Design of Choke in Class-E ZVS Power Amplifier

Agasthya Ayachit, Dalvir K. Saini, and Alberto Reatti Marian K. Kazimierczuk Department of Information Engineering (DINFO) Department of Electrical Engineering University of Florence Wright State University Via Santa Marta 3 1-50139 Florence 3640 Colonel Glenn Hwy., Dayton, OH, 45435, USA [email protected] {ayachit.2, saini. 1 1, marian.kazimierczuk}@wright.edu

Abstract-This paper presents the following for a Class-E current through the choke are non-sinusoidal. The zero-voltage switching (ZVS) power amplifier: (a) design of the losses in the windings as well as in the as choke inductor and (b) theoretical estimation of power losses in caused by the dc, fundamental, and higher order harmonics. the core and a solid round winding. The expressions required Therefore, Fourier analysis is performed and the magnitude of to design the core using the Area-Product (Ap) method are provided. The equations for the dc resistance, ac resistance the harmonic components are evaluated. at high frequencies, and dc and ac power losses are provided The objectives of this paper are as follows: for the solid round winding. A Class-E ZVS power amplifier with practical specifications is considered. A core with air gap 1) To develop a theoretical framework design the choke is selected since the choke inductor carries a dc current in for a Class-E zero-voltage switching power addition to the ac component. The gapped core power loss amplifier. density and power loss are estimated using Steinmetz empirical 2) To present the design equations necessary to calculate equation. Simulation results showing the transient analysis and the dc and ac resistances and dc and ac power losses for Fourier analysis are given. It is shown that, for the given design, the winding power loss due to the fundamental component the choke inductor made up of solid round winding. is dominant and that due to higher order harmonics can be 3) To provide the necessary expressions required to design neglected. In addition, it is also proven that the power loss caused the magnetic gapped core using Area-Product method. by the dc current component is higher than that by the ac current 4) To evaluate the fundamental and higher order harmonic component, which can be neglected. components of the choke inductor current and determine

I. INTRODUCTION their power losses. 5) To verify the presented analysis through circuit simula­ Magnetic components are integral parts in many electronic tions. circuits [1] - [6]. An ideal choke must reject any ac component and conduct only the dc component. However, in practical applications, the choke inductor conducts a fraction of ac II. DESIGN EQUATIONS current in addition to the dc component and the ac component can reach high-frequenies and its waveform is usually not A. Design of Class-E ZVS Power Amplifier sinusoidal. . Thus, the design procedure must consider both Fig. 1 shows the circuit of the Class-E zero-voltage switch­ these components and predict the nature of losses in the ing (ZVS) power amplifier. The amplifier is supplied by a dc core and the winding. This paper presents a comprehensive input voltage source , Lf is the choke inductance, S is the procedure to design the choke inductors used in Class-E ZVS VI power MOSFET, C1 is the voltage-shaping shunt capacitance, power amplifiers. Class-E ZVS power amplifiers belong to which includes the output capacitance of the MOSFET, Land a class of highly efficient electronic circuits. Therefore, the C form the resonant circuit inductor and capacitor, while component design must be optimized to yield the maximum R is the load resistance. efficiency. The expression for the load resistance is The magnetic core for the choke inductor is designed using the Area-Product Ap method. This is a widely adopted method . =_8_Vl to magnetic components in pulse-width modulated converters R 7T2 4 (1) as well as ac resonant inductors [2]. Since a majority of the + Po' current is dc, air gap is essential to avoid core saturation. where Po is the output power. The choke inductor can be In addition, the winding power loss depends on the winding designed using geometry. In this paper, a solid round winding is used and its dc and ac winding power loss are estimated to determine the dominant power loss. Ty pically, in switching circuits, the (2)

978-1-5090-3474-1/16/$31.00 ©20l6 IEEE 5621 ILt - t 21,tm hIm ...... Iml wt ;"�Lt L C

