Design of RF-Choke Using Core Geometry Coefficient

†‡ † Hiroo Sekiya and Marian K. Kazimierczuk †Department of Electrical Engineering, Wright State University, OH ‡Graduate School of Advanced Integration Science, Chiba University, JAPAN

Abstract—This paper presents a derivation of the core core satisfying both the electromagnetic conditions and the K geometry coefficient g for the design of RF-choke inductors restriction of the core window area. The value is and a design example. The design example indicates that the dc winding loss is the dominant component of the total power adjusted by the air-gap length. The Ap method is widely used loss of RF-choke inductors. Additionally, the advantages and for designing the inductors and transformers for dc-dc power K disadvantages of the g method are discussed by using the design converters operating in CCM and DCM. On the other hand, example results. Index Terms— design, RF-choke inductor, core the concept of the Kg approach is to select a proper core geometry coefficient, dc winding loss, ac winding loss, core loss, satisfying the electromagnetic conditions, the restriction of air-gap length, fringing effect, skin effect, Dowell’s equation. the core window area, and the restriction of the winding loss, simultaneously. This method is useful to design inductors and INTRODUCTION transformers with low core and low ac winding losses. By The magnetic component design is a traditional problem, setting the dc winding loss, we can choose the core, which but it is not solved satisfactorily in the power requires shorter length and narrower cross-sectional area of field. For example, the RF-power amplifier, which is one the wire compared to that obtained with the Ap method. This of the important circuits in many systems, has two major leads to a small core volume. magnetic components: a resonant inductor and an RF-choke An RF-choke inductor forces a dc current, which is inductor. In particular, most of the RF-power amplifiers, applicable to dc-current sources and output filters of resonant which are the class A, class B, class AB, class C, class E, converters. Since the current through the RF-choke inductor and class F amplifiers, have an RF-choke inductor [1]. The includes a low ac component, the ac-winding loss and the improvements of some kinds of switching techniques, e.g., core loss are very small, even if the operating frequency is zero-voltage switching (ZVS), zero-current switching (ZCS), high. Therefore, it is possible to focus on the dc-winding and class-E ZVS and zero derivative switching (ZDS), reduce loss when the RF-choke inductor is designed. Additionally, dramatically power losses in switching devices. Therefore, the inductance value of the RF choke inductor is usually high the importance of the design of magnetic components to force the dc current. Therefore, the volume reduction is with low power losses increases for the design of power quite important to make an RF-choke inductor. For the reasons amplifiers and dc-dc converters. Additionally, the reduction described above, we consider that the Kg method is suitable of the magnetic-component volume is also an important to design the RF-choke inductor. aspect because the magnetic components are bottlenecked to This paper presents a derivation of the core geometry determine the circuit size. Generally, the power loss in a coefficients Kg for the design of the RF-choke inductors and magnetic component decreases when the size of the inductor a design example. The design example indicates that the dc increases though both the low power loss and small volume winding loss is dominant in the power losses of the RF-choke are required. This means that there is a trade-off relationship inductor. Additionally, the advantages and disadvantages of the between the power loss and the magnetic component size. Kg method are discussed by using the design example results. Therefore, the design of magnetic components, which are I. BASIC THEORY AND DERIVATION OF CORE GEOMETRY inductors and transformers, is one of the important challenging COEFFICIENT K problems. g There are several well-known strategies for selecting a core A. Expression for Current Through RF-Choke Inductor for the design of magnetic components, for example, the area The purpose of a choke is to supply a dc current. In real product (Ap) method and the core geometry (Kg) method circuits, however, there are a little on the dc current. [2]–[5]. These methods are primarily used in the design of Therefore, we assume that the current through the inductor inductors and transformers for switching-mode power supplies iL is the sum of a Idc and a pure sinusoidal (SMPS). The concept of the Ap approach is to select a proper current with the fundamental frequency equal to the operating frequency f, Designed Inductor L L C = + Ipp sin( )= +( − )sin( ) C Ic 0 0 iL Idc 2 ωt Idc Im Idc ωt , (1) where I is peak-to-peak value of the ac component of the pp vs ics io v current, Im is a maximum value of iL and ω =2πf is the S VDD C angular frequency. In this case, the peak-to peak ripple ratio Dr s RvO to the dc current through the inductor is defined as

