 9

20 Extension of esonant Circuits esonant Condition Analysis: What is the significance of r and o in the previous to equations? Clearly r is a loer frequency of the to. To anser this question, consider the folloing example. Given the folloing circuit ith the indicated parameters. Write a Matlab program that ill determine the frequency response of the transfer function of the voltage to the current as indicated. I + _ C + V _ Basic Electrical Engineering (EE-0) 0

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