JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 1

On the Functional Relation between Quality Factorand Fractional Bandwidth

Miloslav Capek, Member, IEEE, Lukas Jelinek, and Pavel Hazdra, Member, IEEE

Abstract—The functional relation between the fractional band-width and the quality factor of a radiating system is investigatedin this note. Several widely used definitions of the quality factorare compared on two examples of RLC circuits that serve asa simplified model of a single resonant antenna tuned to itsresonance. It is demonstrated that for a first-order system, onlythe quality factor based on differentiation of input impedancehas unique proportionality to the fractional bandwidth, wherease.g. the classical definition of the quality factor, i. e. the ratioof the stored energy to the lost energy per one cycle, is notuniquely proportional to the fractional bandwidth. In addition,it is shown that for higher-order systems the quality factor basedon differentiation of the input impedance ceases to be uniquelyrelated to the fractional bandwidth.

Index Terms—Antenna theory, Electromagnetic theory, Q fac-tor.

I. INTRODUCTION

THE fractional bandwidth (FBW) is a parameter of pri-mary importance in any oscillating system [1], since

it is a relative frequency band in which the system can beeffectively driven by an external source. In the case of anantenna, a fractional bandwidth is a frequency band in whichthe power incident upon the input port can be effectivelyradiated [2].

Based on an analytical evaluation of the basic RLC circuitsin the time-harmonic domain [3], the FBW is believed to beinversely proportional to the quality factor, which is commonlydefined as 2π times the ratio of the cycle mean stored energyand the lost energy, see e.g. IEEE Std. 145-1993, [4]. Thisrelation is known to be very precise for high values of Q,and has been shown to be exact for Q tending to infinity, i.e.a lossless oscillating system cannot be driven by an externalsource, since its FBW is equal to zero. However, this inverseproportionality is known to fail at low values of Q and, infact, it is not clear whether there exists any functional relationof FBW and Q which would be valid in all ranges of Q. Itis however important to stress that if such a relation were toexist, it would be of crucial importance, since there exists afundamental lower bound of Q of a lossless electromagneticradiator [5], [6], which would then imply a fundamentalupper bound of its FBW, an essential theoretical limitationfor electrically small radiators.

Manuscript received March 20, 2014; revised March 20, 2014. This workwas supported by the Czech Science Foundation under project 13-09086S andby the COST IC1102 (VISTA) action.

The authors are with the Department of Electromagnetic Field, Faculty ofElectrical Engineering, Czech Technical University in Prague, Technicka 2,16627, Prague, Czech Republic (e-mail: jelinel1@fel.cvut.cz).

This note serves two purposes. First, a proof is given ofthe non-existence of a general functional relation betweentraditionally defined Q and FBW. The proof is based on ananalytical evaluation of the functional relation for two distinctRLC circuits. It is given by contradiction, and it also coverssome other commonly used prescriptions of Q. Second, it ispointed out that the so-called QZ factor defined in [7] andfurther generalized in [8] is inversely proportional to FBWfor first order systems, but ceases to have this behaviour forhigher order systems.

II. DEFINITION OF THE Q FACTOR

This Section defines several widely used prescriptions ofthe Q factor that will be used later:

• classical Qcl, [4],• modified Qrev, based on the concept of reversible energy,

[9],• QX , based on differentiation of the input reactance, [10],• QZ , based on differentiation of the input impedance, [7],

[8].

A. Classical definition of the Q factor

The classical Q factor is conventionally defined as [4]

Qcl =ω0Wsto

Plost, (1)

in which ω0 is the resonant frequency, Wsto is the cycle meanstored energy, and Plost is the cycle mean power loss. Thisprescription of the Q factor is traditionally encumbered withdifficulties in identifying of the stored energy of a generalelectromagnetic radiator [11]. This problem is however leftaside in this note, as Wsto is used only for non-radiating cir-cuits for which the concept of stored energy is well established[3]. Namely, the cycle mean stored energy of a non-radiatingcircuit can generally be written as

Wsto =1

4

∑n

(Ln |ILn |

2+ Cn |UCn |

2), (2)

and the cycle mean lost power can be written as

Plost =1

2

∑n

Rn |IRn|2 . (3)

In (2) and (3), L, C, R are the inductance, capaci-tance, and resistance of the circuit, and IL, UC, IR arethe corresponding currents and voltages. The conventionF (t) = <{F (ω) exp(ωt)} for time-harmonic quantities hasbeen utilized.

arX

iv:1

411.

