Design Methodology of Bidirectional CLLC Resonant ... - IEEE Xplore

IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 28, NO. 4, APRIL 2013

1741

Design Methodology of Bidirectional CLLC Resonant Converter for High-Frequency Isolation of DC Distribution Systems Jee-Hoon Jung, Member, IEEE, Ho-Sung Kim, Member, IEEE, Myung-Hyo Ryu, and Ju-Won Baek, Member, IEEE

Abstract—A bidirectional full-bridge CLLC resonant converter using a new symmetric LLC-type resonant network is proposed for a low-voltage direct current power distribution system. This converter can operate under high power conversion efficiency because the symmetric LLC resonant network has zero-voltage switching capability for primary power switches and soft commutation capability for output rectifiers. In addition, the proposed topology does not require any snubber circuits to reduce the voltage stress of the switching devices because the switch voltage of the primary and secondary power stage is confined by the input and output voltage, respectively. In addition, the power conversion efficiency of any directions is exactly same as each other. Using digital control schemes, a 5-kW prototype converter designed for a high-frequency galvanic isolation of 380-V dc buses was developed with a commercial digital signal processor. Intelligent digital control algorithms are also proposed to regulate output voltage and to control bidirectional power conversions. Using the prototype converter, experimental results were obtained to verify the performance of the proposed topology and control algorithms. The converter could softly change the power flow directions and its maximum power conversion efficiency was 97.8% during the bidirectional operation. Index Terms—Bidirectional converter, CLLC resonance, dc distribution, high-frequency isolation, soft switching.

I. INTRODUCTION ANY electric power applications such as battery chargers, automobiles, renewable energy sources, uninterrupted power supplies (UPS), and smart grid power systems require bidirectional dc–dc converters (BDCs) to interface between dc voltage buses where energy generation and consumption devices are installed [1]–[4]. Nowadays, plenty of softswitching BDCs focus on eliminating the switching loss, reducing the electromagnetic interference, and achieving an attainable high-frequency operational ability. Therefore, power conversion techniques for high power density without leaking efficiency have been developed and extensively reported [5]–[12]. Several isolated BDC topologies have been suggested for applications of dc power distribution systems [13], [14]. They are smart solutions for galvanic isolation to reduce the weight, size,

M

Manuscript received March 30, 2012; revised July 24, 2012; accepted August 6, 2012. Date of current version October 26, 2012. Recommended for publication by Associate Editor Y.-F. Liu. The authors are with the Korea Electrotechnology Research Institute, Changwon 642-120, Korea (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPEL.2012.2213346

volume, and cost of passive isolators such as line transformers. Isolated BDCs can improve the flexibility of installation and the control of galvanic isolators using high-frequency transformers and active power switches. Especially, the isolated BDCs have been developed for the high-frequency galvanic isolation of a power control interface of dc power distribution systems between dc buses and ac grids [15], [16], and fuel cell and hybrid electric vehicles [17], [18]. For the dc power distribution systems, the isolated BDC can supply clean and stable power, which means high reliability, efficiency, effectiveness, and maneuverability of power systems [19]. A bidirectional phase shift full-bridge converter was proposed with high-frequency galvanic isolation for energy storage systems [20]–[23]. This converter can improve power conversion efficiency using a zero-voltage transition feature; however, it requires input voltage variations to regulate constant output voltage because this topology can only achieve a step-down operation. A boost full-bridge zero voltage switching (ZVS) pulsewidth modulation dc–dc converter was developed for bidirectional high-power applications using passive [17], [24], [25], lossless [26], and flyback [27] snubbers. This topology is proper to the bidirectional power conversion because it has a boost mode for low to high voltage power conversion and a buck mode for high to low voltage conversion. However, this topology requires a snubber circuit to suppress the voltage stress of the switches, which increases circuit complexity and decreases power conversion efficiency. Recently, there has been growing interest in the area of softswitching resonant topologies. Especially, an LLC resonant converter has been proposed and investigated to improve overall power conversion efficiency, since this converter possesses the soft-switching feature as the ZVS for primary power switches and soft commutation for output rectifiers [28]–[34]. If MOSFETs are used as the primary switches, the converter can minimize switching losses because of its ZVS feature. However, the LLC resonant converter can only operate unidirectional power conversion. A bidirectional full-bridge CLLC resonant converter was introduced for a UPS system without any snubber circuits [35]. This topology can operate under soft-switching conditions of primary switches and secondary rectifiers. In addition, the voltage stresses of the power switches are confined to the input and output voltages without any clamp circuits. This converter shows different operations between forward and backward power conversion mode because of the transformer’s asymmetric turn numbers and the asymmetric structure of resonant networks.

