Power Conversion Efficiency of Class-E Power ... - Semantic Scholar
Power Conversion Efficiency of Class-E Power Amplifier Outside Nominal Operation Tomoharu Nagashima1, Xiuqin Wei1 , Hiroo Sekiya1 , and Marian K. Kazimierczuk2 1
2
Graduate School of Advanced Integration Science, Chiba University, Chiba, 263–8522 Japan Department of Electrical Engineering, Wright State University, Dayton, OH, 45435–0001 USA Email: [email protected] [email protected]
Abstract—This paper gives analytical expressions for the output power and the power conversion efficiency of the class-E power amplifier outside the class-E ZVS/ZDS switching conditions. The analytical predictions agreed with the experimental results quantitatively, which indicates the validity of the analytical expressions. Moreover, a design example applying the analytical expressions for the power conversion efficiency is given. By using the analytical expressions, it is possible to design the class-E amplifier, which improves the power conversion efficiency compared with that of the class-E amplifier satisfying the class-E ZVS/ZDS conditions. Index Terms—Class-E power amplifier, power conversion efficiency, output power, nominal operation.
VDD
IDD LC C0 L0 is iCs io vs CS vo S
R
VDD
IDD LC C0 L0 is iCs io v s CS vo S
(a)
R
(b)
Fig. 1. (a) Circuit topology of class-E amplifier. (b) Equivalent circuit of class-E amplifier.
I. Introduction The class-E power amplifier [1]-[7] is expected to be useful for many applications, e.g., radio transmitters, switching-mode dc power supplies, devices for medical applications, and so on. Because of the class-E zero-voltage switching and zero-derivative switching (ZVS/ZDS) conditions, the class-E amplifier can achieve high power conversion efficiency at high-frequencies. Therefore, most papers on the class-E amplifier discuss the class-E amplifier under the class-E ZVS/ZDS conditions, which is called as “nominal conditions”. Reference [4] gives the analytical expressions for the waveforms of the class-E amplifier outside the class-E ZVS/ZDS conditions. The analytical expressions of the output power and the power conversion efficiency outside the class-E ZVS/ZDS conditions, however, have never been derived. Generally, it is known that the conditions for obtaining the maximum power conversion efficiency, which is called as “optimal conditions”, is not the same but approximately the same as nominal conditions. There is, however, no design approach to achieve the optimal conditions of the class-E amplifier except [5]. For determining the element values of the class-E power amplifier with maximum power conversion efficiency, it is important to have the analytical expressions of the power conversion efficiency outside the nominal conditions. This paper presents analytical expressions for the output power and the power conversion efficiency of the class-E amplifier outside the class-E ZVS/ZDS conditions. The analytical predictions agreed with the experimental results quantitatively, which shows the validity of the analytical expressions. Moreover, a design example applying the analytical expressions for the power conversion efficiency is given.
√ 1) A = f0 / f = ω0 /ω = 1/ω L0C0 : The ratio of the operating frequency to the resonant frequency. 2) B = C0 /CS : The ratio of the resonant capacitance to the shunt capacitance. 3) D: The switch-off duty ratio of the switch S . 4) Q = ωL0 /R: The loaded Q-factor. The circuit analysis for obtaining the waveform equations are based on the following assumptions: 1) The MOSFET acts as an ideal switch device, namely, it has zero on-resistance, infinite off-resistance and zero switching times. 2) The dc-feed inductance LC is large enough so that the current through the dc-feed inductor is constant. 3) The loaded Q-factor is high enough to generate a sinusoidal output current io . 4) All the passive elements are linear and have zero equivalent series resistances (ESRs). 5) The circuit operations are considered in the interval 0 ≤ θ ≤ 2π. The switch is in the off-state for 0 ≤ θ < 2πD and in the on-state for 2πD ≤ θ < 2π, where θ = wt represents angular time. By the above assumptions, the equivalent circuit is obtained as shown in Fig. 1(b). B. Waveform Equations The output current is io = Im sin(θ + φ).
II. Waveform Equations Outside Class-E ZVS/ZDS Conditions The analytical expressions for the output power and the power conversion efficiency of the class-E amplifier outside the class-E ZVS/ZDS conditions are derived by using the waveform equations given in [4]. In this section, we introduce only the results, which are obtained in [4]. A. Assumptions and Parameters Figure 1(a) shows a circuit topology of the class-E amplifier. For obtaining the analytical expressions, we define the following parameters:
978-1-4244-9472-9/11/$26.00 ©2011 IEEE
(1)
In (1), Im is the amplitude of the output current Im =
2πVDD β , A2 BQR(2π2 D2 α + γβ)
(2)
where VDD is the dc-supply voltage, 1 1 α= cos 4πD − cos 2πD + cos2 φ 2 2 1 π 1 1 − sin 4πD − sin 2πD sin φ cos φ + 2 − cos 4πD + , 2 A BQ 4 4 (3)
749
VDD
Fig. 2.
IDD LC rLc C0 L0 rL0C0
rS , ESRs of dc-feed inductance rLC , shunt capacitance rCS , and load network rL0 C0 as shown in Fig. 2.
iCs io is vo R S vs CS rCs rS
The output power Po is 2π RI 2 R Po = i2o dθ = m . 2π 0 2
Equivalent circuit of class-E amplifier including ESRs.