5821

v1 [

phys

ics.

clas

s-ph

] 2

1 N

ov 2

014

JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 2

B. Qrev factor based on reversible energy

The original definition of Q (1) can be slightly modified to

Qrev =ω0Wrev

Plost, (4)

in which Wrev (so-called reversible energy) is that part ofthe stored energy Wsto which can be recovered back fromthe input port by a matched load. This reversible energy canin essence be evaluated by bringing the system into a time-harmonic steady state at frequency ω0 by a voltage sourcewith matched internal impedance, afterwards switching off thesource and capturing all the energy returned to the internalimpedance [9]. A detailed description of this method for ageneral radiator can be found in [12].

C. Reactance QX factor

A different approach to defining Q is based on the assump-tion that Foster’s reactance theorem [13] also holds for lossysystems [14], [15]. In that case, Q can be defined by thefrequency derivative of the input reactance as

QX =ω0

2<{Zin}

∣∣∣∣∂={Zin}∂ω

∣∣∣∣ω=ω0

, (5)

where Zin is the input impedance of the circuits. This defi-nition was proposed by Harrington [16], and was refined byRhodes [10], and it is commonly used even nowadays.

D. Impedance QZ factor

A prescription which is widely used in antenna practicegives the Q factor in terms of the input impedance [7], [8].The relation reads

QZ =ω0

2<{Zin}

∣∣∣∣∂Zin

∂ω

∣∣∣∣ω=ω0

(6)

and it is known to correspond well to FBW [8].

III. FUNCTIONAL RELATION OF Q AND FBW

The major purpose of this note is to investigate the func-tional relation

FBW = f (Q) , (7)

where f is an as yet unknown function and

FBW =ω+ − ω−

ω0, (8)

in which ω+ and ω− delimit the operational range of anantenna.

A. First-order systems

Instead of directly analysing relationship (7) for a complexsystem such as an antenna, we start with two single-resonantRLC circuits, depicted in Fig. 1. If it is proved that a givendefinition of Q is not uniquely proportional to FBW for thesimple circuits in Fig. 1, it can be concluded that this Q is notproportional to FBW at all.

C

L

inZ

R0

R

0Z

C

L

inZ

R0

R

0Z

(b)(a)

(a) (b)

Fig. 1. The studied RLC circuits connected to a voltage source with internalresistance R0: (a) R, C, and L in series, (b) C in series with parallel L andR.

Γ

ω0 ω0 ωβ+ωβ−

1

β Γ

FBWβ

zero of the first-orderzero of the second-order

Fig. 2. The course of the reflectance in the vicinity of the circuit’s resonanceω0. The circuit is assumed to be matched (Zin (ω0) = R0) at resonancefrequency ω0, so that |Γ (ω0)| = 0.

We utilize a simple consideration depicted in Fig. 2, whichassumes that the reflectance

|Γ| =∣∣∣∣Zin − Z0

Zin + Z0

∣∣∣∣ (9)

can be expanded to its Taylor series and only the first non-zeroterm can be kept near the circuit’s resonance. Without loss ofgenerality, we also assume that the circuit is matched to theinput resistance (Zin (ω0) = R0). Under such conditions thereflectance can be written as

|Γ| = |ω − ω0|∣∣∣∣∂Γ

∂ω

∣∣∣∣ω=ω0

+O(ω2)

≈ |ω − ω0|ω0

ω0

2<{Zin}

∣∣∣∣∂Zin

∂ω

∣∣∣∣ω=ω0

=FBW

2QZ ,

(10)

which for |Γβ | → 0 gives the required functional relation (7)

FBWβ = 2|Γβ |QZ

. (11)

This means that the QZ factor (6) is uniquely proportional tothe FBW (at least in this differential sense). Furthermore, if theother Q factors are to follow relation (7) they must necessarilybe functionally dependent on QZ . This property is investigatedin the following.