0885-8993/$31.00 © 2012 IEEE

1742


Fig. 2.

Bidirectional full-bridge CLLC resonant converter.

A. Gain Analysis Using the FHA Model Fig. 1. 380-V LVDC power distribution system with bidirectional power converter.

In this paper, operational principles and gain characteristics of the bidirectional full-bridge CLLC resonant converter with the symmetric structure of a primary inverting stage and a secondary rectifying stage will be discussed for a 380-V low-voltage direct current (LVDC) power distribution system shown in Fig. 1. The ZVS and soft commutation conditions of primary switches and output rectifiers are analyzed to design the converter’s resonant network. First harmonic approximation (FHA) will be used to analyze the gain curve of the symmetric resonant network with respect to operating frequencies. In addition, gain properties are analyzed to avoid the gain reduction under high-load and high magnetizing inductance conditions. Intelligent digital control algorithms such as dead-band control and switch control will also be proposed to regulate output voltage and to control bidirectional power conversions. The proposed design methodology and digital control algorithms are experimentally verified using a 5-kW prototype converter.

II. ANALYSIS OF BIDIRECTIONAL CLLC RESONANT CONVERTER The proposed bidirectional full-bridge CLLC resonant converter illustrated in Fig. 2 has the full-bridge symmetric structure of the primary inverting stage and the secondary rectifying stage with a symmetric high-frequency transformer. Power switches in the primary inverting stage converts power from dc to ac to transfer it through the transformer. Using this transformer, the converter can achieve the galvanic isolation between the primary side and the secondary side. In Fig. 2, the transformer Tr is modeled with the magnetizing inductance Lm and the transformer’s turn ratio of 1:1. The leakage inductance of the transformer’s primary and secondary windings is merged to the resonant inductors Lr 1 and Lr 2 , respectively. The resonant capacitors Cr 1 and Cr 2 make automatic flux balancing and high resonant frequency with Lr 1 and Lr 2 . In addition, Fig. 2 shows power flow directions in the converter, which are defined as follows: powering mode (forward power flow direction) and generating mode (backward power flow direction).

From Fig. 2, the electrical circuit of the proposed converter can be simplified to obtain the gain of the resonant network with respect to switching frequency. The input voltage of the resonant network, vr i , is a square waveform which varies from −Vin to Vin because of the full-bridge structure. It can be mathematically expressed using the Fourier series as follows: vr i (t) =

∞ 1 4Vin · sin(2πnfs t) π n =1,3,5,... n

(1)

where t is the time parameter, n is the integer parameter for the Fourier series expansion, and fs is the switching frequency of the power switch, respectively. The fundamental component of vr i , vr i,FHA , can be extracted from (1) using the FHA method as shown in the following equation: vr i,FHA (t) =

4 Vin sin(2πfs t). π

The rms value of vr i,FHA , Vr i,FHA , can be calculated as √ 2 2 Vin . Vr i,FHA = π

(2)

(3)

As the same way, the output voltage of the resonant network, vr o , can be expressed as follows: vr o (t) =

∞ 1 4Vo · sin(2πnfs t − φ) π n =1,3,5,··· n

(4)

where φ is the phase shift with respect to the input voltage. Then, the fundamental component of vr o , vr o,FHA , can be extracted from (4) as follows: vr o,FHA (t) =

4 Vo sin(2πfs t − φ). π

The rms value of vr o,FHA , Vr o,FHA , can be calculated as √ 2 2 Vo . Vr o,FHA = π

(5)

(6)

The fundamental component of the rectifier current irct , irct,FHA , can also be expressed using the same manner of (2) and (5) as follows: √ (7) irct,FHA (t) = 2Irct,FHA sin(2πfs t − φ)

JUNG et al.: DESIGN METHODOLOGY OF BIDIRECTIONAL CLLC RESONANT CONVERTER FOR HIGH-FREQUENCY ISOLATION

Fig. 3.

