a

+

C b

~~).. R

L

Fig. 7-22 The ,erie., f?l.C circuit.

enables us to write' the expression for the"current response

V VLO° ~_I(tJL - l/wCILJ!. = =- = - - tan

Z V RC + (u;L - l/wC)c R

Therefore. the magnitude and phase of the effective current are

17-181 )

and

VI = " ,

\'R" -'- (wL - l/wCt

_I wL - l/wCfJ = - tan ----

R

17-182)

17-183)

The corresponding time solution is

- V

i = Y21 sin (wI + 8) = / ' '" ,sin (wt + &) (7-184)\ R- + (wL - l/wC)'

A study of Eq. (7-183) reveals that the current phasor may either lead orlag the voltage phasor. depending upon the relative values of the inductive andcapacitive reactance wL and l/wC. Whenever wL > l/wC the RLC circuitessentially behaves as an inductive circuit insofar as the current is concerned.It is interesting to note that this condition can be satisfied either by having alarge inductance or else by operating at a high frequency. On the other hand.whenever wL < l/wC the current leads the voltage. thereby indicating that theRLC circuit behaves as a capacitive circuit as far as the current is concerned.However. there are some differences: these are discussed presently.

Appearing in Fig. 7-23 is the phasor diagram for the circuit of Fig. 7-22.

Fig. 7-23 Phasor diagram of the f?/Icircuit for wI. > l/wC.

290 Sinusoidal Steady-State Response of Circuits Cilap. 7

Again note the exact correspondence of this result with those obtained by the othermetbods.

The use of Norton's theorem in solving this problem is left as an ex~rcisefor the reader.

7-10 RESONANCE

Resonance is identified with engineering situations which involve energy-storingelements subjected to a forcing function of varying frequency. Specifically, res­onance is the term used to describe the steady-state operation of a circuit orsystem at that frequency fOTwhich the resultant response is in time phase withthe source function despite the presence of energy-storing elements. Resonancecannot take place when only one type of energy-storing element is present, e.g.,inductance or mass. There must exist two types of independent energy-storingelements capable of interc\1anging energy between one another-for example,inductance and capacitance or mass and spring. Although attention here is confinedto electric circuits, resonance is a phenomenon found in any system involvingtwo independent energy-storing elements be they mechanical, hydraulic, pneumatic,or whatever. In the material which follows, the parallel as well as the seriesarrangement of the energy-storing elements-L and C-are treated as functionsof frequency, with particular emphasis focused on the performance at resonance.

Series Resonance. The series arrangement of Land C along with resistanceR is shown in Fig. 7-22. The expression for the effective current flow causedby a sinusoidal forcing function is given by Eq. (7-179) and is repeated forconvenience. Thus

- V VI = ------ = = (7-201)

R + j(wL - l/wC) Z

What is the effect on 7. of increasing the frequency w of the source function fromzero to infinity? A glance at Eq. (7-201) indicates that a change in frequencymeans a change in the magnitude and phase angle of the comple~ impedance.Note that for w close to zero the inductive reactance is almost zero but thecapacitive reactance approaches infinity. Hence the current magnitude approacheszero. As w increases, the reactance part of Z decreases, thus causing an increasein current. As w continues to increase, a point is reached where the reactanceterm is zero. Calling this frequency Wo we have

1

woL - C = 0 (7-202)Wo

or

296

12 __

Wo - LC

Sinusoidal Steady-State Response of Circuits

(7-203)

Chap. 7

(7-204)

I w, ~. V~c IThe frequency Wo is called the resonant frequency of the circuit. Its value isspecified entirely in terms of the parameters of the two energy-storing elementsof the circuit in the manner call,ed for by Eq. (7-204).

At resonance the impedance of the circuit is a minimum, and specificallyit is equal to R. Consequently; when a series RLC circuit is at resonance, the,current is a maximum and is also in time phase with the voltage. The powerfactor is unity. The complete expression for the current phasor is then

_. V VLQ: VI = = = -- = - /0° (7-205)o Z RLJr. R~

where 10 denotes the current at resonance.