β = (−2πD cos 2πD + sin 2πD) cos φ +(2πD sin 2πD + cos 2πD − 1) sin φ,
(4)
and γ = (sin 2πD − 2πD) cos φ + (cos 2πD − 1) sin φ.
(5)
Moreover, φ, which is the phase-shift between the input voltage and the output current, is expressed as 1 π 3 φ = − arctan cos 4πD − cos 2πD + 2 + 4 A BQ 4 (2πD sin 2πD + cos 2πD − 1) 1 π(1 − A2 ) + sin 2πD − sin 4πD + − πD 4 A2 B (−2πD cos 2πD + sin 2πD) 1 π(1 − A2 ) + πD − sin 4πD − 4 A2 B (2πD sin 2πD + cos 2πD − 1) π 1 1 + 2 + − cos 4πD (πD sin 2πD − sin 2πD) . A BQR 4 4 (6) The switch voltage is given by ⎧ ⎪ ⎪ ⎨ A2 BQR[IDDθ + Im cos(θ + φ) − Im cos φ], vS = ⎪ ⎪ ⎩0,
for 0 ≤ θ < 2πD for 2πD ≤ θ < 2π, (7)
where IDD , which is the dc-supply current, is given by IDD =
2πVDD α . A2 BQR(2π2 D2 α + γβ)
The current through the shunt capacitance is ⎧ ⎪ ⎪ ⎨IDD − Im sin(θ + φ), for 0 ≤ θ < 2πD iCS = ⎪ ⎪ ⎩0, for 2πD ≤ θ < 2π. The current through the switch is ⎧ ⎪ ⎪ for 0 ≤ θ < 2πD ⎨0, iS = ⎪ ⎪ ⎩IDD − Im sin(θ + φ), for 2πD ≤ θ < 2π.
(8)
(9)
(10)
III. Output Power and Power Conversion Efficiency A. Derivation of Analytical Equations In this section, analytical expressions for the output power and the power conversion efficiency are derived by using the waveform equations given in Section II. In real circuits, the power losses occur in the parasitic resistances of each component. It is assumed that the parasitic resistances are small enough not to affect the waveforms [8]. In this paper, we consider the power losses in switch on-resistance
(11)
The conduction loss in the MOSFET on-resistance rS is 2π rS PS = i2 dθ 2π 0 S I2 IDD Im 2 = rS (1 − D)IDD + m (1 − D) + [cos φ − cos(2πD + φ)] 2 π
I2 + m [sin(4πD + 2φ) − sin 2φ] . (12) 8π The power loss in rLC is 2 PLC = rLC IDD .
(13)
The power loss in rCS is obtained as 2π rC PCS = S i2 dθ 2π 0 CS I2 IDD Im 2 cos(2πD + φ) − cos φ = rCS DIDD + mD+ 2 π
I2 + m sin 2φ − sin(4πD + 2φ) . (14) 8π The power loss in rL0 C0 is 2π rL C rL C I 2 i2o dθ = 0 0 m . PL0 C0 = 0 0 2π 0 2
(15)
The analysis in this paper does not assume that the class-E ZVS/ZDS conditions are satisfied. We can classify the switching types of the class-E amplifier into four cases as shown in Fig. 3. The first case is when the switch voltage waveform does not reach zero prior to turn-on switching as shown in Fig. 3(a). In this case, the turn-on switching loss occurs at turn-on instant, which is expressed as 1 Pto = CS f v2S (2πD− ) 2 A2 BQR [(2πDIDD )2 + 4πDIm IDD cos(2πD + φ) = 4π −4πDIDD Im cos φ + Im2 cos2 (2πD + φ) −2Im2 cos(2πD + φ) cos φ + Im2 cos2 φ]. (16) where vS (2πD− ) is the switch voltage just prior to turn-on. The second case is when the switch voltage waveform just reaches zero at turn-on instant as shown in Fig. 3(b), which means that the ZVS condition is satisfied. When ZVS is achieved, the turn-on switching loss is zero. The third case is when the switch voltage waveform reaches zero before MOSFET turns on. In this case, the ZVS can be achieved because of the MOSFET body diode. However, the large conduction loss occurs because the on-resistance of the MOSFET body diode is higher than that of the MOSFET. In case 3, the switch-off duty ratio D in (2) – (10) and (14) must be replaced with the DDON . Therefore, D in (2) – (10) and (14) is expressed as ⎧ ⎪ ⎪ for Cases 1 and 2 ⎨ D, (17) D=⎪ ⎪ ⎩ DDON , for Case 3 DDON is obtained numerically by using Newton’s method. The diode
750
2πD (a)
0
0 (18)
Po (20) Po + Ptotal ⎧ 2 ⎪ α ⎪ ⎪ ⎪ R 2 [rS (1 − D) + rCS D + rLC ⎪ ⎪ ⎪ β ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ +πD A BQR] + R + rS (1 − D) + rCs D + rL0 C0 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ + (rS − rCS )[sin(4πD + 2φ) − sin 2φ] ⎪ ⎪ ⎪ 4π ⎪ ⎪ ⎪ ⎪ 2A2 BQR ⎪ ⎪ ⎪ + [cos(2πD + φ) − cos φ]2 ⎪ ⎪ ⎪ π ⎪ ⎪ ⎪ α ⎪ ⎪ ⎪ rCS − rS + πDA2 BQR −2 ⎪ ⎪ ⎪ πβ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [cos(2πD + φ) − cos φ] , for Cases 1 and 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ =⎪ ⎪ ⎪ α 2 ⎪ ⎪ ⎪ ⎪ ⎪ [rS (1 − D) + rCS DDON + rLC R 2 ⎪ ⎪ ⎪ β ⎪ ⎪ ⎪ ⎪ ⎪ +rDDON (D − DDON )] + R + rS (1 − D) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +r C s DDON + rL0 C0 + rDON (D − DDON ) ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ + {(rS − rDON ) sin(4πD + 2φ) + (rCS − rS ) sin 2φ ⎪ ⎪ ⎪ 4π ⎪ ⎪ ⎪ ⎪ ⎪ +(rDON − rCS ) sin(4πDDON + 2φ)} ⎪ ⎪ ⎪ ⎪ ⎪ α ⎪ ⎪ ⎪ {(rDON − rS ) cos(2πD + φ) + (rS − rCS ) cos φ −2 ⎪ ⎪ ⎪ πβ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩+(rCS − rDON ) cos(2πDDON + φ)} . for Cases 2 and 3