------+ ht o wt VI R � C1

Fig. 2. Waveforms of the ac component of the choke inductor current and its fundamental harmonic component. Fig. I. Circuit of the Class-E power amplifier.

and the amplitude of the third harmonic component is where is is the switching frequency of the MOSFET. The dc current flowing through the choke is equal to the input current ShJm 1ml Im 3= 97f2 = 9 . (10) and is expressed as The even harmonics (n = 2,4, 6, ... ) are zero since the PI Po 1LJ= h = -= -, (3) triangular waveform is assumed to be symmetric about its VI f/VI dc value. By expanding the series, it can be noted that the where T/ is the overall efficiency of the Class-E amplifier. The amplitudes of the harmonics for n :;0. 3, i.e., 1m3, 1m5, ... choke inductor carries an ac component in addition to the dc are much lower than the amplitude of the fundamental 1ml . current. The amplitude of the ac current through the choke is Therefore, the effect of the higher order harmonics of the given by choke inductor current on the winding and core power can VI be neglected. J = (4) h m 4isLJ' B. Design of Core Using Area-Product Method Therefore, the peak value of the choke inductor current is The area-product method relates the geometrical core pa­ Po VI rameters described by the core manufacturers with the inductor hJ(max) = hJ+ hJ m= + (5) magnetic and electric parameters The maximum energy VI 4isLJ' [3]. f/ stored in the choke inductor due to both dc and ac currents is The choke inductor must be designed to withstand the peak current hJ(max) . The winding carrying the choke inductor (11 ) current must satisfy the current density requirement as well as The window utilization factor and the saturation flux exhibit low ac power losses. Ku density are assumed a priori. The current density of the The current through the choke inductor can be approximated Bsat conductor is selected based on the peak current requirement by a symmetrical triangular wave of magnitude h m as shown J and is discussed in the following section. The core area­ in Fig. 2. The Fourier series expansion of the triangular wave product is given by is 4 ( _1 ) (n-I)/2 WLJ -'------'----,,----- (nw t) Ap= WaAc= (12) n 2 sin s . (6) KuJmBsat' where Wa is the core window area and Ac is the core

The fundamental component of the triangular waveform IS cross-sectional area, both of which are mentioned in the core 1 obtained by substituting n= in (6) to get manufacturer datasheet. Further, the window area is estimated from (12) as (7) (13) where Ws is the angular switching frequency and the amplitude of the fundamental of the ac component through the choke From the selected core geometry, the number of turns N inductor is is determined using a specified window area Wa, window - h m - S J I mi 7f2 . (8) utilization factor Ku, and solid round wire current density as Jm as In (8), hJm is the peak value of the triangular waveform of WaKuJm the choke inductor current. By substituting n= 3 in (6), the N= . (14) 2 third harmonic component is hJ(max) However, for the gapped core, to achieve the desired induc­ (9) tance and strength of magnetization, the number of turns is

5622 higher than that calculated in (14). Thus, for a core with air gap with a predetermined inductance L f, the number of turns is [3]

N= (15) where Ac is the cross-sectional area of the core, flo is the permeability of free-space, fLrc is the relative permeability of the core material, N is the number of turns, Ie is the mean magnetic path length, 19 is the assumed length of the air gap. Thus, the actual number of turns is

WaKu max { , l:L (I (16) Fig. 3. Solid round conductor showing a reduced effective area available for N= + . 2Aw(sol) fLoAc 9 fLr�)}c current conduction at high frequencies due to skin-effect.