For operation at a frequency below Wo the resultant j part of Z is capacitiveso that the current leads the voltage. When w > Wo (he inductive reactanceprevails, so that the current then lags the voltage. Depicted in Fig. 7-27 is theplot of the impedance in the complex plane. The magnitude of Z at a frequencyWI is obtained as the length of the line drawn from the origin to the point WI onthe heavy vertical line. This heavy vertical line denotes the value of the reactance

portion of Z. Note that at w = Wo it has a zero value. An alternative way ofrepresenting the information of Fig. 7-27 is illustrated in Fig. 7-28. A plot of thevariation of the magnitude and phase of the current with frequency for two valuesof the ratio woLI R is depicted in Fig. 7-29. Note that when the ratio of inductivereactance to resistance in a series RLC circuit is high, a very rapid rise of currentto the maximum value occurs in the vicinity of the resonant frequency. A char­acteristic of this type is especially useful in radio and other communicationapplications.

JW

Locus of Z

Iw incrEiasmgWI

Fig. 7-27 Locus of the compleximpedance Z as a function of fre­quency w; Z = R + j(wL - l/wC).

Fig. 7-28 Variation of impedance, in­ductive reactance, and capacitive re­actance as a function of frequency.

Sec. 7-10 Resonance 297

(0)

woL _ 1

-R- - 2"

00

+900 ,Current-i-Current

w,rod/sec leads logsvoltage voltage

( b)

w

Fig. 7-29 Variation of current in a series RLC circuit with frequency:(a) magnitude: (b) phase angle.

Bandwidth. To describe the width of the resonance curve the term bandwidth

is used. For the series RLC circuit bandwidth is defined as the range of frequencyfor which the power delivered to R is greater than or equal to Po/2, where Po

is the power delivered to R at resonance. From the shape of the resonance curveit should be clear that there are two frequencies for which the power deliveredto R is half the power at resonance. For this reason these frequencies are referredto as those corresponding to the half-power points. The magnitude of the currentat each half-power point is the same. Hence we can write

I~R = Y~R = nR (7-206)

where subscript I denotes the lower half-power point and subscript 2 the higherhalf-power point. It follows then that

10

II = 12 = V2 = 0.707/0 (7-207)

Accordingly, the bandwidth may be identified on the resonance curve as that

range of frequency over which the magnitude of the current is equal to or greaterthan 0.707 of the current at resonance. In Fig. 7-29(a) for woLI R = t thefrequency at the lower half-power point is denoted WI and that of the upper half­power point is denoted W2' Hence the bandwidth is W2 - WI'

In view of the fact that the current at the half-power points is 10/V2, it

follows that the magnitude of the impedance must be equal to V2 R to yieldthis current. This information can now be used to obtain an expression for thebandwidth in terms of the parameters of the series circuit. Calling the reactanceat the lower half-power frequency XI' we have

IXI = wlL - - = -R (7-208)

wlC

298 Sinusoidal Steady-State Response of Circuits Chap. 7

The minus sign appears on the right side of the equation because below resonance

the capacitive reactance exceeds the inductive reactance. Rearran~in~ Eq.(7-208) leads to

, Rwi + ZWI

I- LC = 0 (7-209)

The roots are therefore

R J (R r I .~(WI)I.2 = - 2L ± 2L + LC = - a ± Vwhere

(7-210)

Ra == - (7-211)2L

Although Eq. (7-210) provides two solutions to Eq. (7-209), only one of these

is physically realizable. The negative frequency is meaningless and so may bediscarded. Hence the expression for the lower half-power frequency is'

• ,/ 2 2WI = -a + ya + Wo (7-212)

The upper half-power frequency is found in a similar fashion. In this instancewe have

which leads to

R _I = 0w~ - Y-W2 - LC

(7-213)

(7-214)

The expression for the useful W2 is then

,/2 2W2 = a + ya + Wo (7-215)

Therefore, the expression for the bandwidth becomes, from Eqs. (7-212)and (7-215),

(7-216)

This expression is significant because it reveals that the bandwidth of the seriesRLC circuit depends solely upon the R/ L ratio. Note that it is not the individualvalues of R or L but rather their ratio that is important. Note too that bandwidthdepends not at all upon the capacitance parameter C.