1743

FHA model of the bidirectional CLLC resonant converter.

where Irct,FHA is the rms value of irct,FHA . Therefore, the average output current Io can be derived using (7) as follows: √ Ts 2 2 2 2 Io = Irct,FHA . |irct,FHA (t)|dt = (8) Ts 0 π Since vr o,FHA and irct,FHA are in phase, the resistive load of the resonant network, Ro,e , is equal to the ratio of the instantaneous voltage and current as shown in (9) with the load resistance, Ro Ro,e

Vr o,FHA 8 = = 2 Ro . Irct,FHA π

(9)

Fig. 3 shows the equivalent circuit of the proposed full-bridge bidirectional CLLC resonant converter using the FHA method. The resonant network of the converter is composed of the series resonant capacitor Cr = Cr 1 = Cr 2 , the series resonant inductance Lr = Lr 1 = Lr 2 , and the magnetizing inductance Lm . Since the turn ratio of the transformer is 1:1, it does not affect to the circuit model. The forward transfer function Hr of the resonant network can be derived as follows: Hr (s) =

Zo (s) Ro,e Vo,FHA (s) = · Vi,FHA (s) Zin (s) (sCr )−1 + Ro,e

(10)

where Zin is the input impedance of the resonant network Zin (s) =

1 + sLeq (s) + Zo (s) sCr

(11)

where Leq is the equivalent value of the resonant inductance in the primary side and Zo is the impedance of the output stage as shown in (12) and (13), respectively Leq (s) = Zo (s) =

Lr (12) s2 Lm Lr [Ro (sLm + sLr + Ro )]−1 + 1 Lm Cr−1 + sLm Ro,e . sLm + (sCr )−1 + Ro,e

(13)

From (3), (6), and (10), the dc–dc voltage conversion ratio from input to output can be calculated as Vr o,FHA Vo = = Hr (j2πfs ). Vin Vr i,FHA

Fig. 4. Three-dimensional gain curve according to load current and normalized frequency.

where A(fn ) and B(fn ) are the function components as shown in (16) and (17), respectively A(fn ) = Q2 + k + 1 − B(fn ) = Qfn −

k + (2 + k)Q2 kQ2 + 4 2 fn fn

3Q + 2Qk 2Qk + 3 . fn fn

(16) (17)

In addition, fn can be described using the first resonant frequency fr , Q is the quality factor, and k is the inductance ratio between the magnetizing inductance and the leakage inductance, respectively, given by fs 1 √ , fr = , fr 2π Lr Cr Lr Lr Zr = , k= . Cr Lm fn =

Q=

Zr , Ro,e (18)

Fig. 4 shows a 3-D gain curve according to load current and normalized frequency, which is derived using (15). The gain has a peak value at the low resonant frequency which contains the magnetizing inductance as a resonant component. At the series resonant frequency fr , the gain of the converter is slightly higher than unity gain. In the light-load condition, the gain is high and the slope of the gain curve is sharp. However, the overall value of the gain curve is decreasing under the heavy-load condition. The higher load induces the lower gain in the proposed bidirectional CLLC resonant converter. B. Operating Principles

(14)

Using (10)–(13), the gain of the converter, i.e., Hr , can be reformulated with the normalized frequency fn as follows: −1 Qfn A(fn ) − B(fn ) + j[A(fn ) + Qfn−1 B(fn )] Hr (fn )= −2A(fn )B(fn ) + j[A2 (fn ) − B 2 (fn )] (15)

Fig. 5 shows operational modes of the proposed bidirectional CLLC resonant converter. This converter has six operational modes during a single switching cycle. Mode 1, 2, and 3 repeat to Mode 4, 5, and 6 with different switch and rectifier pairs. Mode 1 and 4 are dead-time durations, Mode 2 and 5 are resonance and power transfer modes, and Mode 3 and 6 are out of resonance, respectively. While power is transferred from the primary side to the secondary side, the primary switches operate in the inverting

1744

Fig. 5.


