THE INSTITUTE OF ELETONIS, INFOMATION AND OMMUNIATION ENGINEES TEHNIAL EPOT OF IEIE. MATLAB SPIE 34- - E-mail: {kawata,ushida}@fe.bunri-u.ac.jp, {yamagami,nishio}@ee.tokushima-u.ac.jp F HB ) HB Spice MATLAB HB MATLAB Spice Spice-oriented frequency-domain intermodulation analysis combining with MATLAB omparisons between Volterra series and harmonic balance methods Junji KAWATA, Yoshihiro YAMAGAMI, Yoshifumi NISHIO, and Akio USHIDA Faculty of Eng., Tokushima Bunri University 34- Shido, Sanuki, 79-93 JAPAN Faculty of Eng., Tokushima University - Minami-josanjima, Tokushima, 77-85 JAPAN E-mail: {kawata,ushida}@fe.bunri-u.ac.jp, {yamagami,nishio}@ee.tokushima-u.ac.jp Abstract It is very important to analyze F circuits, mixers and modulators driven by multiple input frequencies. Volterra series methods are widely used for the frequency-domain analysis of nonlinear circuits, because they give the solutions in analytical forms. On the other hand, HB (harmonic balance) method is well-known that can be applied even for relatively strong nonlinear circuits. In this article, we propose a Spice-oriented HB method combining with MATLAB. Firstly, for the nonlinear devices modeled by the special functions, their characteristics are approximated by the Taylor series, and the Fourier coefficients to the input and output relations can be calculated by MATLAB in the symbolic forms. Thus, the determining equation of HB method can be formulated by the equivalent circuit and/or net-list. It can be efficiently solved by the D analysis of Spice. Thus, the frequency-domain solutions such as frequency response curves can be easily obtained by the D sweep. We found from the examples that, although Volterra series method can be efficiently applied to weakly nonlinear circuits, it becomes erroneous for strong nonlinear circuits. On the other hand, our HB method can be stably applied to relatively strong nonlinear circuits. Key words Volterra series method, harmonic balance method, MATLAB, Spice modulator and mixer circuit
i. d = I S exp v d /V T, I S = [A], V T = 5. () v d I v d = V d V d cos ωt, V d =.58 () i d 3 D, cos ωt, cos ωt, [-3] cos 3ωt i d = k k v d k vd k 3 vd, 3 (3) k = 3, k =.599, k =.39, k 3 = 38. 3 V d HB) 3 3 [5-8] V d 3 3 5 3 3 5 3 5 95 43 79 55 59 39 3 43 7 538 859 49 9 383 7 55 9.4.54.4 87 47.5 5 347 [9,] [-4] V d V d =. 3 3 HB 5 7 Spice ABM (V d =.) 3.459.538.7 87 [5-] 3 4.559 59 347 5..5. 9 7.397.483.559 83 MATLAB V d =. 7 [7] Spice HB HB. Spice [3] v in(ω t) Linear v G v(t) v (ω t) - (3) 3 s 3. (a) r = /k [3]. U (s), U (s) H (s), H (s) H k (s) = H (s) H (s) (4) (i d, v d ) 3 [4] in 3 i G
H (s), H (s) v in (ω t), v in (ωt) I N (±jω, ±jω ) = k ((H (±jω ) H (±jω )) ((H (±jω ) H (±jω )) (3) (4) = k [H (jω )H (jω ) H (jω )H (jω ) I N H (jω )H (jω ) H (jω )H (jω ) I N (s, s ) = k H k (s )H k (s ) (5) r Y (s) ω 3 ω 4 H k (s, s ) Y (s s )H k (s, s ) = I N (s, s ) () () 3 3 3 3 I 3N (s, s, s 3) = k 3H k (s )H k (s )H k (s 3) 3 k [H k(s ) (7) H k (s, s 3 ) H k (s )H k (s, s 3 ) H k (s 3 )H k (s, s )]. 