0
2πD (b)
θ 2π
0
2πD
2πDDon (c)
θ 2π
θ 2π
(d)
vg (V)
0
0
π
θ 2π
0
π
θ 2π
0
π
θ 2π
vS (V)
0 6 0
-6
vo (V)
vg (V)
5
20
From (11) and (19), the power-conversion efficiency can be obtained analytically as η=
2πD
θ 2π
Fig. 3. Switching types of class-E amplifier. (a) vS (2πD) > 0 and vS (θ) > 0 for ∀θ ∈ 0 ≥ θ < 2πD (Case 1). (b) vS (2πD) = 0 and DDON = D (Case 2). (c) DDON < D (Case 3). (d) vS (2πD) > 0 and DDON < D (Case 4).
vS (V)
where rDON is the on-resistance of the body diode and the body diode turns on at θ = 2πDDON as shown in Fig. 3(c). Finally, there is also the case when the switch voltage waveform returns to positive values after the body diode conduction during the on-state as shown in Fig. 3(d). It is unnecessary to consider this type of the switching because this type of the switching is a rare case and never appears in the results of this paper. From the above considerations, the total power loss is ⎧ ⎪ PS + PCS + PLC + PL0 C0 + Pto , for Case 1 ⎪ ⎪ ⎪ ⎨ Ptotal = ⎪ (19) P + P + P + P , for Case 2 S C L L C S C 0 0 ⎪ ⎪ ⎪ ⎩P + P + P + P for Case 3 S CS LC L0 C0 + PDON .
vo (V)
conduction loss is 2π 2πD rD rD i2S dθ = ON i2 dθ PDON = ON 2π 0 2π 2πDDON S I2 2 + m (D − DDON ) = rDON (D − DDON )IDD 2 IDD Im [cos(2πD + φ) − cos(2πDDON + φ)] + π
2 I + m [sin(4πDDON + 2φ) − sin(4πD + 2φ)] , 8π
(a)
(b)
Fig. 4. Waveforms for the nominal operation. (a) Analytical waveforms. (b) Experimental waveforms, vertical of vg : 5 V/div, vS : 20 V/div, vo : 10 V/div. horizontal: 200 ns/div.
The design parameters are obtained as LC = 34.67 μH, L0 = 7.96 μH, CS = 5.84 nF, C0 = 3.60 nF, which means A = 0.9406, and B = 0.6156. Moreover, the switch on-resistance is rS = 0.16 Ω from the data sheet of the IRF530 MOSFET and the body diode on-resistance is rDON = 1.54 Ω from the Spice model of the IRF530 MOSFET. All element values including ESR values were measured by HP4284A LCR impedance meter. In the experiment, the shunt capacitance was composed of some fabricated capacitances connected in parallel. Therefore, the ESR of CS can be ignored in these experiments. Table I gives the values from the analytical predictions and experimental measurements. Fig. 4 shows the analytical and the experimental waveforms for the designed conditions. It is seen from Fig. 4 that the class-E ZVS/ZDS conditions were achieved in this state. We define the state in Table I and Fig. 4 is the “nominal state”. Figures 5 and 6 show the output power and the power conversion efficiency of the class-E amplifier as functions of A and B, respectively, which are obtained by changing the parameters A and B from the nominal state. By comparing Figs. 5 and 6, it is seen that the parameter A is much more sensitive to the output power and the power conversion efficiency than the parameter B. It is seen from Figs. 5 and 6 that the analytical predictions agreed with the experimental results quantitatively, which validates the analytical expressions. IV. Design Example
B. Experimental Verification For validating the analytical expressions, circuit experiments were carried out. The design specifications are given as follows: operating frequency f = 1 MHz, dc-supply voltage VDD = 5 V, output resistor R = 5 Ω, switch-off duty ratio D = 0.5, and loaded quality factor Q = 10. First, we carry out the design of the class-E amplifier with the class-E ZVS/ZDS conditions [6], [7], namely, ⎡ ⎤ ⎢⎢⎢ vS (2πD) ⎥⎥⎥ ⎢⎢⎣⎢ dvS (θ) ⎥⎥⎥ = 0. (21) |θ=2πD ⎦ dθ
A design example of the class-E amplifier applying the analytical expressions for the power conversion efficiency is shown in this section. The design specifications are the same as those in Section III. The design conditions are given as ⎡ ⎤ ∂η ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢⎣ ⎥⎥⎥⎦ = 0, (22) ∂A vS (2πD) where ∂η/∂A = 0 implies the peak values of the power conversion efficiency against variation of A, which is a sensitive parameter to the
751
Analytical Experimental
4
80
3
60
2
40 20 0 0.5 0.75 1 Nominal condition A (b)
1.5
1.25
1.5
Fig. 5. Output power and power conversion efficiency as functions of A. (a) Output power. (b) Power conversion efficiency.
