ECCTD 2005 - European Conference on Circuit Theory and Design, Cork Ireland, 29 August - 2 September 2005
Optimization Procedure of Class E Amplifiers Using SPICE Yuichi Tanji£,
Hiroo SekiyaÝ,
Abstract — Class E amplifiers are known as a switching type power amplifier and a good candidate for fully integrated CMOS transceivers which are used for Global System for Mobile Communication and Bluetooth. However, the design of Class E amplifiers is difficult, because the passive elements and the device parameters of CMOS switch must be optimized so that the switching losses are minimized. In this paper, an optimization procedure of Class E amplifiers is proposed, where all the design parameters are automatically assigned repeating the SPICE transient analysis. In the numerical examples, the proposed procedure is demonstrated using the Berkeley SPICE.
1
Introduction
A Class E amplifier is the best choice as the power amplifier which is capable of combining a high efficiency ) with a resonant output power (30 dBm) [1]. ( It is a good candidate for CMOS transceiver, because the switching type amplifiers do not obstruct the constant envelope modulation like Global System for Mobile Communication and Bluetooth. On the other hand, it is difficult to design a Class E amplifier on an integrated circuit, where the designers have to adjust the passive elements including the circuit and the device parameters of transistor in order to minimize the switching losses. To overcome its difficulty, an optimization procedure based on a technique for analyzing the nonlinear phenomena occurring in circuits is proposed in [2]. In this method, the Class E amplifiers are idealized by two linear circuits on the on/off state of the transistor switch and the passive elements are determined so that the requirements as Class E amplifier are satisfied. This method, however, can not consider the influences of semiconductor devices which affect the efficiency due to the idealization of the circuit equations. In this paper, we propose an optimization procedure of Class E amplifiers. Both the passive elements including the circuit and device parameters of the transistor switch are determined because of use of SPICE. The requirements as Class E amplifier must be constrained on the steady state of the circuits. Hence, we propose first a numerical method for finding the steady-state responses and implement it on SPICE. Next, the method is applied to the optimization of Class E amplifiers. In the numerical examples, the proposed procedure is demonstrated using the Berkeley SPICE. £ Dept. of Reliability-based Info. Systems Eng., Kagawa University, Japan, e-mail:
[email protected] Ý Graduate School of Science and Technology, Chiba University, Japan, e-mail:
[email protected] Þ Dept. of Systems Eng., Shizuoka University, Japan, e-mail:
[email protected]
0-7803-9066-0/05/$20.00 ©2005 IEEE
Hideki AsaiÞ
2 Finding Steady State Responses on SPICE 2.1 Time-Domain Shooting Method Consider a general network containing an arbitrary number of linear/nonlinear components. The MNA formulation of such a network can be written as [4],
(1)
where is the vector consisting of node voltages, independent voltage source currents, linear inductor currents, nonlinear capacitor charges, nonlinear inductor fluxes, and currents and voltages of nonlinear compoand are matrices describing the lumped nents; memory and memoryless elements of the network, respectively; is a vector consisting of the independent voltage/current sources; is a nonlinear function matrix which describes the nonlinear elements of the circuit. Assuming an independent voltage/current source with a period , we find the steady-state responses of the circuit written by (1). The steady-state solution is satisfied with
¼
(2)
Hence, finding the steady state response is reduced to the nonlinear optimization problem for determining the initial solution of with respect to (2). The Newton Raphson method is applied to solving this problem [5] as
ÜÜ ¼ (3) Each element of the Jacobian is calculated by giving a small perturbation to the initial values and applying a numerical integration algorithm. Suppose that a small perturbation is appended to the initial value as . Applying a numerical integration algorithm for solving the the initial value problem of (1), we calculate the element of
the Jacobian as
(4) where is the variation of after a small perturbation is appended to .