3 Spice 3 H 3k (s, s, s Spice MATLAB HB) 3) HB Y (s s s 3 )H 3k (s, s, s 3 ) = I 3N (s, s, s 3 ). (8) V BE V B U (s) Linear i i B i E Linear U (s) Linear (a) st order r r H (s) H (s) (b) nd order r I N(s,s ) H (s,s ) k Linear r H 3k(s,s,s 3) I 3N(s,s,s 3) (c) 3rd order 3 H out (s, s, s 3 ) = H k (s ) H k (s, s ) H 3k (s, s, s 3 ). (9) D, ω, ω, ω, ω, 3ω, 3ω ω ω, ω ω, ω ω, ω ω. (4) ω ω, ω ω 3 v in(t) = A sin ω t, v in(t) = A sin ω t () 3 [3] D A H k(jω, jω ) A H k(jω, jω ) ω A H k (jω ) 3 A A H 3k(jω, jω, jω ) 3 4 A3 H 3k(jω, jω, jω ) ω A H k (jω ) 3 A A H 3k (jω, jω, jω ) 3 4 A3 H 3k(jω, jω, jω ) ω ω A A H k (jω, jω ) ω ω A A H k (jω, jω ) ω A H k(jω, jω ) ω A H k(jω, jω ) ω ω 3 4 A AH 3k(jω, jω, jω ) ω ω 3 4 A A H 3k (jω, jω, jω ) ω ω 3 4 A A H 3k(jω, jω, jω ) ω ω 3 4 AA H 3k(jω, jω, jω ) 3ω 4 A3 H 3k(jω, jω, jω ) 3ω 4 A3 H 3k(jω, jω, jω ) s i = {±jω, ±jω }, i =,, 3 () (5) (4) 3 H (jω )H (jω ) H (jω )H (jω )] () 5 ω ω ω ω MATLAB [7] i = I S (α F exp(v BE /V T ) exp(v B /V T )) i B = I S ((α F ) exp(v BE /V T )(α ) exp(v B /V T )) i E = I S (exp(v BE /V T ) α exp(v B /V T )) (3) I S = [A], V T =, α F =.99, α =.3 () (3) HB HB HB V d 3 i d = k (V d ) k (V d )v d k (V d )v d k 3 (V d )v 3 d (5) (4) K v d (t)=v d (V d,k cos ν k tv d,k sin ν k t) k=, () K i d (t)=i d {I d,k cos ν k ti d,k sin ν k t) k= () (5) i d MATLAB 5 5 5 3 = 7. (7) (4) Spice HB 3
i d v - d (a) Vd Vdcosν t Vd sinνt V dk sinνkt Diode HB model (b) Id I cosν d t I d sinνt I dk sinνkt 3 (a) (b) HB L i = v v L = L i L dt, dt, v = i, v = V cos ν k t V sin ν k t i L = I cos ν k t I sin ν k t i = I cos ν k t I sin ν k t V L = ν k LI, V L = ν k LI V = I, V = I (8) Sine-osine [5,] I = ν k V, I = ν k V (9) e (t) e (t) (a) Q L Vout E V (jω ) V (jω ) α F I d I d g d (b) L V out - 5 (a) (b) = 5[kΩ], = [kω], =.[kω] = [nf], = [nf], = [nf], L = [µf], E = v = sin ω t, v = sin ω t, for ω = 8. [rad/sec].5..5..5 ω ω ω (Volterra) ω ω ω ω 5 5 5 3 ω ω (c) [x rad/s] (c). ω osine Sine il I I. ω Inductor v L L V L V L V L= ν k LI V L = ν k LI.8.4 ω ω ω ω ω apacitor esistor v v i i I I = ν k V V V V I I I = ν k I V V (d) 5 5 5 3 ω ω [x rad/s] (d) 4.. Steady-state waveform V =I V =I 4 L Sine-osine -. L Sine-osine -4. 4 8 [µs] Spice D (e) ω = 7 [rad/sec],. 4.4. HB 5(a) 3 i [8] B = I B, (V B, )I B, (V B, )v B I B, (V B, )v B 3 I B,3(V B, )v 3 B () i BE = I BE, (V BE, )I BE, (V BE, )v BE 3 I BE, (V BE, )v BE I BE,3(V BE, )v 3 BE.4. HB 5(a) 5(d) v d =.58 3 HB i d = 3.5v d.4v d 38.v 3 d. () ω = 8. [rad/sec] ω 4 HB 78.38[sec] k =.5(= g d ), k =.4, k 3 = 38. () L.3 3 3 5(e) MATLAB.4.3 ω ω 5(c) HB 4 4 V B < 3 HB I S exp(v B /V T ) = 4
V v L out 4 ω ω (ω = 7 [rad/s], ω = 8. [rad/s]) HB V = V ω ω ω ω ω ω.95.93.99.9.93.98..4.7.3.8.9 3 3.9.3.5.9.7.7 4.5 3.35 3.35.59.5. 5 9.8 4.55 4.3 3.34.75.38 V V (a) = [kω], = [Ω], L = [µh], (3) V =, V =, V = 5 V = V = ω. ω ω HB.4 [x]. ω 7 (b) [x rad/s] (b) = [pf], v 3 = sin ω t, v =. sin 57 t. HB HB 3. -. 3 (a) -3. [8] (c) v (t) v (t) (c) v (t) = sin 5 t, 4 L (3) 3 ω.8. v (t) - Q Q Q 3 ω [ω =57x rad/s] v (t) ω ω [x] 3 4 5 7 8 9 3... -. Output waveform 3 4 5 [µsec] ω=.[x rad/s] v (t) =. sin 57 t i d = I S e λv d, for I S = [A], λ = 4, α =.99 (3) ω ω [x] ω ω [x] i d k (v d )k (v d )v k (v d )v k 3(v d )v 3, v d = v d v (4) (d) ω [x rad/s] (4) 3 MAT- (d) = [nf], v LAB = sin ω t, v =. sin. t. HB L 4. Sine-osine 4. Spice (b) ω = 57 -. [rad/s] ω ω = 5 [rad/s] -4. (c) 4 8 (e) [µsec] HB (e) v (t) = sin 7 t, v (t) =. sin. t V out, =.799 (HB) V =.83 (Tran. DFT) (a) V out, =.93 (HB) V =.9585 (Trans. DFT) L 7 [rad/s] ω ω = 5 [rad/s], ω = 57 [rad/s] ω 7 [rad/s] (d) (ω ω ) HB ( ω ω ) ω = ω = 57 [rad/s] ω ω 5[s] ω [x] 4 8 4 8 5