ΔiL Ipp 2(Im − Idc) γr ≡ = = . (2) Idc Idc Idc

B. Core Geometry Coefficient Kg for RF-Choke Inductors Fig. 1. Circuit topology of the class E amplifier. The inductance of an inductor with an air gap is given by 2 2 N μ0N A L = = c , (3) The core window utilization factor is defined as the ratio of lg lc lc + lg + the total cross-sectional area of the winding bare wire ACu to μ0Ac μrμ0Ac μr the window cross-sectional area of a core Wa where N is the number of wire turns, lg is the are-gap ACu NAw length, lc is the core length, Ac is the cross-sectional area Ku = = . (12) −7 Wa Wa of the , μ0 =4π × 10 H/m is the free-space permeability, and μr is the relative permeability of the core From (11) and (12), we obtain material. 2 KuWaPwdc In general, it can be written that N = 2 (13)   ρwlT Idc l l Ni = B g + c , (4) μ0 μ0μr and K W ρ l I2 2 = u a w T dc where B is the flux density. Hence, the maximum flux density Aw . (14) Pwdc Bm is μ0NI μ0NI (1 + γ /2) B = m = dc r . (5) These relations include both the dc winding loss condition m l l c + l c + l in (11) and core-area condition in (12). The number of μ g μ g r r turns should also satisfy the electromagnetic condition in (6). From (3) and (5), the number of turns is Equating of the right-hand sides of (6) and (13), the core LI LI (1 + γ/2) geometry coefficient [2] is obtained as N = m = dc , (6) AcBm AcBm 2 WaAcKu which is obtained from electromagnetic point of view. The Kg = lT length of the winding wire is 2 2 2 2 4 (1 + 2)2 (15) = ρwL ImIdc = ρwL Idc γ/ ( 5) = 2 2 m , lw NlT , (7) PwdcBm PwdcBm where lT is the mean length of a single turn (MLT). The dc It is convenient to express the dc winding loss as the ratio of winding resistance is the output power Pwdc = αPo, resulting = ρwlw = ρwNlT 2 2 2 2 4 (1 + 2)2 Rwdc , (8) = ρwL ImIdc = ρwLIdc γ/ Aw Aw Kg 2 2 . (16) αPoBm αPoBm where Aw and ρw are the cross-sectional area of the winding bare wire and the resistivity of the copper, respectively. The core geometry coefficient Kg provides the core Therefore, with a good combination of Wa, Ac,andlT satisfying NρwlT electromagnetic condition in (9), dc-winding-loss condition Aw = . (9) Rwdc in (11), and core-area restriction in (12), simultaneously. The The dc winding loss is concrete values of Kg are presented for Ku =0.4 in [3], 2 or can be calculated from the core dimensions provided by Pwdc = RwdcI . (10) dc the core manufacuturers. By using the proposed expressions Therefore, (9) is rewritten as for Kg in (15) and (16), we can select the core satisfying 2 the conditions (9), (11), and (12) using only the electrical NρwlT Idc Aw = . (11) Pwdc parameters. II. DESIGN EXAMPLE 0.879 mm. The maximum current density of the wire is

A. Design Specifications and Conditions Idc(1 + γr/2) Jm = This section presents a design example of an RF-choke Aw 0.807 × (1 + 0.01/2) inductor for the class-E resonant power amplifier [6] whose = (19) 0.518 × 10−6 topology is shown in Fig. 1. The specifications of the class-E =1.56 A/m2 < 5. amplifier are: operating frequency f =1MHz, load resistance =10Ω =5 RL , loaded quality factor QL , and supply voltage It is confirmed that the wire satisfies the =15 VDD V. By using the design procedure in [6], the maximum-current-density condition. In the inductor inductance value of the RF-choke to achieve the ripple ratio design using the core geometry coefficient, the =1 =113 γr %isLc . mH. In this case, the dc-current and maximum-current-density condition is not guranteed. =0807 the maximum current through the inductor is Idc . A Therefore, we need to confirm this condition. From (12), the = (1 + 2) and Im Idc γr/ = 0.811 A. The output power is number of turns is Po =11.8 W. K W 0.4 × 0.6 × 10−4 The RF-choke inductor is designed to achieve the following N = u a = =46.2. (20) A 0.518 × 10−6 specifications: 1) maximum current density of the wire is w 2 Jm < 5 A/m , 2) core window utilization factor is Ku = 0.4, We pick N =46. 3) maximum flux density is less than the saturated flux density, Bm