Alternatively, low iron powder cores with a low relative permeability fLr can be utilized. However, in both of the dc current density. Assuming a copper conductor, the skin cases the core permeability is reduced, while the number of depth as a function of frequency is [3], turns required to achieve a certain inductance is increased. This affects the winding design. The dc current through the p 66.2 ' (21) choke as well as the ac current are functions of the load vITo resistance of the amplifier. In several applications, for example, where 1.7 10-8 [l·m is the resistivity of the conductor, Class-E ZVS inverter in wireless power charging systems, p= 24 x is the operating frequency, is the relative permeability the load resistance is a function of the distance between fo ILr of the conductor 1 for metals), and is the absolute the transmission coil and the receiver coil. In such operating (fLr= flo permeability of free space. Let 1 be the current density of conditions, the dc and ac current may exceed a critical value m the winding. For the selected choke inductor, current density causing the core to saturate. This requires an air gap in the is given by core. For the gapped core, the peak value of the magnetic flux

density in the core of the choke inductor is given as � � hf(max) hf Jm :::::; (22) ILrclLoN I Lf(max) Aw(sol) Aw(sol)' Bpk = . (17) Mn,lq 2 Ie (1 + ) where is in A/mm and A is the cross-sectional area L, 1m w(sol) . of the winding. Thus, the inner diameter of the solid round However, the amplitude of magnetic flux density due to the winding must be fundamental of the ac component of the choke inductor current is di(sol) = (23) (18) A corresponding wire for the inductor winding is chosen from the AW G datasheet. Its inner and outer diameters are noted Thus, for the gapped core, the Steinmetz empirical equation along with the bare wire area and the resistance per unit length. for the power loss density at frequency fs is [4], [5] In addition, the average length per turn IT of the winding b around the core is also estimated. Thus, the total length of the 8fLrcfLON hJm winding is l = NIT. The dc resistance of the solid round P vg = kr B emb = kr s (19) w [ M lq winding is 21 1 + ' 1 'if c ( l:, ) where k, and are the Steinmetz empirical constants plw 4pN1T a, b [4]. = = . (24) Rwdc(sol) A- -- d2 Consequently, the total power loss in the gapped core is w(sol) 'if i(sol)

The conduction power loss due to the dc component is

(20) 4pN1TIJ = 2 = (25) PRwdc(sol) ILfRwdc(sol) d2 ' 'if i(sol) Winding Design C. At high operating frequencies, the available area for current This section provides a methodology to design the inductor conduction is smaller than that at dc or low frequencies. This with a solid round winding. The solid round winding is phenomena occurs due to the skin effect phenomena at high designed to withstand the maximum current stress at a uniform frequencies, thereby increasing the overall resistance of the

5623 winding This effect is illustrated in Fig. The effective Relative permeability of the core material = [3]. 3. • ILrc 3000. area is determined as In addition, in order to avoid distortion due to core satura­ 2 )2 tion, the maximum choke inductor current is assumed as For 7rd _ 7r( di - w A - i O -7r0 (d 0 ) (26) e 4 4 Wt- W· hjmax = 1.2 A. Therefore, using (22), the total area of the winding is Therefore, the solid round winding ac resistance can be approximated by 1.2 - - hj(max) 0.24 mm2 . Aw -A w(sol) - 5 106 = (32) pl 4pNlT Jm X w ) . Rwac(sol) = - = s: S: (27) A e Uw( di - Uw From the AW G datasheet, the selected winding is AW G The inner diameter of the selected wire is The power loss due to the ac component 23. di(sol) = 0.573 mm, the outer diameter is 0.632 and do(sol) = mm, I';'1 64IZjm the bare wire area is A = 0.258 mm2. pwac(sol) = --Rwae(sol) = Rwae(sol)· (28) w(sol) 2 27r4 Using (13), the window area is Wa = 13.45 mm2. There­ 2 The number of layers is a factor, is related to the area product fore, Wap = Wa/2 = 6.72 mm . From (14), the number of Ap and the window utilization factor K". The selected core turns is allows for a single layer winding to be wound, and, therefore, 0.4 6. 5 10-6 _ K"Wa _ x 72 x 10.38. the proximity effect is neglected in this paper. N - - 0.258 6 = (33) Aw(sol) x lO- D. Total Power Losses In comparison, using (15), the number of turns assuming 19 = For the inductor with solid round winding, the total power 0.1 mm is loss in the designed inductor is the sum of the core power loss given in (20), the power loss due to dc current given in (25), N= and the winding loss given in (28), and is expressed as 40 10-6 (0.1 9.567 10-3) 10-3 X x + x x = + + (29) -----,- ---'c-::-----;::;------::-:--:: --:-::- 9.68 PLj(sol) Peg Pw(sol) pwac(sol)· __ _ 10-7 3 . _10-6--+-- __ = 47r x x 7 2 X III. DESIGN EXAMPLE (34)