The QZ factors of the two circuits under consideration can

JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 3

easily be calculated from the input impedances, which yields

Q(a)Z =

ω(a)0 L

R, ω

(a)0 =

1√LC

, (12a)

Q(b)Z =

R

ω(b)0 L

1

ω0

√LC

, ω(b)0 =

R

L

√CR2

L− 1

, (12b)

where superscripts (a) and (b) refer to the two circuits in Fig. 1.The other Q factors can be evaluated in a straightforwardmanner as

Q(a)cl = Q

(a)Z , (13a)

Q(a)X = Q

(a)Z , (13b)

Q(a)rev =

Q(a)Z

2, (13c)

and

Q(b)cl = ξclQ

(b)Z , (14a)

Q(b)X = ξX Q

(b)Z , (14b)

Q(b)rev = ξrev

Q(b)Z

2, (14c)

where

ξcl =√χ, (15a)

ξX =√χ

χ

χ+(Q

(b)Z

)−2 , (15b)

ξrev =√χχ+

(Q

(b)Z

)−2

χ+ 2(Q

(b)Z

)−2 , (15c)

χ =1 +

√1 + 4

(Q

(b)Z

)−2

2. (15d)

The above results offer a simple interpretation. Since theQZ factor has been shown to have a unique functional relationto FBW, see (11), the other quality factors could have sucha unique functional relation only if the functional relationscorresponding to circuit (a) and circuit (b), see (13) and (14),are the same, i.e. if the corresponding ξ coefficients in (15a),(15b), (15c) are equal to unity. That this is not the case is clearfrom their analytical prescription and also from their graphicalrepresentation in Fig. 3. By means of contradiction, it mustthen be stated that there is no general functional relationbetween FBW and Qcl, QX , Qrev, the only exception beingQ→∞.

B. Higher-order systems

Section III-A has shown that, in the case of first-ordersystems, only QZ is a potential candidate for having a generalfunctional relation to FBW. The purpose of this subsection isto test this property on higher-order systems.

Higher-order systems offer more degrees of freedom. Thisin general makes approximation (10) invalid. In fact, it can beshown [17] that a circuit of order n can always be tuned so

QZ

ξ

ξrev

ξX

0 1 2 3 4 50

0.5

1

1.5

2ξcl

Fig. 3. The ξ factors of (15a)–(15c) as a function of QZ .

10−1

100

10110

−4

10−2

100

102

KP / KS

QZ ® 0: KS=KP

QZ

R0

QZ = | KS - KP |

KP

ω0R0 ω0KP

R0 1ω0KSR0

KSR0

ω0

Fig. 4. The quality factor QZ of a selected second-order RLC circuit.

that first n−1 terms of the Taylor expansion vanish (binomialtransformer).

An example of such a second order system [18], [19] isdepicted in Fig. 4, which for KP = KS results in QZ = 0.Another example [20] is a thin-strip dipole of length L andwidth w = L/100, see Fig. 5. If the dipole is fed by a voltagegap placed at h ≈ 0.228L, we realize that QZ = 0 at ka ≈6.171, in which k = ω/c0 is the wavenumber, c0 is the speedof light, and a is the radius of the smallest circumscribingsphere.

This awkward property of a possibly zero value of QZ in thecase of circuits with clearly finite FBW unfortunately excludeQZ from prescriptions with a possibly unique relation to FBW.

IV. CONCLUSION

It has been shown that, contrary to common belief, theclassical quality factor defined by the stored and lost energyis not related to the fractional bandwidth by a general andunambiguous functional relation. This is also true for Qfactors resulting from reversible energy and input reactance.Considering the first-order system, only the Q factor basedon differentiation of the input impedance has been shownto be a possible candidate for such a general functionalrelation. It has however been demonstrated that for higher-order systems, including elementary radiators like dipoles,

JOURNAL OF LATEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 4

6.00 6.05 6.10 6.15 6.20 6.25 6.300

2

4

6

8

10

ka

QZ

w

hL

h1 = 0.225 L

h2 = 0.228 L

h3 = 0.232 L

h4 = 0.235 L

QZ ® 0: ka » 6.171, h » h2

Fig. 5. The quality factor QZ of a thin-strip dipole as a function of electricalsize and as a function of feeding position.

no quality factor has in general exact proportionality to thefractional bandwidth.