5 0
0
π
2πθ
0
π
2π θ
vS (V)
1.25
0
π
2πθ
vo (V)
0 0.5 0.75 1 Nominal condition A (a)
vg (V)
1
Theoretical 0.8916 0.6777 4.00 nF 5.91 nF 1.19 W 93.6 %
A B C0 CS Po η
vS (V)
20
100
4
0
Measured 0.8874 0.6799 4.01 nF 5.90 nF 1.13W 93.1 %
Difference −0.47 % 0.32 % 0.24 % −0.07 % −4.93 % −0.53 %
vg (V)
Analytical Experimental
η (%)
Po (W)
TABLE II Theoretical Predictions and Experimental Measurements for (22)
100
5
80
η (%)
3 Po (W)
vo (V)
6
2 Analytical Experimental
1
60
-6
0.6
0.7
B (a)
0.8
1
0
0.5
0.6
0.7
B (b)
0.8
0.9
1
Fig. 6. Output power and power conversion efficiency as functions of B. (a) Output power. (b) Power conversion efficiency.
Analytical 5.0 V 0.5 1 MHz 5.00 Ω 34.7 μH 7.96 μH 5.84 nF 3.60 nF 2.88 W 92.2 %
Measured 5.0 V 0.5 1 MHz 4.99 Ω 43.0 μH 8.01 μH 5.81 nF 3.53 nF 0.16 Ω 0.0 Ω 0.30 Ω 0.01 Ω 2.87 W 91.8 %
which was 1.3 % higher than that for the nominal conditions. This means that the 13 % power-loss rate reduction can be achieved. V. Conclusion This paper has given the analytical expressions for the output power and the power conversion efficiency for the class-E amplifier outside the class-E ZVS/ZDS conditions. The output power and the power conversion efficiency obtained from the analytical expressions agreed with the experimental results quantitatively, which indicates the validity of the analytical expressions. Moreover, the design example applying the analytical expressions for the power conversion efficiency is given in this paper. The designed amplifier achieved the 13 % power-loss rate reduction under the ZVS condition.
TABLE I Analytical Predictions and Experimental Measurements for Nominal Condition
VDD D f R LC L0 CS C0 rS rCS rL0 C0 rLC Po η
(b)
Fig. 7. Waveforms. : (a) From analytical expressions. (b) From circuit experiment. Vertical of vg : 5 V/div, vS : 20 V/div, vo : 10 V/div. Horizontal: 200 ns/div.
Analytical Experimental
20
Nominal condition 0.9
(a)
40
Nominal condition 0 0.5
0
Difference 0.00 % 0.00 % 0.00 % −0.20 % 24.1 % 0.68 % −0.65 % −1.88 % −0.35 % −0.46 %
References
power conversion efficiency. From this condition, the power conversion efficiency can be improved compared with the class-E amplifier satisfying ZVS/ZDS conditions. Moreover, the ZVS condition is also given for the EMI reduction. We set A and B as design parameters. The equations in (22) are solved numerically by using Newton’s method. Table II gives the theoretical predictions and experimental measurements. The other component values including the ESR values are the same as those in Table I for both the theory and the experiment. Figure 7 shows the theoretical and experimental waveforms. From Fig. 7, we can confirm that the class-E amplifier achieved the ZVS condition. The measured power conversion efficiency was 93.1 %,
752
[1] N. O. Sokal and A. D. Sokal, “Class E - A new class of high-efficiency tuned single-ended switching power amplifiers,” IEEE Jounal of Solid State Circuits, vol. 10, no. 3, pp. 168–176, Jun. 1975. [2] F. H. Raab, “Idealized operation of the class E tuned amplifier, ” IEEE Trans. Circuits Syst., vol. 24, no. 12, pp. 725–735, Dec. 1977. [3] M. Albulet and R. E. Zulinski, “Effect of switch duty ratio on the performance of class E amplifiers and frequency multipliers, ” IEEE Trans. Circuits Syst., vol. 45, no. 4, pp. 325–335, Apr. 1998. [4] T. Suetsugu and M. Kazimierczuk, “Steady-state behavior of class E amplifier outside designed conditions, ” in Proc. IEEE ISCAS, Kobe, Japan, May, 2005, pp. 708–711. [5] F. del Aguila, P. Pala, J. Bonet and R. Giralt, “A technique for maximum efficiency class E amplifier design, ” European Conference on Circuit Theory and Design, Sept, 2003, pp. 281–284. [6] D. J. Kessler and M. K. Kazimierczuk, “Power losses and efficiency of class-E power amplifier at any duty ratio,” IEEE Trans. Circuits Syst., vol. 51, no. 9, pp. 1675–1689, Sept. 2004. [7] M. K. Kazimierczuk, RF Power Amplifiers, New York, NY: John Wiley & Sons, 2008. [8] H. Sekiya, N Sagawa, and M. K. Kazimierczuk, “Analysis of class DE amplifier with nonlinear shunt capacitances at any grading coefficient for high Q and 25 % duty ratio,” IEEE Trans. Power Electron., vol.25, no.4, pp.925–932, Apr. 2010.