A. Selection of Magnetic Core and Winding Thus, using (16), the number of turns chosen for this design A choke inductor is designed for the following specifica­ is tions: supply voltage = 10 V, output power = 10 W, VI Po N= max{10.38, 9.68}= 10.38. (35) switching frequency is = 1 MHz. The efficiency of the amplifier is assumed as T/ = 0.9. Using (I) and (2), the Select the number of turns as N = 10. The maximum values of the load resistance is R = 5.76 n and the choke thickness of the winding is equal to the diameter of the solid inductance is L j = 40 ILH. The dc input current using (3) round wire, di(sol) = 0.552 mm. From the core datasheet, is Ir = hj = 1.11 A. The amplitUde of the ac current the height of the window for the selected core geometry 0.0625 through the choke inductance using (4) is hjm = A. is H = 2D = 7.4 mm. For 10 turns, the overall height 5.52 Therefore, the peak value of the choke inductor current is occupied by the windings is H = N di(sol) = < 2D. hj(max) = hj +h jm = 1.1725 A. It must be noted that Therefore, the winding arrangement consists of only one layer, hjm/ hj= 5%. Thus, the magnitude of the ac current i.e., NI = 1. Further, the wire length per single turn wound through the choke inductor is only 5% of its dc value. The around the center-post of the selected core is maximum energy stored in the choke inductance using (11) is 7.45 10-3 23.40 mm. IT = 7rF = 7r x X = (36) 1 40 10-6 1.22 2 X X 0.0288 mJ. WLj= L = = Therefore, the total length of the wire is 0.234 m. 2 jILj(max) 2 lw= NIT= (30) For the given core and electrical parameters, the peak value Assume the following core parameters: current density Jm= of the magnetic flux density for the gapped core as given in 2 5 A/mm , window utilization factor K" = 0.25, saturation (17) is magnetic field density 0.25 The core area product Bsat = T. ILOlLrcN1Lj(max) using (12) is Bpk = Ie +IL relg 0.028 10-3 4W 4 x X 10-7 3000 10 1.2 Lj - 0.0364 cm4 47r X X x x (37) Ap= -,----,-0.25-,-,------,--5 ---:-10;;-- 6 ----:-0.--,--,-,-25 = . " B t x x x 10-3) (3000 0.1 10-3) K Jm sa (28.72 x x x (31) + 0.137 T. A Magnetics® HS gapped pot P-type core 41811 was = selected, which has: For the selected core material, the saturation flux density is Core area product 0.05 cm4. 0.45 T at 25°C and Thus, the chosen • Ap= Bsat = Bpk < Bsat. Core cross-sectional area 3 . mm2. air gap avoids core saturation. The solid round winding has a • Ac= 7 2 Mean magnetic path length single layer wound over the center post of the selected core. • lc= 28.72 mm.