Quality Factor Qo of the RLC Circuit. By forming the ratio of the resonant

frequency to the bandwidth we obtain a factor which is a measure oqhe selectivityor sharpness of tuning of the series RLC circuit. This quantity is called thequality factor of the circuit and is denoted by Qo. Thus

Sec. 7-10 Resonance

(7-217)

299

Moreover, when Eq. (7-216) is inserted into the last equation an alternativeexpression results:

.....••..... ~

Q _ ~ _ woL _ XLOo - R/ L - R - R

(7-218)

A glance at the curves appearing in Fig. 7-29(a) should make it clear as to whythe quality factor Qo is used as a measure of the selectivity or sharpness oftuning. Note how much narrower the resonance curve is for Qo = 10 than forQo = !. Equation (7-216) shows that the value of C in no way influences thebandwidth but it does alter the value of Wo0 Hence if C is changed, Wo may bemade to occur at a different position along the frequency scale in Fig. 7-29(a)without in any way affecting the .sharpness of tuning. This feature of the high­Q circuit is used often in communiC'ation networks. For example, in a radio theantenna may be considered in terms of an equivalent RLC circuit where C isadjusted by means of the dial tuning knob. If the dial is turned to the positionwhich tunes in the frequency (wo) of a given radio transmitting station,' thesharpness of tuning (i.e .. small bandwidth) allows only signals from that station

-to produce large resonant current signals. The signals from other broadcastingstations. although present in equal strength at the antenna, produce little or nosignal strength in the circuit because the dial is tuned to a frequency considerablyoff resonance relative to their broadcasting frequencies. Values of Qo Of the orderof 100 are typical in radio circuits.

By making use of the knowledge that Qo is very large in many resonantcircuits, we can express the lower and upper half-power points in terms of theresonant frequency and the bandwidth. Returning to Eq. (7-212), we can rewriteit as

R--+2L -

(7-220)

(7-219)

R J (R r R2- 2L + 2L + (fo L 2

For values of Qo which are 5 or greater ve: y little error is made by writingR R . R

WI = - 2L + IQo = - 2L + Wo

Inserting Eq. (7-216) into Eq. (7-220) yields(7-221-)

Similarly, it can be shown that

W2 = Wo + !Wbw ; (7-222)

Another characteristic of a series RLC circuit at resonance is worth noting.It has to do with the magnitude of the voltage drop' appearing across Land C.In terms of a phasor formulation the phasor voltage drop across L at resonance

300 Sinusoidal Steady-StateResponse01Circuits Chap. 7

is given by

(7-223)

Therefore the magnitude of the voltage across the terminals of the inductor is

Qo times the rms value of the applied voltage. For high-Q circuits this representsa considerable amplification of the source voltage. By proceeding in a similarfashion for the capacitor we find

- -(I) V(I)V co = 10 jwC F R woC / - 90° (7-224)V' .

= -.(woL)/-90° = QoVL=90°R

Again note the resonant voltage-rise effect across the capacitor terminals. Thephasor voltage diagram at resonance is depicted in Fig. 7-30.

Vco

Fig. 7·30 Voltage phasor diagram· of10 the series RLCcircuit at resonance.

On the basis of Eqs. (7-223) and (7-224) ,mother definition of the qualityfactor Qn is possible. It represents the extent to which the voltage across L orC rises at resonance expressed as a multiple of the applied voltage. Statedmathematically. we have

VLO _ VcoQo = V - V (7-225)

where Vu) and Vco are both measured at resonance.

The definitions of Qo given by Eqs. (7-218) and (7-225) are restrictive inthe sense that they apply specifically to the series RLC circuit. The definition

of Eq. (7-217) is more generally applicable because it is based on the frequency­selectivity characteristic of the circuit. However, the most universally applicabledefinition is one that is expressed in terms of energy. In this connection, then,let us assume that the instantaneous expression for the forced solution of currentat resonance is

io = Imo cos wot (7-226)

The total energy stored by the circuit in both Land.C can be expressed as

W = !Li~ + Kv~o (7-227)

Inserting the appropriate expressions for io and VcO yieldsI

Sec. 7-10 Resonance 301

122 Icr2 rW = Z LI mO COS wof + zC c Jo I mO COS wof

1 2 2 I~o. 2

= -LImO cos wof + 2 2C SIn wof2 Wo

1 2 2 1 2 . 2

= ZLI mO cos wof + ZLlmo SIn wof

(7-228)

--~------

1 2 2

= ZLImo = LIo

where 10 is the rms current at resonance. It is interesting to note that at resonancethis energy is a constant quantity.