(a)

(b)

(c)

(d)

(e)

(f)

Operational modes of the bidirectional CLLC resonant converter: (a) Mode 1, (b) Mode 2, (c) Mode 3, (d) Mode 4, (e) Mode 5, and (f) Mode 6.

mode; however, the secondary switches turn OFF and operate as rectifiers in the rectifying mode. Fig. 6 shows theoretical waveforms of the proposed converter for all modes during a single switching cycle. In Fig. 6, there are only the waveforms of the powering mode. However, in the case of the generating mode, the waveforms are exactly same as the waveforms of the powering mode with the stage inverted each other, since the power stage and components of the primary side and the secondary side are symmetric. This symmetric structure can guarantee that the switch control algorithm and the power conversion efficiency of the proposed converter do not change without reference to the direction of the power flow. Detail descriptions and explanations of the operational modes are as follows: Mode 1 [ta − tb ]: This mode is a dead-time duration; there is no power transferred to the secondary rectifying stage. The primary current charges the output capacitance of the primary switches Si3 and Si4 , and discharges the output capacitance of Si1 and Si2 . After the charge and discharge processes, the primary current will pass through the antiparallel diode of Si1 and Si2 , which makes the switches operate under the ZVS condition. Mode 2 [tb − tc ]: Si1 and Si2 turn ON and power will be transferred to the secondary rectifying stage through the transformer Tr . The primary current changes its direction to positive according to Si1 and Si2 because the input voltage source Vin forces the primary current to positive direction through Si1 and

Si2 . During Mode 2, Tr sees the output voltage on the secondary side, then the magnetic energy of the magnetizing inductance Lm is built up linearly. Therefore, Lm does not participate in the resonance of the primary stage. Assuming negligible dead-time duration, the primary current ip can be obtained as ip (t) = ip (tb ) cos ωr (t − tb ) + −

t

Vin − VC r 1 (tb ) sin ωr (t − tb ) Zr

VT r (t − τ ) cos ωr τ dτ.

(19)

tb

Each component used in (19) can be calculated as follows: Ts Δ Vin − ip (tb ) ≈ − (20) Lm + Lr 4 2 Δ = t d − t c = tg − t f ,

ωr = 2πfr

(21)

where VT r is the voltage across the transformer winding and VC r 1 is the voltage across Cr 1 , respectively. In addition, the magnetizing current im can be obtained as follows: t 1 VC r 1 (τ )dτ Vin (t − tb ) − im (t) = im (tb ) + Lm + Lr tb (22) where im (tb ) = ip (tb ) and Ts is the switching period, respectively. Mode 2 will end when ip meets im , which means the end of the resonance operation which transfers power from the


1745

of Mode 2 with the different switch pair of Si3 and Si4 ip (t) = ip (te ) cos ωr (t − te ) − −

t

VT r (t − τ ) cos ωr τ dτ

te

ip (te ) ≈

Vin + VC r 1 (te ) sin ωr (t − te ) Zr

Vin Lm + Lr

Ts Δ − . 4 2

(25) (26)

In addition, the magnetizing current im can be calculated as follows: t 1 im (t) = im (te ) − VC r 1 (τ )dτ Vin (t − te ) + Lm + Lr te (27) where im (te ) = ip (te ). Mode 6 [tf − tg ]: After Mode 5, the resonance and the power transfer are stopped. Without power transfer through Tr , is becomes zero and the antiparallel diodes of So3 and So4 are softly commuted. As the same manner of Mode 3, ip can be calculated as Fig. 6.

Theoretical waveforms of the bidirectional CLLC resonant converter.

primary side to the secondary stage. When the converter operates at fs = fr , Δ becomes zero. Mode 3 [tc − td ]: After ip meets im , the resonance is stopped and power will not be transferred to the secondary side any more. Therefore, the secondary current is becomes zero, and the output capacitor could not be charged by is any more. The primary current ip equals to the magnetizing current im during this mode and the magnetizing energy will be built up until the switches Si1 and Si2 are turned OFF. During this period, since the output stage is separated from the primary side, Lm participates in the resonant operation. It will form the resonant tank to contain Lm in series with Lr 1 and Cr 1 . In Mode 3, ip follows im with large magnetizing inductance enough to ignore its resonance. If not, the primary current can be calculated as follows:

Vin + VC r 1 (tf ) sin ωm (t − tf ). Zm (28) When Modes 3 and 6 appear in the operational waveform, it is evident that the output rectifiers are softly commuted after the power transfer.