5624 From the fundamental harmonic approximation discussed in

Section II-A, the triangular waveform of the ac component of Rwac the choke inductor current is approximated by its fundamental sinusoidal component. The amplitude of the fundamental of Ct ac component through the choke inductor is 8 . 6 A hjm1 = 2 hjm= 0.8105 x 0.0625 = 0 050 (3S) 7r Fig. 4. High-frequency lumped-circuit model of choke inductor. The amplitude of the sinusoidal magnetic flux density using a (IS) is

o rcN1Ljm1 One can observe that the winding dc power loss is dominated Bem= JL JL lc +JL rclg by the power loss due to dc component of the choke inductor 7 . 6 current. Since the number of layers the proximity 47r X 10- X 3000 x 10 x 0 050 (39) Nl = 1, 3 3 effect is absent. Thus, neglecting the ac winding loss, the total (28.72 x 10- ) +(3000 x 0.1 x 10- ) = 5.77 mT. power loss in the choke inductor is mW. B. Power Losses in the Choke PLj(sol) = PCg +P Rwde(sol) = 0.04 +20.328 = 20.368 (4S) For the selected core material and the specified operating frequency f = fs = 1000 kHz, the Steinmetz parameters are 7 C. AC Characteristics of Choke Inductor k= 7.36 X 10- , a= 3.47, and b= 2.54. Thus, the total core power loss density for Bem= 5.77 mT using (19) is The high-frequency model of choke inductor is shown in Fig. 4. The turn-to-turn capacitance Ctt formed by the kr Pvg = (Bcm)b conductor turns of a single-layer inductor is expressed as [6] 7 ( 3 3.47 3 2.54 = 7.36 x 10- x 1 X 10 ) x (5.77 X 10- ) mW 2 .lIn D.. wO 2EolT +E.,-· = 0.0389 . Ctt = �======aretan D'w'in. ) • em3 .lIn Dw() t'r D'W'tn -- - (40) (1+ t in g,::'i�'r 1 ( The volume of the core A x Vc = cle = 37.2 28.72 = (49) l.068 em3. Therefore, the total core power loss using (20) is The equivalent capacitance Ct for N turns assuming the turn­ mW to-turn capacitances are in series is given as Ct = i!�1' _ _ em3 _ t": mW. Pc -Pvg Vc -0.0389 em3 x l.068 -0.0410 At high frequencies, the displacement current bypasses the -- (41) inductance and flows through the capacitance. The resistance For the copper winding, the skin depth is estimated using (21) Rwae is the ac series resistance. The impedance of the model as shown in Fig. 4 is s 6 66.2 66.2 0.0662 mm. 1 l+_ _ w= IT= � = (42) Wzi fs 106 Z= (Rwae+.5Lj)II- = Rwac 2' (50) v v .5 Ct 1+ _ s_ +(...L) The dc resistance of the solid round winding is Qowo Wo 4pNlT where the frequency of the zero is Wzi = RZ"", the self­ mn (43) f Rwdc(sol) = 2 = 16.8 resonant frequency is fo = , )L;c;,and the quality factor 7rd i(sol) 27r LfCr 1 �. yielding the conduction winding loss due to the dc current as of the choke is Qo = 21;= H,:", Fig. 5 shows the 2 magnitude of the impedance of the high-frequency model PRwde(sol) = I Rwde(sol) = 20.328 mW. (44) Lj of choke inductor as a function of frequency. Fig. 6 shows The ac resistance of the solid round winding is the phase of the impedance of the high-frequency model of choke inductor as a function of frequency. 4pNlT n. Rwac(sol) = 6 _ 6 = 0.048 (45) w(di w) IV. SIMULATION RESULTS The power loss due to the ac component The components in the Class-E amplifier component were 1 designed using the procedure presented in [1]. The values of = {;n = 0.0615 mW. (46) pwae(sol) -2-Rwac(sol) the circuit components are: choke inductance Lj = 40 JLH The ratio of the winding dc power loss to the winding ac (as calculated earlier), shunt capacitor C1= 5.1 nF, resonant power loss is capacitor C = 4.7 np, resonant inductor L = 6.5 JLH, and load resistor R= 6 n. The circuit was built on SABER circuit PRwdc(sol) = 33l. (47) simulator. Fig. 7 shows the results obtained by transient anal­ pwac(sol) ysis. For the designed circuit, the maximum current hj(max)

5625 Waveform of choke inductor curent. 108 r------.------�------��--� (A) :1(5) 1.2 ,r------�

1.15

106 1.1

105 1.05

104 §: 1.0 N

0.95 J�=====i======oi======!

94u 96u 98u 100u 1(5) 10'

Fig. 7. Waveform of the choke inductor current iLj obtained through SABER. 106 Fourier analysis of choke induclor current. f (Hz) (A) :f(Hz) 1.25 Fig. 5. Impedance magnitude as a function of frequency. ,---==----==----==--==----==---- liLiI

1.0 .... � ...... � ...... ;...... ;......