Returning to Eq. (7-218) and multiplying both numerator and denominatorby I~ we can write for Qo

_ woLl~ _ 21TfoLl~ _ 21T I~LQo - R/~ - RI~ - I~R/fo (7-229)

Now since I~L represents the total stored energy at resonance and I~R/fo is theenergy dissipated per cycle, Eq. (7-229) may be written more generally as

Q _ total stored energyo - 21T d" d Ienergy lsslpate per cyc e

(7-230)

(7-232)

Equation (7-230) is applicable to any resonant system regardless of its composition.As long as the quantities on the right side can be determined the quality factorcan, in turn, be found.

Parallel Resonance. The circuit configuration for the study of parallel res­onance appears in Fig. 7-31. The expression for the total current is

] = Vy = V(C + jB) (7-231)

where] is a fixed, known quantity and V is the nodal voltage, which varies asy varies with frequency. For the circuit of Fig. 7-31 the expression for theadmittance is

- 1 1 1 ( 1)Y = C + jB = - + jwC + -.- = - + j wC - -

R JwL R wL

I

R

Fig. 7-31 Circuit configuration forstudying parallel resonance.

302 Sinusoidal Steady-StateResponseof Circuits Chap. 7

Hence

1 = V[~. + j( wC - W~)](7-233)

A study of this equation reveals that for small values. of W the susceptance B

will be large. Then to keep the product of the right side constant and equal to

1, the nodal voltage V must be correspondi~gly small. As W increases, B decreases,and so V increases. When B equals zero, V has its maximum value. At this pointthe output nodal voltage is in time phase with the current source, and the circuitis said to be in parallel resonance. The. frequency at which resonance occurs isfound from

or

IB. = woC - - = 0

woL(7-234)

IWo = , /- (7-235)yLC

which is identical to the expression which applies to the series RLC circuit. AsW increases beyond wo, the susceptance gets ever larger, thus causing the nodalvoltage to diminish toward zero. A sketch of V as a function of frequency isdepicted in Fig. 7-32. The bandwidth is identified as the frequency range lyingbetween WI and W2' For the parallel case, the lower frequency is found from

I Iw1C - - = -- (7-236)

w,J.. R

which leads to

2 I IWI + -WI - - = 0RC LC

Ignoring the negative solution, the desired expression is

I J ( I r I --WI = - 2RC + 2RC + LC = -a + Va2 + w~

where

Ia = 2RC

(7-237)

(7-238)

(7-239)

v

va' Y R

0.707 I R

01 w, Wn w, w,rod/sec

Sec. 7-10 Resonance

Fig. 7-32 Variation of the nodal volt­age II of Fig. 7-31 as a function offrequency.

303

Similarly for'the upper half-power frequency pointI I

W2C - - = -W2L R

This leads to

I J( I )2 I •/. 2

W. = -- + -- + - = a + va- + Wu- 2RC 2RC LC

Therefore, the expression for the bandwidth is

W.w =. W, - W, = 20 = fc: I

(7-240)

(7-241 )

(7-242)

Employing the definition of the quality factor given by Eq. (7-217) leadsto

Wo WoQo =.- = -- = woRC (7-243)

Wbw IIRC

It can be shown that Eq. (7-230) also applies with equal validity in determiningQo·

Just as there occurs a resonant rise in voltage associated with Land C inthe series-resonant case, so too there occurs a resonant rise in current in the Land C elements when the circuit is at parallel resonance. This is readily dem­onstrated. Recall that the value of the nodal voltage Vo at resonance is

Vo = IRLJr. (7-244)

Since this voltage appears across L we can write

lLO= .Vo = IR /-90°= IwoRC/-900= IQo/-90° (7-245)JWoL woL

For th'(' capacitor

- Vo

Ico = TC = jlRwoC = Qol/90° (7-246)I JWo

The subscript 0 denotes the resonant condition. An inspection of each of thelast two expressions shows that the magnitude of the current through L or C isQo times the source current I. It follows then that another way of expressing Qo

in the parallel-resonance case is to write

I IQo = ~ = ~ (7-247)I I

Equation (7-247) is analogous to Eq. (7-225) for the series-resonant case.Appearing in Fig. 7-33 is the phasor diagram for the currents using the

nodal voltage at resonance, Vo, as the reference phasor. Note thatlLO lags andIco leads Vo by 90°, as called for by Eqs. (7-245) ar.d (7-246). Of course, theresultant of the three currents is the source current I.

304 Sinusoidal Steady-StateResponseof Circuits Chap. 7