RF HB ) HB. MATLAB Spice

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1 THE INSTITUTE OF ELETONIS, INFOMATION AND OMMUNIATION ENGINEES TEHNIAL EPOT OF IEIE. MATLAB SPIE {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, JAPAN Faculty of Eng., Tokushima University - Minami-josanjima, Tokushima, JAPAN {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

2 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 = V d HB) 3 3 [5-8] V d [9,] [-4] V d V d =. 3 3 HB 5 7 Spice ABM (V d =.) [5-] 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

3 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 ) () () 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 = 7. (7) (4) Spice HB 3

4 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] ω ω ω (Volterra) ω ω ω ω 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) ω ω [x rad/s] (d) 4.. Steady-state waveform V =I V =I 4 L Sine-osine -. L Sine-osine [µs] Spice D (e) ω = 7 [rad/sec], 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 (e) MATLAB.4.3 ω ω 5(c) HB 4 4 V B < 3 HB I S exp(v B /V T ) = 4

5 V v L out 4 ω ω (ω = 7 [rad/s], ω = 8. [rad/s]) HB V = V ω ω ω ω ω ω 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 (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] Output waveform [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 = [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]

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