P P P P Core Volume t wdc wac c (38) =50.2+0.04 + 0.203 = 50.4 mW

(mH) 10 Inductance Value Lc 100

c Core Volume Vc

The equivalent series resistance (ESR) for the inductor is L expressed as

Pt 1 10 V Resr = c I2 (cm dc (39) 50 4 × 10−3 . 3 = (Ω) = 62.5 mΩ. ) 0.8072 0.1 1

F. Confirmation of The Specified Conditions of Inductance Value From (6), the maximum flux density is calculated as (1 + 2) 0.01 0.1 = LIdc γr/ 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Bm γr NAc Ripple Ratio −3 1.33 × 10 × 0.807 × (1 + 0.01/2) (40) (a) = 250 46 × 0.58 × 10−4 =0.404 T, Total loss Pt Dc winding loss Pwdc 200 Ac winding loss Pwac which is less than the saturated flux density Bsat =0.5 T, Core loss Pc but 33 % larger than the specified value of Bm =0.3 T. This difference occurs due to the normalization of the air-gap length 150 and the difference in the Kg value of the selected core. From this result, it can be stated that we should give the value of 100 Bm with a sufficient margin from Bsat. Power Loss (W) Power The ratio of the dc winding loss to output power is 50 P α = t P o −3 (41) 62.5 × 10 0 = =0.423 %, 11.9 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Ripple Ratio γr which is 15 % lower than the required value because of the (b) difference of the cross-sectional area of the wire and the Kg L value. Fig. 2. Characteristics of the designed RF-choke inductor c as a function of the ripple ratio of the current γr for α = 0.01, Idc = 0.807 A, and f = The core window utilization factor is 1 MHz. (a) Inductance value Lc and core volume Vc of the designed inductor. (b) Dc winding losses P , ac winding loss Pwac, core loss Pc, and total NAw wdc K = power loss Pt of the designed inductor. u W a (42) 46 × 0.518 × 10−6 = =0.397, 0.6 × 10−4 wire l . However, the increase of the ac component of the which is almost the same as the required value. This is because w current make the large I and B . Therefore, it is difficult we fix the number of turn N for adjusting the fringng effect. m pp to predict what happens in the inductor when the ripple ratio of Fundamentally, the design method using core geometry the current varies. In this section, we investigate this problem. coefficient gurantees inductance value, maximum flux density, The all designed inductor in this section consists of Magnetics and ratio of the dc winding loss to output power as shown in from References [3] and [7] and AWG standardized Section II. However some differences apperar due to air-gap wire. length normalization, difference of selected wire size, and Figure 2 shows the characteristics of the RF-choke inductor difference in the Kg value of the selected core. of the class E amplifier designed by Kg method as a function III. DISCUSSION OF RESULTS γr. For the design of the class E amplifier, the specifications, The inductance value of the RF-choke inductor Lc of the which are f =1MHz, VDD =15V, a n d R =10Ω, are given. 2 class E switching circuit becomes small when the ripple ratio Additionally, α = 0.01, Bm=0.3T,andJmax =5A/mm are of the current through it becomes large. Small Lc value also given as specifications of the inductor design. Figure 2(a) provides the small core volume Vc and a short length of the shows the characteristics of the inductance value Lc and the 6 core volume Vc. As ripple ratio γr decreases, the inductance value increases. Because the high-inductance value prevent ac current from flowing through the inductor, high inductance 5 ) value is necessary to generate the current with low ripple 3 ratio. Therefore, the core volume Vc becomes large with the 4 (cm decrease in the ripple ratio γr. The core volume is constant c 0 075 V when γr > . . This is because this core volume is the 3 smallest for all prepared cores. It is seen from Fig. 2(b) that the dc winding loss and core loss are almost constant regardless 2 of γr. The concept of Kg method is to achieve the required dc winding loss. The output power is constant against the ripple Core Volume ratio variation. Therefore, the dc winding loss is also constant 1 for all the ripple ratio. The designer can achieve the required maximum flux density Bm by using Kg method. Therefore, 0 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 the core loss is also constant regardless of γ , which is small r Ratio of Output Power to Dc Winding Loss α enough to neglect. When the ripple ratio increases, the ac (a) component of the current through the inductor also increases. 800 Therefore, ac winding loss increases with the increase in the 700 ripple ratio. For large γr, the ac winding loss is almost same as Total loss Pt Dc winding loss Pwdc the dc winding loss. This means that it is important to consider 600 Ac winding loss Pwac Core loss Pc the ac winding loss when the ripple ratio γr is high. For γr > 0.32, we cannot design the inductor by using Kg 500 method. In this range, the maximum current density is higher 400 than Jm. This means that the cross-sectional area of the wire is too narrow. For avoiding this problem, we need to set a 300 smaller value of α because a small power loss requires a wide

