Steady state analysis of class-E amplifier with non-linear capacitor by means of discrete-time techniques

Steady State Analysis of ,Class-E Amplifier with non-linear capacitor ‘ Pala Schgn waldeer, dor& Bofiet Dalmau and Rosa Giralt Mas RIA DEL SENYAL 1 COMUNICACIONS aguila@,tsc.upc.es A~STRACT thod to steady S of swit circuits used to, describe, the ci I. INTRODUCTION During the,. last years switched circuits have become widely used due to its efficiency in power supply amplifiers. Analysis to accurately. In many knowledge , of transient beh secondayy, . the main interest be steady state waveforms. In this paper a new method for the steady state switched circuits with non-linea is presented. The technique is b time-domain ,discretization of the resulting non-linear eleme is easily determined. - \! ’ Non=linear i aviour in terms DOFFQ9 -t 40&)QD(q +”, (4y,(4 = 0 Where 06). and Nkfs,! are polynomials. The control variable of the .non-linearity Yfi) is the only unknown in these equatio A (?P derivatives o approximated using gear dicretizations. For 0-7803-6542-9/00/$10.00 0 2000 IEEE 895 and v, - vn-1 A Gear-1 discretization V, = 3 v, - 4 vn-, + vn-2 for Gear-2 V, = 2A As a result, equation (2) may be rewriter in the form 2 'n-, + 2 'n-i%('n-,) + '= (3) 0 0 Where the only unknowns are the samples Numeric resolution of (3) allows computing the waveform in each state, once the initial conditions are determined. When changing from one topology to the other, the state variables \vcD, vc, iL , iLy are continuous. Next, we derive a relation between ivcD, vc, iL, iLY and (v, , qD( v,). . .) that allows expressing state variable continuity in terms of (v, , qD( v,) . . .) . Consider the equations describing the circuit behaviour for the ON state: of VP). Repeatedly deriving (4) the relations expressed in (5) (see below) may be obtained. The next step consists in computing the discretization matrix D relating v, V, . . . to the samples of v and qD,qD, .. . to the samples of qD. For the Gear-1 discretization scheme, the discretization matrix D=D1 is obtained from: 0 0 0 1 vn-3 P A' 3 A' .!I[ A' (6) -1 331 -- A3 A3 A3 A' For higher order discretization schemes there are several degrees of freedom when computing v from G. In this case, additional restrictions may be set. One approach is to consider that the behaviour of the derivatives at the current and the previous sample is the same (7). This yields a matrix D=D22 as: 0 0 0 0 0 0 1 0 0 OL322-249 0 0 0 0 0 0 0 0 1 0 0 0 0 0 I =i 2 0 0 0 L 2 2 2 2 qt) 0000~~+0 .qt) v'(t) L'(t) v+(t) .:+@) 0 I 3 11 3 8d SA' SA' SA' SA' -lop SA' I SA' 4A' 46' 4A' 4A' 4A' 2A 2A 2A 4A' 4A' 4A' 4A' 4A' i = D22~ Another approach to obtain the discretization matrix is to set the degree of freedom with equations obtained from low order discretization (8), thus Dzl is obtained for a Gear-2 approach also. Now, defining the state vector s as s = [vcD, vc, iL , iLq p, we may solve fiom (5) s=-M,-'M,Dv-M,-'M,Dq,(v)-M,-'m= A,v+Aqgq,(v)+a (9) From this equation we may compute the state vector at the end of one switching period, using the last samples of v a qD. If the state vector is continuous when switching occurs, 'ON = 'OFF (10) Or equivalently, (1 1) Where a vector of '7nitiaZ sampZes " v' is obtained from the last samples of the previous topology. Transient analysis could be carried out by repeatedly solving (2) and using (11) at each switching. We may now introduce the periodicity assumption, i.e. vN+, = v,, vN+, = v2,. .. This leads to the nonlinear algebraic system of N = N' + NON + N' +NoFF equations described by (1 2) where the vector of unknowns contains the initial samples V'ON and V'OFF along with VON and VOFF. The NON equations describe the topology of circuit during the ON interval. The N&F equations describe the topology during the OFF interval. In this way we have defined the steady-state circuit behaviour in a single matrix. Note that the system is extended with N~N and N~FF equations from (5) due to the added unknowns: the initial samples of v# at every switching instant. V'ON+AqDonq'DoN "ON = *vOFp 'OFF + AqDOPFqDOFF + 'OFF AV, -AV, q c The efficient resolution of the nonlinear system of equations is obtained using global convergent methods based on modifications of Newton's method [2]. 111. RESULTS This method has been applied to the resolution of the circuit of figure 1 based on [3]. The circuit element values are: VDD=2v, L, =200nH, Rx+.25R, L=2.71nH, C=14pF, R=l.85R., n=3, C,o=40.3pF and Y,=lV. To avoid convergence problems solving the resulting system of nonlinear equations, the original expression of the nonlinear capacitor qD( v) = (n + l)C,, vb,[ 1 + < r1 should be modified [4]. It has shown to be convenient to linearize the behaviour for high levels of v. The singularity has also been removed with a linear extension of qD(v) for low levels of v. Thus the non-linear charge of capacitor CO is defined in (7). qdv) = %V+bd [d%, ' 44 (_('I =(n+l)c,OVb,[l+$~' l'dVh, < dt) < LVb, (13) ,AV) = ""V+4, /uvbi < dt) We consider lu=40 and ld=-0.85 the points of linealizing by tangent straight line. The parameters of these lines are defined in (1 4). I I I NI",, =O (12) 897 70 . 60 . F 50 - - Q40 - 30 - 20 - 10. b The result of v(t/ with a discretization of Gear 2 with 20 samples is showed in figure, 2. This result agrees with Pspice with 60 samples. 0 10 20 30 40 50 60 70 80 90 100 I [1190 “$1 Fzgure 2. Waveform of vQ). Pspice( ). Gear-2 with D22(0). .Gear-Z with D21(A). The result of v(t) with more samples is very accurate and the error isn’t appreciate at the chart. 111. CONCLUSIONS determine the steady ear switched circuits e method is based on scretization of the be the circuit [I], transforming * -the initial problem, the solution of a nonlinear difference differential system of equations, into the solution 06 a nonlinear algebraic system of equations, where the only unknowns to be determined are the samples of the control variables. In’ switched circuits, it is necessary to preserve state variable continuity at each switching instant. So, an exact anhytic relation between the unkriowns ,and the state variables of the circuit has’ been described in detail. To validate the method, it has been applied to the determination of the steady state response of the class-E amplifier, an example of the-kind of circuits to which this paper‘ refers. The results coincide with those obtained using integration techniques, having to compute the response until the transient dies out. .- I 3 .- -IV. REFERENCES [ 11 J. Bonet-Dalmau and P. Pala-Schonwalder, “A Discrete-Time Approach to the Steady-State and Stability Analysis of Distributed Nonlinear Autonomous -Circuits” IEEE Trans. Circuits Systems. I, vol47, pp.23 1-235, 2000. [2] J. E. Dennis and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Englewood Cliffs, NJ, Prentice Hall (1983). [3] P. Alinikula, K. Choi and S. I. Long, “Design of Class E Power Amplifier with Nonlinear Parasitic Output Capacitance”, IEEE Transactions on Circuits and Systems, 46,2, 114-1 19 (1999). [4] P. J. C. Rodrigues, Co~~~ter-~i~e~~~a~sis of Nonlinear Microw&e Circuits, Artech House (1998).