2
Graduate School of Advanced Integration Science, Chiba University, Chiba, 263–8522 Japan Department of Electrical Engineering, Wright State University, Dayton, OH, 45435–0001 USA Email: [email protected] [email protected]
Abstract—This paper gives analytical expressions for the output power and the power conversion efficiency of the class-E power amplifier outside the class-E ZVS/ZDS switching conditions. The analytical predictions agreed with the experimental results quantitatively, which indicates the validity of the analytical expressions. Moreover, a design example applying the analytical expressions for the power conversion efficiency is given. By using the analytical expressions, it is possible to design the class-E amplifier, which improves the power conversion efficiency compared with that of the class-E amplifier satisfying the class-E ZVS/ZDS conditions. Index Terms—Class-E power amplifier, power conversion efficiency, output power, nominal operation.
VDD
IDD LC C0 L0 is iCs io vs CS vo S
R
VDD
IDD LC C0 L0 is iCs io v s CS vo S
(a)
R
(b)
Fig. 1. (a) Circuit topology of class-E amplifier. (b) Equivalent circuit of class-E amplifier.
I. Introduction The class-E power amplifier [1]-[7] is expected to be useful for many applications, e.g., radio transmitters, switching-mode dc power supplies, devices for medical applications, and so on. Because of the class-E zero-voltage switching and zero-derivative switching (ZVS/ZDS) conditions, the class-E amplifier can achieve high power conversion efficiency at high-frequencies. Therefore, most papers on the class-E amplifier discuss the class-E amplifier under the class-E ZVS/ZDS conditions, which is called as “nominal conditions”. Reference [4] gives the analytical expressions for the waveforms of the class-E amplifier outside the class-E ZVS/ZDS conditions. The analytical expressions of the output power and the power conversion efficiency outside the class-E ZVS/ZDS conditions, however, have never been derived. Generally, it is known that the conditions for obtaining the maximum power conversion efficiency, which is called as “optimal conditions”, is not the same but approximately the same as nominal conditions. There is, however, no design approach to achieve the optimal conditions of the class-E amplifier except [5]. For determining the element values of the class-E power amplifier with maximum power conversion efficiency, it is important to have the analytical expressions of the power conversion efficiency outside the nominal conditions. This paper presents analytical expressions for the output power and the power conversion efficiency of the class-E amplifier outside the class-E ZVS/ZDS conditions. The analytical predictions agreed with the experimental results quantitatively, which shows the validity of the analytical expressions. Moreover, a design example applying the analytical expressions for the power conversion efficiency is given.
√ 1) A = f0 / f = ω0 /ω = 1/ω L0C0 : The ratio of the operating frequency to the resonant frequency. 2) B = C0 /CS : The ratio of the resonant capacitance to the shunt capacitance. 3) D: The switch-off duty ratio of the switch S . 4) Q = ωL0 /R: The loaded Q-factor. The circuit analysis for obtaining the waveform equations are based on the following assumptions: 1) The MOSFET acts as an ideal switch device, namely, it has zero on-resistance, infinite off-resistance and zero switching times. 2) The dc-feed inductance LC is large enough so that the current through the dc-feed inductor is constant. 3) The loaded Q-factor is high enough to generate a sinusoidal output current io . 4) All the passive elements are linear and have zero equivalent series resistances (ESRs). 5) The circuit operations are considered in the interval 0 ≤ θ ≤ 2π. The switch is in the off-state for 0 ≤ θ < 2πD and in the on-state for 2πD ≤ θ < 2π, where θ = wt represents angular time. By the above assumptions, the equivalent circuit is obtained as shown in Fig. 1(b). B. Waveform Equations The output current is io = Im sin(θ + φ).
II. Waveform Equations Outside Class-E ZVS/ZDS Conditions The analytical expressions for the output power and the power conversion efficiency of the class-E amplifier outside the class-E ZVS/ZDS conditions are derived by using the waveform equations given in [4]. In this section, we introduce only the results, which are obtained in [4]. A. Assumptions and Parameters Figure 1(a) shows a circuit topology of the class-E amplifier. For obtaining the analytical expressions, we define the following parameters:
978-1-4244-9472-9/11/$26.00 ©2011 IEEE
(1)
In (1), Im is the amplitude of the output current Im =
2πVDD β , A2 BQR(2π2 D2 α + γβ)
(2)
where VDD is the dc-supply voltage, 1 1 α= cos 4πD − cos 2πD + cos2 φ 2 2 1 π 1 1 − sin 4πD − sin 2πD sin φ cos φ + 2 − cos 4πD + , 2 A BQ 4 4 (3)
749
VDD
Fig. 2.
IDD LC rLc C0 L0 rL0C0
rS , ESRs of dc-feed inductance rLC , shunt capacitance rCS , and load network rL0 C0 as shown in Fig. 2.
iCs io is vo R S vs CS rCs rS
The output power Po is 2π RI 2 R Po = i2o dθ = m . 2π 0 2
Equivalent circuit of class-E amplifier including ESRs.
β = (−2πD cos 2πD + sin 2πD) cos φ +(2πD sin 2πD + cos 2πD − 1) sin φ,
(4)
and γ = (sin 2πD − 2πD) cos φ + (cos 2πD − 1) sin φ.