V out, = 3.987 (HB) V = 4.8 (Tran. DFT) V out, =.37 (HB) ω = 7 [rad/s], V =.337 (Tran. DFT) ω =. [rad/s] 5 9.3[s] [] K.S.Kundert, J.K.White and A.Sangiovanni-Vincentelle, Steady- State methods for simulating Analog and Microwave ircuits, 4. Kluwe Academic, Pub. 99. Spice MATLAB [3] Y.Yamagami,Y.Nishio, A.Ushida, M.Takahashi and K.Ogawa, HB Analysis of communication circuits based on multidimensional Fourier transformation, IEEE Trans. omput.-aided Des. Interg. ircuits Syst., vol.8, pp.5-77, 999. [4] A.Ushida,T.Adachi and L.O.hua, Steady-state response of HB MATLAB nonlinear circuitsbased on hybrid method, IEEE Trans. on ircuits and Systems-I, vol.39, pp.49-, 99. [5] A.Ushida, Y.Yamagami and Y.Nishio, Frequency responses of nonlinear networks using curve tracing algorithm, ISAS, vol.i, pp.4-44,. [] J.Kawata, Y.Taniguchi, M.Oda,Y.Yamagami and Y.Nishio and A.Ushida Spice-oriented frequencydomain analysis of nonlinear electronic circuits, IEIE Trans. Fundamentals, vol.e9- HB A, pp.4-4, 7. MATLAB [7] MATLAB 4. [8] K.K.larke and D.T.Hess, ommunication ircuits: Analysis 3 and Design, Addison-Wesley Pub. o., 97. [9].Telichevesky and K.Kundert, SpectreF Primer, adence 3 HB MATLAB Design Systems, San Jose, alifornia, 99.. Spice 4 3 4 HB 5. 7 [-3] 3 4 3 HB 5 7 MATLAB [] M.Schetzen, The Volterra and Wiener Theorems of Nonlinear Systems, John Wiley and Sons, 978. [] J.Wood and D.E.oot, Fundamentals of Nonlinear Behavioral Modeling for F and Microwave Design, Artech House, 5. [3] P.Wambacq and W.Sansen, Distortion Analysis of Analog Integrated ircuits, Kluwer Academic Pub., 998. [4] B.J.Leon and D.J.Shaefer, Volterra series and Picard iteration for nonlinear circuits and systems, IEEE Trans. ircuits and Systems, vol.5, pp.789-793, 978. [5].Hayashi, Nonlinear Oscillations in Physical Systems, McGraw-Hill, 94. [] Y.Ueda, The oad to haos-ii, Aerial Press. Inc.,. [7].J.Gilmore and M.B.Steer, Nonlinear circuit analysis using the method of harmonic balance-a review of the Art. Part I. Introductory concepts, Int. Jour. of Microwave and Millimeter- Wave omputer-aided Eng. vol., pp.-37, 99. [8].J.Gilmore and M.B.Steer, Nonlinear circuit analysis using the method of harmonic balance-a review of the Art. Part II. Advanced concepts, Int. Jour. of Microwave and Millimeter- Wave omputer-aided Eng. vol., pp.59-8, 99. [] A.S.Sedra and K..Smith, Microelectronic ircuits, Oxford Univ. Press, 4. [9] 3. [].Telichevesky, K.S.Kundert and J.K.White, Efficient steadystate analysis based on matrix-free Krylov-subspace methods, AM, pp.48-485, 995. HB [5-] 3 Spice 4 5 7 3 HB 3 Spice 4 5 7 4 [-3] 3 FFT 4 5 7 5 [3] 3 FFT 4 5 7 SpectreF[9] 3 4 5 7 5