Circuit Q and Field Energy

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Hortense Lloyd
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1 1 Problem Circuit Q and Field Energy Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ (April 1, 01) In a series R-L-C circuit, as sketched below, the maximum power is delivered to the load resistor R at the resonant (angular) frequency ω 0 =1/ LC, wherel is the inductance and C is the capacitance of the circuit. 1 The frequency bandwidth δ is defined by the relation that the power delivered into the resistor at frequencies ω 0 ± δ is one half the maximum. The Q of this circuit is defined as the reciprocal of the relative bandwidth, ω 0 /δ. Deduce the Q of this circuit, and relate it to the electromagnetic field energy of the circuit. Generalize this result to a circuit with frequency-dependent reactance X(ω) for which X(ω 0 )=0. Solution.1 Series R-L-C Circuit The impedance Z of the circuit when operated at angular frequency ω is ( Z = R + ix = R + i ωl 1 ) ωc where X = ωl =1/ωC is the reactance. The resonant frequency is = R + i ω LC 1, (1) ωc ω 0 = 1 LC, () 1 If the circuit represents an antenna (built from perfect conductors), the load power is radiated to infinity and R is the radiation resistance of the antenna. 1

2 at which the reactance vanishes, the impedance is simply Z = R and the drive voltage V 0 e iωt leads to current I 0 e iωt given by I 0 = V 0 R. (3) The time-average power delivered from the source to the load resistor is = Re(VI ) The time-average (magnetic) energy stored in the inductor is = V 0 R = I 0 R. (4) U B (ω 0 ) = LI 0 4 = LV 0 4R, (5) while the time-average (electric) energy stored in the capacitor is U E (ω 0 ) = Q 0 4C = ω 0I0 4C = LI 0 4C = U L(ω 0 ) = U 0, (6) noting that I = dq/dt = iωq, sothati and Q are 90 out of phase, as are the magnetic and electric energies (5)-(6), and hence their equality at resonance implies that each of these are equal to the total (time-average) stored energy U 0. Away from resonance the current is given by I = V 0 Z, (7) and the power delivered to the load resistor is P = Re(VI ) = I R = V 0 Re 1 Z = V 0 Re 1 R ix = V0 R (R + X ) = (8) 1+X /R The power P is one half of at frequencies where X = ±R. Thatis,for ± R = X = ω LC 1 ωc = ω ω 0 ωω 0C, (9) which leads to the quadratic equation The solutions are ω ± ω 0RCω ω 0 =0. (10) ω = ±ω 0RC ± ω 4 0R C +4ω 0. (11) If the circuit is small compared to the wavelength λ =πc/ω 0,wherec is the speed of light, then the current is the same throughout the circuit. Otherwise, I 0 is strictly just the current at the terminals of the voltage source.

3 We now restrict the analysis to cases where the τ = RC time constant is small compared to the period π/ω 0, and we take the second ± sign in eq. (11) to be +, finding ω ω 0 ± ω 0τ ω 0 ± δ. (1) The quantity δ ω 0τ is called the bandwidth of the circuit. The reciprocal of the relative bandwidth is called the Q of the resonance, δ 1 ω 0 τ = ω 0 ω 0 RC = ω 0I0L = ω 0 U 0 = ω 0L R = X L(ω 0 ) = X C(ω 0 ). (13) R R This demonstrates the usual lore that the Q of a series R-L-C circuit can be expressed in terms of (time-average) quantities at resonance, or in terms of component impedances, as Q ω 0 bandwidth ω 0 stored energy at resonance power delivered to the load at resonance reactance of L or of C at resonance. (14) R. Circuit with Impedance Z = R + ix(ω) and X(ω 0 )=0 For a more general circuit only some of the analysis of sec..1 holds, namely eqs. (3)-(4) and (7)-(8). In general, we cannot identify an effective inductance L and effective capacitance C such that the reactance has the form X(ω) =ωl 1/ωC (see footnote 4 of [1]). If the reactance vanishes at some resonant frequency ω 0 then for nearby frequencies it is a good approximation to write X(ω) (ω ω 0 ) dx(ω 0). (15) dω Then, eq. (9) becomes ± R = X(ω ± δ) ±δ dx(ω 0), (16) dω and hence, and δ R dx(ω 0 )/dω, (17) δ ω 0 dx(ω 0 )/dω = πf 0 dx(f 0 )/dω, (18) R R where ω 0 =πf 0. The form (18) is often used to evaluate the Q of an antenna system via computation of the (terminal) reactance near resonance. 3 3 Expression (18) can be generalized to a potential Q of a circuit at angular frequency ω 0 even if X(ω 0 ) 0, in that if the circuit were tuned by adding (lossless) inductors or capacitors as needed to bring X(ω 0 ) to 0, the (potential) Q of the tuned circuit would be [] Q tuned ω 0 dx(ω 0 )/dω + X(ω 0 ). (19) R 3