2.2 Restricted Steady State Condition
iC
The time-domain shooting method finds steady state responses, repeating the transient analysis of a network. Thus, one may think that this method is easily implemented on SPICE-like simulators. However, the simulators do not provide us with all informations of the MNA matrix. Node voltages, inductor currents, capacitor voltages, voltages and currents for the nonlinear elements are only available for the initial condition and we can not know the informations of parasitic elements including semiconductor device models. Therefore, we can not implement the time-domain shooting method without modifications of SPICE source code. Alternatively, we develop the frame work for finding the steady state responses, rewriting the SPICE netlist and repeating the SPICE transient analysis. Assume to be the vector which corresponds to the node voltages and linear inductor currents of of (1). The steady state condition only for the node voltages and inductor currents is written by
¼
L0
VD
C0
v
S CS
vS
R
vout
Dr
Figure 1: Topology of basic Class E amplifier. Dr
T
2T
T
2T
vs
(5)
The shooting method provided in the previous subsection is carried out to find the solution of (5). In a circuit, memory elements such as inductor and capacitor are included in the inside and outside of semiconductor devices. When the voltages and currents of the memory elements which exist in the outside of devices satisfy (5), the voltages and currents in the inside of the devices almost reach the steady state. Therefore, we can obtain the almost steady-state responses using the condition (5), nevertheless (5) is not complete as the steady state condition for (1). 2.3 Implementation The shooting method may sometimes fail to converge into the appropriate solution. To improve the convergence, we use a damping parameter as
i
LC
ÞÞ
(6)
If , then, the iteration (6) is corresponding to the Newton Raphson method. In our implementation, although a fixed value of is taken, more sophisticated determination is presented in [6]. The implementation of our shooting method on Berkeley SPICE is summarized as: 1. Carry out the operating point and transient analysises. Get the transient responses for from the SPICE output file.
Figure 2: Switching voltage of Class E amplifier. 2. Rewrite ’.ic’ statement for initial node voltages and ’ic=’ values in linear inductor statements using the responses for . 3. Add a small perturbation to a node voltage or an initial linear inductor current. 4. Carry out the transient analysis. Get a column of the Jacobian (4). 5. Repeat steps 3 and 4 until all the elements of Jacobian are calculated. 6. Carry out the damping Newton method (6). 7. Repeat from steps 3 to 6 until a stopping condition is satisfied. 3 Optimization procedure of Class E Amplifier The shooting method provided in the previous section is incorporated into the design of the Class E amplifier [2]. A topology of basic class E amplifier is shown in Fig. 1, where the circuit consists of input voltage , dc-feed inductor , MOS switch , shut capacitor to the switch, a series resonant circuit composed of inductor and capacitor , and output resistor R. In order to attain the high-efficiency, all the losses occur during switching should be minimized, which demands that the drain-source voltage when the switch closes is zero. Furthermore, it is necessary that the time derivative of the switch voltage is also equal to zero at the switching moment [1]. As a result, the requirements as
[V]
Eb R7
R8 0
R4
C1
C3
v(1)
R9 e
R5
R6
vout −0.5
C2
v(7)
0
Figure 3: CR amplifier, where ,
, , , , , , , , and . Class E amplifier are obtained by
0.0001
0.0002 [sec.]
(a) [V]
0 v(1)
(7)
(8)
Figure 2 shows the typical waveform of the switch voltage of the Class E amplifier in Fig. 1, the switch voltage smoothly lands into the ground at and without switching losses. To satisfy the requirements (7) and (8), the design parameters such as the values of the passive elements and device parameters of the switch should be adjusted. Further, the requirements must be fulfilled on the steady state. Thus, the restricted steady state condition (5) is enforced together with (7) and (8). We also apply the damping Newton method to solving the composite problem. First, the derivative of the switching voltage is approximated by the 1-st order difference1 :
(9) where is the time step size of the numerical
−0.5 v(7)
0.0998
0.0999
0.1 [sec.]