ACKNOWLEDGEMENT

The authors would like to thank to Mats Gustafsson fromLund University, Sweden for a fruitful discussion about thetopic, and for pointing out the possibility of minimizing QZby off-center feeding of a dipole.

REFERENCES

[1] P. M. Morse and H. Feshbach, Methods of Theoretical Physics.McGraw-Hill, 1953.

[2] C. A. Balanis, Antenna Theory Analysis and Design, 3rd ed. JohnWiley, 2005.

[3] E. Hallen, Electromagnetic Theory. Chapman & Hall, 1962.[4] Standard Defintions of Terms for Antennas 145 - 1993, IEEE Antennas

and Propagation Society Std.[5] L. J. Chu, “Physical limitations of omni-directional antennas,” J. Appl.

Phys., vol. 19, pp. 1163–1175, 1948.[6] R. C. Hansen and R. E. Collin, “A new Chu formula for Q,” IEEE

Antennas Propag. Magazine, vol. 51, no. 5, pp. 38–41, Oct. 2009,[7] D. Kajfez and W. P. Wheless, “Invariant definitions of the unloaded Q

factor,” IEEE Antennas Propag. Magazine, vol. 34, no. 7, pp. 840–841,1986.

[8] A. D. Yaghjian and S. R. Best, “Impedance, bandwidth and Q ofantennas,” IEEE Trans. Antennas Propag., vol. 53, no. 4, pp. 1298–1324, April 2005.

[9] C. A. Grimes, G. Liu, F. Tefiku, and D. M. Grimes, “Time-domainmeasurement of antenna Q,” Microwave and Optical Technology Letters,vol. 25, no. 2, pp. 95–100, April 2000.

[10] D. R. Rhodes, “Observable stored energies of electromagnetic systems,”J. Franklin Inst., vol. 302, no. 3, pp. 225–237, 1976.

[11] G. A. E. Vandenbosch, “Reply to “Comments on ‘Reactive energies,impedance, and Q factor of radiating structures’”,” IEEE Trans. Anten-nas Propag., vol. 61, no. 12, p. 6268, Dec. 2013.

[12] M. Capek, L. Jelinek, G. A. E. Vandenbosch, and P. Hazdra, “Ascheme for stored energy evaluation and a comparison with contem-porary techniques,” IEEE Trans. Antennas Propag., 2014, submitted(arXiv:1309.6122).

[13] R. M. Foster, “A reactance theorem,” Bell System Tech. J., vol. 3, pp.259–267, 1924.

[14] D. R. Rhodes, “A reactance theorem,” Proc. R. Soc. Lond. A., vol. 353,pp. 1–10, Feb. 1977.

[15] W. Geyi, P. Jarmuszewski, and Y. Qi, “The Foster reactance theoremfor antennas and radiation Q,” IEEE Trans. Antennas Propag., vol. 48,no. 3, pp. 401–408, March 2000.

[16] R. F. Harrington and J. R. Mautz, “Control of radar scattering by reactiveloading,” IEEE Trans. Antennas Propag., vol. 20, no. 4, pp. 446–454,July 1972.

[17] R. E. Collin, Foundations for Microwave Engineering, 2nd ed. JohnWiley - IEEE Press, 1992.

[18] M. Gustafsson and B. L. G. Jonsson. (2013) Stored electromagneticenergy and antenna Q. eprint arXiv: 1211.5521v2. [Online]. Available:http://arxiv.org/abs/1211.5521

[19] M. Gustafsson and S. Nordebo, “Bandwidth, Q factor and resonancemodels of antennas,” Progress In Electromagnetics Research, vol. 62,pp. 1–20, 2006.

[20] M. Gustafsson, D. Tayli, and M. Cismasu. (2014) Q factors forantennas in dispersive media. eprint arXiv: 1408.6834v2. [Online].Available: http://arxiv.org/abs/1408.6834