4 Comparison of eq. (18) with the last form of eq. (14) permits us to say that close to the resonant frequency ω 0 the general circuit is equivalent to a series R-L-C circuit with 4 L 0 = 1 dx(ω 0 ), and C 0 = 1 = dω ω 0L 0 ω 0 dx(ω 0 )/dω. (1) In addition, we can deduce from the complex version of Poynting s theorem (see, for example, [1]) that X(ω) = 4ω[ U B(ω) U E (ω) ] I(ω), () where now U B and U E are the (time-average) energies associated with the magnetic field B and electric field E. 5 Using eq. () in (18), we have that Alternatively, eq. (9) can be written as d(ω[ U B (ω) U E (ω) ]) dω. (4) ω0 ± R = X(ω 0 ± δ) = 4(ω 0 ± δ)[ U B (ω 0 ± δ) U E (ω 0 ± δ) ] I(ω 0 ± δ) = 4(ω 0 ± δ)r[ U B (ω 0 ± δ) U E (ω 0 ± δ) ], (5) noting that since P (ω 0 ± δ) = /, eq. (8) tell us that I(ω 0 ± δ) = /R. Thetwocases of eq. (5) can now be written as ω 0 + δ = 4[ U B (ω 0 + δ) U E (ω 0 + δ) ], (6) ω 0 δ = 4[ U B (ω 0 δ) U E (ω 0 δ) ]. (7) For δ small compared to ω 0 it is a good approximation that U B (ω 0 + δ) U E (ω 0 + δ) U B (ω 0 δ) U E (ω 0 δ), (8) 4 It is proposed in [3] that the equivalent inductance and capacitance at any frequency be defined by L(ω) = 1 ( dx(ω) dω + X(ω) ), and C(ω) = ω ω (dx(ω)/dω X(ω)/ω), (0) for which L(ω) =L and C(ω) =C when X = ωl 1/ωC. The utility of such a definition is unclear. 5 In [8] it is also proposed that we identify L = 4( U B U B,rad ) I, and C = I 4ω ( U E U E,rad ), (3) but as discussed in footnote 4 of [1], this is unsatisfactory since in general U B,rad U E,rad (see the Appendix of [1]), so that we do not recover eq. () when writing X = ωl 1/ωC. That is, the reactive field energy U B U E includes contributions from radiation field energy (contrary to the assumptions of many workers [3, 5, 4, 6, 7], who may have been misled by the special case of a single, small dipole radiatior). 4

5 such that δ 4[ U B (ω 0 + δ) U E (ω 0 + δ) ], (9) and the Q of the circuit is given by δ ω 0[ U B (ω 0 + δ) U E (ω 0 + δ) ]. (30) The forms (4) and (30) differ from that of eq. (14) in that the energy in the numerator is the so-called (time-average) reactive field energy U B U E (see, for example, sec. 3 of [1]), 6 rather than the total stored energy at resonance. 7,8 References [1] K.T. McDonald, Reactance of a Sinusoidally Driven Antenna (Nov. 15, 010), [] J. Audet, Q Calculations of L-C Circuits and Transmission Lines: A Unified Approach, QEX Sep./Oct., 43 (006), [3] W. Geyi, Calculation of Element Values of Antenna Equivalent Circuit, ISAP005 (Seoul, Korea), [4] R.F. Harrington, Effect of Antenna Size on Gain, Bandwidth, and Efficiency, J. Res. Nat. Bur. Stand. 64D, 1 (1960), [5] R.E. Collin and S. Rothschild, Evaluation of Antenna Q, IEEE Trans. Ant. Prop. 1, 3 (1964), [6] D.R. Rhodes, Observable Stored Energies of Electromagnetic Systems, J.FranklinInst. 30, 5 (1976), 6 Evaluated on resonance in eq. (4) and off resonance by the half-bandwidth δ in eq. (30). 7 The concept of circuit Q originated in circuit examples where radiation is ignored, and the stored energy does not include the (infinite) energy stored in the far ( radiation ) field of a circuit that has been oscillating forever. When the reactance () is evaluated via computations of field energies (as might be done for antenna systems), the infinite, total stored energy is not useful for calculating the Q. 8 A different definition of the Q of an antenna system, not intrinsically related to the relative fractional bandwidth, is often used, following [4, 5], max[ U B (ω 0 ) U B,rad (ω 0 ), U E (ω 0 ) U E,rad (ω 0 ) ]. (31) Still other definitions for the Q of antenna systems exist (see, for example, [6, 7, 8]), based on the notion that the Q of a system can be defined independently of considerations of bandwidth (which seems misquided to this author). 5

6 [7] A.D. Yaghjian and S.R. Best, Impedance, Bandwidth, and Q of Antennas, IEEE Trans. Ant. Prop. 53, 198 (005), [8] W. Geyi, Tribute to Professor Bob Collin (May 18, 011), 6

Chapter 33. Alternating Current Circuits

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