Power Loss (mW) Power 200 cross-sectional area of the wire. This process, however, is a nonsense. In this case, we should not use Kg method. This is 100 because it is guaranteed that the dc winding loss is lower than the required power loss, and we do not need to consider dc 0 winding loss. Namely, the Ap method should be used in this 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 situation. Ratio of Output Power to Dc Winding Loss α Figure 3 shows the characteristics of the RF-choke inductor (b) of the class E amplifier designed by K method as a function g Fig. 3. Characteristics of the designed RF-choke inductor L as a function =133 =0807 c of α. In this case, Lc . mH, Idc . Aand of the ratio of the output power to the dc winding loss α for γr =0.01, Idc = 0.807 A, and f = 1 MHz. (a) Core volume Vc of the designed inductor. γr =0.01 are fixed for any α. When the cross sectional (b) Dc winding losses Pwdc, ac winding loss Pwac, core loss Pc, and total area of the wire is narrow, the dc winding loss becomes high. power loss Pt of the designed inductor. Therefore, the increase of α tends to the narrow cross-sectional area of the wire. Therefore, the core volume Vc becomes small as the α increases as shown in Fig. 3(a). Since the large Figure 4 shows the characteristics of the RF-choke inductor α allows the large dc-winding loss, the dc winding loss is of the class E amplifier designed by Kg method as a function proportional to the α as shown in Fig. 3(b). Conversely, the of the Idc. When we vary the dc-supply voltage VDD of the variations of the ac winding and the core losses are very small class E amplifier, the output power Po becomes high. The compared with the variation of the dc winding loss because element values, however, do not vary when the dc-supply the Im and Bm are fixed when the α is changed. Strictly, the voltage VDD varies. Therefore, Lc =1.33 mH, γr =0.01,and core loss becomes small with the increase in the α since the α =0.01 are fixed in Fig. 4. The maximum amplitude of the core volume Vc becomes small. It is seen from Fig. 3(b), both current Im is high when the dc-current is high. For avoiding the ac winding loss and the core loss are negligible for all α. the maximum flux density Bm is larger than the saturated For α>0.078, we cannot the design the inductor by using magnetic density Bsat, the core volume should be large with Kg method. Because the cross-sectional area of the wire is the increase in Idc as shown in Fig. 4(a). The output power too narrow, the current density problem occurs in this range. increases with the increase in Idc, namely, VDD. Because of 50 10 100

Lc Core Volume

) 40 Inductance Value

3 1 Core Volume Vc (mH)

c 10 L (cm c 30 V

0.1 V c (cm 20 1 3 ) 0.01 Core Volume 10 of Inductance Value

0.001 0.1 0 0.1 1 10 100 1000 0246810 Operating Frequency (MHz) Dc-current Idc (A) (a) (a)

14

12 100

Total loss Pt Total loss Pt 10 Dc winding loss Pwdc Dc winding loss Pwdc Pwac Ac winding loss Pwac Ac winding loss P Core loss Pc Core loss c 8 50