(5)
Moreover, φ, which is the phase-shift between the input voltage and the output current, is expressed as 1 π 3 φ = − arctan cos 4πD − cos 2πD + 2 + 4 A BQ 4 (2πD sin 2πD + cos 2πD − 1) 1 π(1 − A2 ) + sin 2πD − sin 4πD + − πD 4 A2 B (−2πD cos 2πD + sin 2πD) 1 π(1 − A2 ) + πD − sin 4πD − 4 A2 B (2πD sin 2πD + cos 2πD − 1) π 1 1 + 2 + − cos 4πD (πD sin 2πD − sin 2πD) . A BQR 4 4 (6) The switch voltage is given by ⎧ ⎪ ⎪ ⎨ A2 BQR[IDDθ + Im cos(θ + φ) − Im cos φ], vS = ⎪ ⎪ ⎩0,
for 0 ≤ θ < 2πD for 2πD ≤ θ < 2π, (7)
where IDD , which is the dc-supply current, is given by IDD =
2πVDD α . A2 BQR(2π2 D2 α + γβ)
The current through the shunt capacitance is ⎧ ⎪ ⎪ ⎨IDD − Im sin(θ + φ), for 0 ≤ θ < 2πD iCS = ⎪ ⎪ ⎩0, for 2πD ≤ θ < 2π. The current through the switch is ⎧ ⎪ ⎪ for 0 ≤ θ < 2πD ⎨0, iS = ⎪ ⎪ ⎩IDD − Im sin(θ + φ), for 2πD ≤ θ < 2π.
(8)
(9)
(10)
III. Output Power and Power Conversion Efficiency A. Derivation of Analytical Equations In this section, analytical expressions for the output power and the power conversion efficiency are derived by using the waveform equations given in Section II. In real circuits, the power losses occur in the parasitic resistances of each component. It is assumed that the parasitic resistances are small enough not to affect the waveforms [8]. In this paper, we consider the power losses in switch on-resistance
(11)
The conduction loss in the MOSFET on-resistance rS is 2π rS PS = i2 dθ 2π 0 S I2 IDD Im 2 = rS (1 − D)IDD + m (1 − D) + [cos φ − cos(2πD + φ)] 2 π
I2 + m [sin(4πD + 2φ) − sin 2φ] . (12) 8π The power loss in rLC is 2 PLC = rLC IDD .
(13)
The power loss in rCS is obtained as 2π rC PCS = S i2 dθ 2π 0 CS I2 IDD Im 2 cos(2πD + φ) − cos φ = rCS DIDD + mD+ 2 π
I2 + m sin 2φ − sin(4πD + 2φ) . (14) 8π The power loss in rL0 C0 is 2π rL C rL C I 2 i2o dθ = 0 0 m . PL0 C0 = 0 0 2π 0 2
(15)
The analysis in this paper does not assume that the class-E ZVS/ZDS conditions are satisfied. We can classify the switching types of the class-E amplifier into four cases as shown in Fig. 3. The first case is when the switch voltage waveform does not reach zero prior to turn-on switching as shown in Fig. 3(a). In this case, the turn-on switching loss occurs at turn-on instant, which is expressed as 1 Pto = CS f v2S (2πD− ) 2 A2 BQR [(2πDIDD )2 + 4πDIm IDD cos(2πD + φ) = 4π −4πDIDD Im cos φ + Im2 cos2 (2πD + φ) −2Im2 cos(2πD + φ) cos φ + Im2 cos2 φ]. (16) where vS (2πD− ) is the switch voltage just prior to turn-on. The second case is when the switch voltage waveform just reaches zero at turn-on instant as shown in Fig. 3(b), which means that the ZVS condition is satisfied. When ZVS is achieved, the turn-on switching loss is zero. The third case is when the switch voltage waveform reaches zero before MOSFET turns on. In this case, the ZVS can be achieved because of the MOSFET body diode. However, the large conduction loss occurs because the on-resistance of the MOSFET body diode is higher than that of the MOSFET. In case 3, the switch-off duty ratio D in (2) – (10) and (14) must be replaced with the DDON . Therefore, D in (2) – (10) and (14) is expressed as ⎧ ⎪ ⎪ for Cases 1 and 2 ⎨ D, (17) D=⎪ ⎪ ⎩ DDON , for Case 3 DDON is obtained numerically by using Newton’s method. The diode
750
2πD (a)
0
0 (18)
Po (20) Po + Ptotal ⎧ 2 ⎪ α ⎪ ⎪ ⎪ R 2 [rS (1 − D) + rCS D + rLC ⎪ ⎪ ⎪ β ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ +πD A BQR] + R + rS (1 − D) + rCs D + rL0 C0 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ + (rS − rCS )[sin(4πD + 2φ) − sin 2φ] ⎪ ⎪ ⎪ 4π ⎪ ⎪ ⎪ ⎪ 2A2 BQR ⎪ ⎪ ⎪ + [cos(2πD + φ) − cos φ]2 ⎪ ⎪ ⎪ π ⎪ ⎪ ⎪ α ⎪ ⎪ ⎪ rCS − rS + πDA2 BQR −2 ⎪ ⎪ ⎪ πβ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ [cos(2πD + φ) − cos φ] , for Cases 1 and 2 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ =⎪ ⎪ ⎪ α 2 ⎪ ⎪ ⎪ ⎪ ⎪ [rS (1 − D) + rCS DDON + rLC R 2 ⎪ ⎪ ⎪ β ⎪ ⎪ ⎪ ⎪ ⎪ +rDDON (D − DDON )] + R + rS (1 − D) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +r C s DDON + rL0 C0 + rDON (D − DDON ) ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎪ + {(rS − rDON ) sin(4πD + 2φ) + (rCS − rS ) sin 2φ ⎪ ⎪ ⎪ 4π ⎪ ⎪ ⎪ ⎪ ⎪ +(rDON − rCS ) sin(4πDDON + 2φ)} ⎪ ⎪ ⎪ ⎪ ⎪ α ⎪ ⎪ ⎪ {(rDON − rS ) cos(2πD + φ) + (rS − rCS ) cos φ −2 ⎪ ⎪ ⎪ πβ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩+(rCS − rDON ) cos(2πDDON + φ)} . for Cases 2 and 3
0
2πD (b)
θ 2π
0
2πD
2πDDon (c)
θ 2π
θ 2π
(d)
vg (V)
0
0
π
θ 2π
0
π
θ 2π
0
π
θ 2π
vS (V)
0 6 0
-6
vo (V)
vg (V)
5
20
From (11) and (19), the power-conversion efficiency can be obtained analytically as η=
2πD
θ 2π
Fig. 3. Switching types of class-E amplifier. (a) vS (2πD) > 0 and vS (θ) > 0 for ∀θ ∈ 0 ≥ θ < 2πD (Case 1). (b) vS (2πD) = 0 and DDON = D (Case 2). (c) DDON < D (Case 3). (d) vS (2πD) > 0 and DDON < D (Case 4).