ip (t)=ip (tf ) cos ωm (t − tf ) −

III. DESIGN METHODOLOGY OF POWER STAGE

In this section, the design methodology of the power stage of the proposed converter will be discussed. This section is composed of three parts: the design of the magnetizing inductance using soft-switching conditions, the design considerations of the resonant network, and the design of the resonant network with limited gain curves using a linear control for regulating the output voltage. Consequently, the design parameters of the resonant components such as the magnetizing inductance Lm and the resonant inductance Lr will be addressed for the proper operation of the proposed converter. In addition, other imporVin − VC r 1 (tc ) tant design factors such as the inductance ratio k, the quality ip (t) = ip (tc ) cos ωm (t − tc ) + sin ωm (t − tc ) factor Q, and the output current Io in the power stage will be Zm discussed. (23) Lm + Lr 1 ωm =

, Zm = . (24) A. Magnetizing Inductance Using Soft-Switching Conditions Cr (Lm + Lr )Cr The ZVS operation of the primary power MOSFETs and the Mode 4 [td − te ]: This mode is also a dead-time duration with soft commutation of the output rectifiers are significant factors the switch pair of Si3 and Si4 . The operation during this mode for the efficiency-optimal design of the bidirectional full-bridge is similar to Mode 1; however, the charging and discharging CLLC resonant converter. In Fig. 6, the primary current is alcapacitance of the switch pair is changed on the contrary to ways negative at ta , which makes Si1 and Si2 operate under Mode 1. The primary current passing through the antiparallel the ZVS condition. As the same manner, Si3 and Si4 can aldiode of Si3 and Si4 can make those switches turn ON under ways operate under the ZVS condition because of the positive the ZVS condition. direction of the primary current at td . Therefore, the primary Mode 5 [te − tf ]: Si3 and Si4 turn ON and the converter starts current should discharge the output capacitance of four primary to transfer power from the primary side to the secondary side. switches during the dead time for their ZVS turn-on. It means During this mode, ip changes its direction because of the same that the ZVS condition of the switches depends on the magnereason of Mode 2. Actually, Mode 5 shows the same operation tizing inductance Lm and the dead-time duration tdt (tb −ta and

1746


Also, the square rms value of is can be calculated as 2 2 Vo2 1 Ts 5π − 48 2 + 2 . Is,rm s = 16 12π 2 Lm Ro

(36)

Equations (35) and (36) show that the larger magnetizing inductance induces the smaller rms value of the primary and secondary current. It means that the biggest value of the magnetizing inductance satisfied with (31) can reduce the conduction loss and improve the power conversion efficiency of the converter. However, large Lm can make a gain reduction below the unity gain in the proposed bidirectional CLLC resonant converter. It will be discussed in the next section. Fig. 7.

Mechanism of ZVS operation of the primary switches in Mode 1.

te −td ) as 4Vd · max{CS i 1 , CS i 2 , CS i 3 , CS i 4 } = 16CS fs,m ax Lm . min{|ip (ta )|, ip (td )} (29) Fig. 7 shows the ZVS mechanism of the primary switches at Mode 1. In addition, the lower operating frequency than the resonant frequency can guarantee the soft commutation of the output rectifiers because the difference between the switching frequency fs and the resonant frequency fr makes the duration of Modes 3 and 6. Therefore, Lm is limited by the maximum switching frequency fs,m ax as shown in the following equations: tdt ≥

fs ≤ fr Lm ≤

(30)

tdt tdt = . 16CS fs,m ax 16CS fr

(31)

Inequality (31) shows that small Lm can guarantee the ZVS of the primary switches. However, small Lm also increases the conduction loss of the MOSFETs, the transformer windings, and the output rectifiers in the primary and secondary side. Assuming that the converter operates under a four-mode case except Modes 3 and 6, and tdt and Vf are negligible, (19) can be reformed as √ (32) ip (t) = 2Ip,rm s sin (ωr t + φ) where Ip,rm s is the rms value of the primary current ip and V T √ o s φ = sin−1 . (33) 4 2Lm Ip,rm s In addition, the average output current Io can be calculated as √ Vo 2 2 T s /2 Io = = Ip,rm s sin (ωr t + φ)dt. (34) Ro Ts 0 From (33) and (34), the square rms value of ip can be obtained as 2 Ip,rm s

V2 = o 8

Ts 2Lm

2 +

π Ro

2 .