0.75 .... � ...... � ...... ;...... ;...... 90 r------�----�======>�---, , , . , � , , . , 0.5

0.25 45 0.0

-1 meg 0.0 1 meg 2meg 3meg 4meg 5meg f(Hz)

Fig. 8. Frequency spectrum of the choke inductor current obtained through

-45 SABER showing its dominant dc component.

choke current components have been determined. It has been

- ______L- ______shown that the ac winding loss is significantly lower than the 90 L- � �====� 102 104 106 108 dc winding loss and can be neglected for the design presented f (Hz) in this paper. The dc winding loss is also higher than the Fig. 6. Impedance phase as a function of frequency. core power loss for this design example. Thus, the power loss due to the dc choke inductor current is the dominant loss was recorded as 1.171 A. The average value was measured as component. Simulation results have been presented to show the frequency spectrum of the choke inductor current. The h = hj = 1.0925 A. The ac magnitude was measured as Fourier analysis result has revealed that the dc component of hjm = 0.0618 A. The power loss in the choke inductor was the current through the choke is dominant compared to the measured and is equal to PLj = 0.15 W. From the Fourier analysis result shown in Fig. 8, it was noted that the the dc fundamental and the consecutive harmonics. Therefore, the component of the choke inductor current was dominant in power loss due to the harmonics in the choke inductor current comparison with the fundamental component and consecutive can be neglected. The self-capacitance has been determined harmonic components. Therefore, one may conclude that the and can be included by the shunt capacitor of the amplifier. power loss in the choke is primarily due to the dc current. REFERENCES

V. CONCLUSIONS [1] M. K. Kazimierczuk,Radio-Frequency Power Ampl!fiers, 2nd. Ed.,Wiley, Chichester,UK, 2015. This paper has presented the design procedure for the [2] M. K. Kazimierczuk and H. Sekiya, "Design of ac resonant inductors using area product method," in IEEE Energy Conversion Congress and choke used in Class-E ZVS power amplifiers using area Exhibition, San Jose,CA, Sept. 2009,pp. 994-1001. product method. A gapped core has been selected to avoid [3] M. K. Kazimierczuk, High-Frequency Magnetic Components, 2nd. Ed., core saturation. The power loss has been evaluated through a Wiley,Chichester, UK, 2014. [4] C. P. Steinmetz, "On the law of hysteresis,", Proc. IEEE, vol. 72, pp. modified Steinmetz equation for gapped cores. The core losses 197221,1984. are low because the core is air gapped and the ac component [5] A. Ayachit and M. K. Kazimierczuk, "Steinmetz equation for gapped of the current is also low and results in a limited ac component magnetic cores," IEEE Magnetics Letters, vol. 7,May 2016. [6] S. W. Pasko,M. K. Kazimierczuk, and B. Grzesik, "Self-capacitance of of the flux density. The ac resistance of solid round winding coupled toroidal inductor for EMI filters," IEEE Trans. Electromagnetic have been determined using the effective area available for Compat., vol. 57,no. 2, pp. 216-223,Apr. 2015. current conduction at high operating frequencies. The dc and [7] American Wire Gauge (AWG) Datasheet. https:llwww.micrometals.com. [8] Magnetics Ferrite Cores,2013 Catalog. htttps:llwww.mag-inc.com. ac winding power losses due to the dc and ac fundamental

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