6 Loss (mW) Power Power Loss (W) Power 4 0 2 0.1 1 10 100 1000 Operating Frequency (MHz) 0 (b) 0246810 L Idc Fig. 5. Designed inductor characteristics of RF-choke inductor c as a Dc-current (A) f γ =0.01 α I dc] (b) function of the operating frequency for r , = 0.01, and [ = 0.807 A. (a) Inductance value Lc and core volume Vc of the designed inductor. (b) Dc winding losses Pwdc, ac winding loss Pwac, core loss Pc, L Fig. 4. Designed inductor characteristics of RF-choke inductor c as a and total power loss Pt of the designed inductor. function of the magnitude of dc current Idc for γr =0.01, α = 0.01, and f = 1 MHz. (a) Core volume Vc of the designed inductor. (b) Dc winding losses Pwdc, ac winding loss Pwac, core loss Pc, and total power loss Pt of the designed inductor. of the inductors in this figure. A higher frequency makes the inductors be small. Therefore, the Lc decreases as increase of f as shown in Fig. 4(a). The core volume Vc also decreases. =001 = the fixed α . , the dc winding loss, which is Pwdc αPo, The designers should concern the core loss at high-frequency is also increases with the increase in Idc. The dc winding loss operations. The core loss for RF-choke inductor, however, is the dominant factor of the total power loss Pt as shown generates low core loss as shown in Fig. 4(b). The ac-winding in Fig. 4(b), and the ac winding loss and the core loss are loss is also problem for high-frequency current because of the negligible. This figure shows the effectiveness to use the core skin and proximity effects. It is seen from Fig. 4(c) that the geometry coefficient for the design of RF-choke inductors. ac-winding loss is also generated in RF-choke inductor. This is Figure 4 shows the characteristics of the RF-choke inductor because the current has few ac component because of small γr. of the class E amplifier designed by Kg method as a function As a result, the dc-winding loss Pwdc is the dominant factor of the operating frequency f. Dc-current through the inductor of the total loss Pt regardless of f. This result shows we can Idc, γr =0.01,andα =0.01 are fixed for all the designs use ferrite core in spite of the high-frequency operations when the RF-choke with low ripple-ratio current is designed. Technical University of Warsaw, Poland. In 1984, he was a Project Engineer for Design Automation, Inc., Lexington, IV. CONCLUSION MA. In 1984-85, he was a Visiting Professor with the This paper has presented the fundamental theory to derive Department of Electrical Engineering, Virginia Polytechnic the Kg factor for the design of RF-choke inductor and design Institute and State University, VA. Since 1985, he has been example. The design example indicates that the dc winding with the Department of Electrical Engineering, Wright State loss is dominant in the power losses of the RF-choke inductor. University, Dayton, OH, where he is currently a Professor. Fundamentally, it can be stated that the Kg method is useful His research interests are in high-frequency high-efficiency for the design of RF-choke inductors because the required dc switching-mode tuned power amplifiers, resonant and PWM winding loss can be achieved. In some cases, however, the dc/dc power converters, dc/ac inverters, high-frequency current density is a bottleneck for the design of the inductor. rectifiers, electronic ballasts, ,modeling and control of In the cases, the Ap method is better than the Kg method. This converters, high-frequency magnetics, power semiconductor is because the coefficient of the Ap method takes into account devices. E-mail: [email protected] the maximum current density instead of the dc winding loss.

REFERENCES [1] M. K. Kazimierczuk, RF Power Amplifiers, Chichester, UK: A John Wiley and Sons, 2008 [2]C.W.T.McLyman,Transformer and Inductor Design Handbook, 3rd Ed., New York, NY: Marcel Dekker, 2004. [3]C.W.T.McLyman,Magnetic Core Selection for Transformer and Inductors, 2nd Ed., New York, NY: Marcel Dekker, 1997. [4] M. K. Kazimierczuk and H. Sekiya, “Design of ac resonant inductors using area product method,” IEEE Energy Conversion Congress and Expo., San Jose, CA, Sep. 2009. [5] C. W. T. McLyman, “Designing a continuous current buck-boost converter using the core geometry,” Proceedings of Electrical Insulation Conference and Electrical Manufacturing Expo., Indianapolis, IN, Oct. 2005, pp. 329 – 336. [6] H. Sekiya, I. Sasase and S. Mori, “Computation of design values for class E amplifier without using waveform equations,” IEEE Trans. Circuits and Systems I, vol. 49, no.7, pp. 966–978, July, 2002 [7] Magnetics Inc., Ferrite Cores, 2006.

Hiroo Sekiya was born in Tokyo, Japan, on July 5, 1973. He received the B.E., M.E., and Ph. D. degrees in electrical engineering from Keio University, Yokohama, Japan, in 1996, 1998, and 2001 respectively. Since April 2001, he has been with Chiba University and now he is an Assistant Professor at Graduate School of Advanced Integration Science, Chiba University, Chiba, Japan. Since Feb. 2008, he has been also with Electrical Engineering, Wright State University, Ohio, USA as a visitng scholar. His research interests include high-frequency high-efficiency tuned power amplifiers, resonant dc/dc power converters, dc/ac inverters, and digital signal processing for wireless communication. E-mail: [email protected]

Marian K. Kazimierczuk received the M.S., and Ph.D., and D.Sci. degrees in electronics engineering from the Department of Electronics, Technical University of Warsaw, Warsaw, Poland, in 1971, and 1978, and 1984, respectively. He was a Teaching and Research Assistant from 1972 to 1978 and Assistant Professor from 1978 to 1984 with the Department of Electronics, Institute of Radio Electronics,