(35)

B. Design Considerations of Resonant Network Design of the resonant components such as Lr , Lm , and Cr should consider maximum load changes, maximum acceptable operating frequency excursion, maximum input voltage range, circulating energy in the resonant network, and short-circuit characteristic. The quality factor Q and the inductance ratio k of the converter are selected by considering input voltage range, maximum load, and gain curve of the converter. In the selection of the component values of the resonant network, following concepts are important to power conversion efficiency and proper operating range to regulate the output voltage. The quality factor Q will significantly affect gain characteristic and available operating range. If Q is high, the gain variation will be small and the operating frequency range can be wide. If Q is too high, however, the resonant impedance will be high and the gain can be decreased lower than the unity gain. It means that the output voltage cannot be regulated because of the insufficient voltage gain. On the other hand, low Q requires low Zr , which makes the converter operate under poor efficiency because of high circulation current. The inductance ratio k will significantly influence the shape of gain characteristics and ZVS region borders. In addition, k is strongly related to the value of Lm . If k is low, high Lm can be possible considering the fixed fr , which reduces the conduction loss of the converter. However, this low k can make the gain be lower than the unity gain within the operating frequency range. In practice, the conventional LLC resonant converter is usually designed under a fixed fr for a rated load and input voltage range. When the load increases or the input voltage decreases, the operating frequency is decreased by a feedback loop to keep the output voltage regulated to a reference voltage. The gain of the proposed bidirectional CLLC resonant converter, however, can be decreased according to decreasing the operating frequency under high-load condition. Since the resonant capacitor Cr 2 is added to the resonant network in the secondary side, the resonance in the resonant network is more complicated than the resonance of the conventional converter’s resonant network, and one more resonant peak is added to the gain curve with respect to the switching frequency. Therefore, the resonant components of Lm , Lr , and Cr should be selected considering proper fr for the operating frequency, the ZVS condition of the switches derived in (31), and proper Q and k for the available gain curve higher than the unity gain.


1747

In particular, in the case of the proposed bidirectional CLLC resonant converter, the gain reduction lower than the unity gain should be avoided using the proper resonant network. The resonant network in the proposed bidirectional CLLC converter should be designed to keep the gain of the converter higher than the unity gain with respect to entire load range. There is one more thing to be considered in the view point of the converter’s gain: limited gain curve for linear control. It will be discussed in the next section. C. Limited Gain Curve for Linear Control The gain curve of (15) has to monotonically decrease according to the normalized frequency fn within the operating frequency range of the bidirectional CLLC resonant converter because the output voltage is regulated employing a linear feedback controller. Using the polarity of the gain curve’s derivative with respect to fn , the gain can be decided to whether it is a monotonically decreasing function or not. Equation (15) can be simplified as follows: fn3 Hr (fn ) =

fn2 C 2 (fn ) + Q2 D2 (fn )

(a)

(37)

where C(fn ) and D(fn ) are the function components as shown in (38) and (39), respectively C(fn ) = −fn2 (1 + k) + k

(38)

D(fn ) = fn4 − fn2 (2 + k) + k.

(39)

The derivative of (37) can be derived as follows: d 2kf 4 C(fn ) − Q2 fn2 D(fn )E(fn ) Hr (fn ) = n 1.5 dfn fn2 C 2 (fn ) + Q2 D2 (fn )

(40) (b)

where E(fn ) is the function component as shown in the following equation: E(fn ) = fn4 + fn2 (2 + k) − 3k.

(41)

The derivative of the gain curve must be negative if the gain is a monotonically decreasing function within the operating frequency range. The polarity of (40) can be decided by only the value of its numerator because the denominator of (40) is always positive. According to the quality factor Q, the maximum value of Q can be derived using the numerator of (40) as

2kfn2 ,m ax C(fn ,m ax ) , 0 < k < 1. (42) Q< D(fn ,m ax )E(fn ,m ax ) The inequality (42) is the design condition of the load impedance under specific values of the resonant impedance and the maximum switching frequency. From (42), the maximum load current Io,m ax can be calculated as follows: 8Vo Qm ax (43) π 2 Zr where Qm ax is the maximum value of the quality factor under the specific resonant impedance as shown in the following equation:

2kfn2 ,m ax C(fn ,m ax ) . (44) Qm ax = D(fn ,m ax )E(fn ,m ax ) Io,m ax