vS (V)
where rDON is the on-resistance of the body diode and the body diode turns on at θ = 2πDDON as shown in Fig. 3(c). Finally, there is also the case when the switch voltage waveform returns to positive values after the body diode conduction during the on-state as shown in Fig. 3(d). It is unnecessary to consider this type of the switching because this type of the switching is a rare case and never appears in the results of this paper. From the above considerations, the total power loss is ⎧ ⎪ PS + PCS + PLC + PL0 C0 + Pto , for Case 1 ⎪ ⎪ ⎪ ⎨ Ptotal = ⎪ (19) P + P + P + P , for Case 2 S C L L C S C 0 0 ⎪ ⎪ ⎪ ⎩P + P + P + P for Case 3 S CS LC L0 C0 + PDON .
vo (V)
conduction loss is 2π 2πD rD rD i2S dθ = ON i2 dθ PDON = ON 2π 0 2π 2πDDON S I2 2 + m (D − DDON ) = rDON (D − DDON )IDD 2 IDD Im [cos(2πD + φ) − cos(2πDDON + φ)] + π
2 I + m [sin(4πDDON + 2φ) − sin(4πD + 2φ)] , 8π
(a)
(b)
Fig. 4. Waveforms for the nominal operation. (a) Analytical waveforms. (b) Experimental waveforms, vertical of vg : 5 V/div, vS : 20 V/div, vo : 10 V/div. horizontal: 200 ns/div.
The design parameters are obtained as LC = 34.67 μH, L0 = 7.96 μH, CS = 5.84 nF, C0 = 3.60 nF, which means A = 0.9406, and B = 0.6156. Moreover, the switch on-resistance is rS = 0.16 Ω from the data sheet of the IRF530 MOSFET and the body diode on-resistance is rDON = 1.54 Ω from the Spice model of the IRF530 MOSFET. All element values including ESR values were measured by HP4284A LCR impedance meter. In the experiment, the shunt capacitance was composed of some fabricated capacitances connected in parallel. Therefore, the ESR of CS can be ignored in these experiments. Table I gives the values from the analytical predictions and experimental measurements. Fig. 4 shows the analytical and the experimental waveforms for the designed conditions. It is seen from Fig. 4 that the class-E ZVS/ZDS conditions were achieved in this state. We define the state in Table I and Fig. 4 is the “nominal state”. Figures 5 and 6 show the output power and the power conversion efficiency of the class-E amplifier as functions of A and B, respectively, which are obtained by changing the parameters A and B from the nominal state. By comparing Figs. 5 and 6, it is seen that the parameter A is much more sensitive to the output power and the power conversion efficiency than the parameter B. It is seen from Figs. 5 and 6 that the analytical predictions agreed with the experimental results quantitatively, which validates the analytical expressions. IV. Design Example
B. Experimental Verification For validating the analytical expressions, circuit experiments were carried out. The design specifications are given as follows: operating frequency f = 1 MHz, dc-supply voltage VDD = 5 V, output resistor R = 5 Ω, switch-off duty ratio D = 0.5, and loaded quality factor Q = 10. First, we carry out the design of the class-E amplifier with the class-E ZVS/ZDS conditions [6], [7], namely, ⎡ ⎤ ⎢⎢⎢ vS (2πD) ⎥⎥⎥ ⎢⎢⎣⎢ dvS (θ) ⎥⎥⎥ = 0. (21) |θ=2πD ⎦ dθ
A design example of the class-E amplifier applying the analytical expressions for the power conversion efficiency is shown in this section. The design specifications are the same as those in Section III. The design conditions are given as ⎡ ⎤ ∂η ⎢⎢⎢ ⎥⎥⎥ ⎢⎢⎢⎣ ⎥⎥⎥⎦ = 0, (22) ∂A vS (2πD) where ∂η/∂A = 0 implies the peak values of the power conversion efficiency against variation of A, which is a sensitive parameter to the
751
Analytical Experimental
4
80
3
60
2
40 20 0 0.5 0.75 1 Nominal condition A (b)
1.5
1.25
1.5
Fig. 5. Output power and power conversion efficiency as functions of A. (a) Output power. (b) Power conversion efficiency.