(b) Figure 4: Steady state responses of CR amplifier obtained from (a)the proposed shooting method and (b)the transient analysis with appropriate initial condition, where and are corresponding to and in Fig. 3. On the other hand, assuming design parameters, the elements of Jacobian are written by
(12)
integration. (13) Appending a small variation into , we calculate the elements of Jacobian associated with (7) and where and are the variations of and , respectively, when a small perturba(8), respectively, as tion is appended to the design parameter . If the number of the design parameters is larger than (10) 2, the Jacobian becomes a rectangular matrix. Then, we use QR decomposition [7] in each Newton iteration (11) instead of LU one.
where and are the variations of 4 Numerical Results and , respectively, when a small perturba- 4.1 CR amplifier tion is appended to . To confirm that the proposed shooting method can cap1 Instead of considering the numerical difference, we can evaluate the current through the capacitor ¼ . Then, an independent voltage ture the initial state which provides the steady state sosource with ¼ is connected to the capacitor in series. lution, this method was applied to the CR amplifier
v(2)
20 10 0 −10 0
0.5
1
1.5
2
[micro−sec.] 10
v(4)
5
where the switch was treated as the ideal switch with 0.16 on-resistor. As a result, # and $
were obtained and all passive elements were determined as follows, mH, nF, H, nF, and . Figure 5 shows the steady state waveforms of the Class E amplifier with these values of passive elements. We can see that the requirements as Class E amplifier are satisfied since the capacitor voltage ) and its time derivative are almost zero at 0, 1, and 2 micro seconds.
0
5 Conclusions
−5 −10 0
0.5
1
1.5
2
1.5
2
[micro−sec.]
v(5)
6 4 2 0 0
0.5
1
[micro−sec.]
Figure 5: Voltage waveforms of Class E amplifier, where , , and are corresponding to , , and , respectively. shown in Fig. 3, where level 1 was used as the device model. Using NGSPICE [8], which is the latest version of Berkeley SPICE, we calculated the steady state responses. Figure 4(a) shows the waveforms obtained from the proposed method. For a comparison, the transient analysis was carried out until 0.1 [sec.]. We show the responses from 0.998 [sec.] to 0.1 [sec.] in Fig. 4(b). The waveforms obtained from the proposed method are in agreement with the transient responses which reach to the steady state. 4.2 Class E amplifier To design the Class E amplifier shown in Fig. 1, we define the following parameters [2]: 1) ! 2) ! . 3) " . 4) # ! ! . 5) $ . 6) % . Specification; ! MHz, V, , " , and % , was given, and # and $ were selected as the design parameters. The procedure provided in Sect. 3 was applied to this problem,
The optimization procedure of Class E amplifiers using SPICE has been proposed. Fist, the time-domain shooting method for finding the steady state responses has been presented and the SPICE netlist based implementation has been provided. Next, the time-domain shooting method is incorporated with the design of Class E amplifiers. In the numerical example, it has been confirmed that the passive components including in the Class E amplifier can be automatically determined. Since the proposed method is implemented on SPICE, this procedure can also assign the device parameters. In the future, we will report about the details. References [1] P. Reynaert, K. L. R.Mertens, M. S. J. Steyaert, “State-space behavioral model for CMOS class E power amplifiers,” IEEE Trans. TCAD, vol. 22, no. 2, pp. 132-138, Feb. 2003. [2] H. Sekiya, I. Sasase and S. Mori, “Computation of design values for class E amplifiers without using waveform equations,” IEEE Trans. Circuits and Systems, vol. 49, no. 7, pp. 966-978, July 2002. [3] H. Sekiya, Y. Tanji, J. Lu and T. Yahagi, “Design Procedure for Generalized Class E Amplifiers with Implicit Circuit Equations,” Proc. NOLTA2004, pp. 306-309, Dec. 2004. [4] C. Ho, A. Ruehli, and P. Brennan, “The modified nodal approach to network analysis,” IEEE Trans. Circuits Syst., vol. CAS-22, pp. 504-509, June 1975. [5] T. J. Aprille, Jr., and T. N. Trick, “Steady-state analysis of nonlinear circuits with periodic input”, Proc. IEEE, vol. 60, pp. 108114, 1972. [6] M. Kakizaki and T. Sugawara, “A modified netwon method for the steady-state analysis,” IEEE Trans. TCAD, vol. CAD-4, no. 4, pp. 662-667, Oct. 1983. [7] G. H. Golub and C. F. V. Loan, Matrix Computations, The Johns Hopkins University Press, 1996. [8] http://ngspice.sourceforge.net/index.html