5 0
0
π
2πθ
0
π
2π θ
vS (V)
1.25
0
π
2πθ
vo (V)
0 0.5 0.75 1 Nominal condition A (a)
vg (V)
1
Theoretical 0.8916 0.6777 4.00 nF 5.91 nF 1.19 W 93.6 %
A B C0 CS Po η
vS (V)
20
100
4
0
Measured 0.8874 0.6799 4.01 nF 5.90 nF 1.13W 93.1 %
Difference −0.47 % 0.32 % 0.24 % −0.07 % −4.93 % −0.53 %
vg (V)
Analytical Experimental
η (%)
Po (W)
TABLE II Theoretical Predictions and Experimental Measurements for (22)
100
5
80
η (%)
3 Po (W)
vo (V)
6
2 Analytical Experimental
1
60
-6
0.6
0.7
B (a)
0.8
1
0
0.5
0.6
0.7
B (b)
0.8
0.9
1
Fig. 6. Output power and power conversion efficiency as functions of B. (a) Output power. (b) Power conversion efficiency.
Analytical 5.0 V 0.5 1 MHz 5.00 Ω 34.7 μH 7.96 μH 5.84 nF 3.60 nF 2.88 W 92.2 %
Measured 5.0 V 0.5 1 MHz 4.99 Ω 43.0 μH 8.01 μH 5.81 nF 3.53 nF 0.16 Ω 0.0 Ω 0.30 Ω 0.01 Ω 2.87 W 91.8 %
which was 1.3 % higher than that for the nominal conditions. This means that the 13 % power-loss rate reduction can be achieved. V. Conclusion This paper has given the analytical expressions for the output power and the power conversion efficiency for the class-E amplifier outside the class-E ZVS/ZDS conditions. The output power and the power conversion efficiency obtained from the analytical expressions agreed with the experimental results quantitatively, which indicates the validity of the analytical expressions. Moreover, the design example applying the analytical expressions for the power conversion efficiency is given in this paper. The designed amplifier achieved the 13 % power-loss rate reduction under the ZVS condition.
TABLE I Analytical Predictions and Experimental Measurements for Nominal Condition
VDD D f R LC L0 CS C0 rS rCS rL0 C0 rLC Po η
(b)
Fig. 7. Waveforms. : (a) From analytical expressions. (b) From circuit experiment. Vertical of vg : 5 V/div, vS : 20 V/div, vo : 10 V/div. Horizontal: 200 ns/div.
Analytical Experimental
20
Nominal condition 0.9
(a)
40
Nominal condition 0 0.5
0
Difference 0.00 % 0.00 % 0.00 % −0.20 % 24.1 % 0.68 % −0.65 % −1.88 % −0.35 % −0.46 %
References
power conversion efficiency. From this condition, the power conversion efficiency can be improved compared with the class-E amplifier satisfying ZVS/ZDS conditions. Moreover, the ZVS condition is also given for the EMI reduction. We set A and B as design parameters. The equations in (22) are solved numerically by using Newton’s method. Table II gives the theoretical predictions and experimental measurements. The other component values including the ESR values are the same as those in Table I for both the theory and the experiment. Figure 7 shows the theoretical and experimental waveforms. From Fig. 7, we can confirm that the class-E amplifier achieved the ZVS condition. The measured power conversion efficiency was 93.1 %,
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[1] N. O. Sokal and A. D. Sokal, “Class E - A new class of high-efficiency tuned single-ended switching power amplifiers,” IEEE Jounal of Solid State Circuits, vol. 10, no. 3, pp. 168–176, Jun. 1975. [2] F. H. Raab, “Idealized operation of the class E tuned amplifier, ” IEEE Trans. Circuits Syst., vol. 24, no. 12, pp. 725–735, Dec. 1977. [3] M. Albulet and R. E. Zulinski, “Effect of switch duty ratio on the performance of class E amplifiers and frequency multipliers, ” IEEE Trans. Circuits Syst., vol. 45, no. 4, pp. 325–335, Apr. 1998. [4] T. Suetsugu and M. Kazimierczuk, “Steady-state behavior of class E amplifier outside designed conditions, ” in Proc. IEEE ISCAS, Kobe, Japan, May, 2005, pp. 708–711. [5] F. del Aguila, P. Pala, J. Bonet and R. Giralt, “A technique for maximum efficiency class E amplifier design, ” European Conference on Circuit Theory and Design, Sept, 2003, pp. 281–284. [6] D. J. Kessler and M. K. Kazimierczuk, “Power losses and efficiency of class-E power amplifier at any duty ratio,” IEEE Trans. Circuits Syst., vol. 51, no. 9, pp. 1675–1689, Sept. 2004. [7] M. K. Kazimierczuk, RF Power Amplifiers, New York, NY: John Wiley & Sons, 2008. [8] H. Sekiya, N Sagawa, and M. K. Kazimierczuk, “Analysis of class DE amplifier with nonlinear shunt capacitances at any grading coefficient for high Q and 25 % duty ratio,” IEEE Trans. Power Electron., vol.25, no.4, pp.925–932, Apr. 2010.
