Definition of Stability Transfer function of a linear time-invariant (LTI) system Fs () = b 2 1 0+ b1s+ b2s + + b m m m 1s - - + bms a0 + a1s+ a2s2 + + an-1sn- 1+ ansn Characteristic equation and poles Stability of Dc-to-Dc Converters a0 + a1s+ a2s2 + + an-1sn-1+ ansn = 0: characteristic equation Roots of the characteristic equation: poles, natural modes, or eigenvalues The LTI system is stable if and only if all roots of the characteristic equation are located in the left-half plane(lhp) of the s-plane jω σ 15
Pole Location and Unit pulse sponse jω Stability of Dc-to-Dc Converters σ 16
Stability Assessment using Average Model sponse of average model and switch model Stability of Dc-to-Dc Converters Inductor current (A) The average model always preserves the converter dynamics regardless of the magnitude of external/internal disturbances. However, the average model does not accept the classical stability theory, because it is time-invariant but nonlinear model. 17
Stability Analysis using Small-Signal Model sponse of small-signal model and switch model 10 8 6 switch model Stability of Dc-to-Dc Converters 4 2 0 0.0 0.5 1.0 1.5 Time (ms) The small-signal model preserves the converter dynamics if the disturbance is not too large. The small-signal model accepts all the classical stability theories. The stability defined on the small-signal model is called local stability or small-signal stability. small-signal model 18
Nyquist (Stability) Criterion Nyquist (stability) criterion: graphical method to determine the number of RHP roots in the equation 1+T(s)=0 Polar plot of Ts () Unit circle N = 2 (-1, 0) s=+ j N = 0 Z = N + P s=+ j0 Z: number of RHP roots in 1 + T() s = 0 N: number of encirclements of (-1, 0) point made by the polar plot of T( s) P: number of RHP poles in T() s s=+ j0 19
Application Example Fs () = b0+ b1s+ b2s2+ + bm-1sm- 1+ bmsm a0+ a1s+ a2s2+ + an-1sn- 1+ ansn Characteristic equation: a 2 n 1 n 0 + a1s+ a2s + + an 1s - - + ans = 0 as2 1 2 + + a n n 1 n 1s - - + ans + = 0 a0 + a1s Ts () 1+ Ts () = 0 Polar plot of Ts () Z = N + P Z: number of RHP roots in a 2 0 + a1s+ a2s + + an-1sn - 1 + a n ns = 0 N: number of encirclement of (-1, 0) point made by the polar of T( s) P: number of RHP roots in a0 + a1s= 0 20
Stability Analysis using Small-Signal Model vˆ () s s Gvs( s) + + + vˆ () s o ˆ ı () s o Zp( s) Gvd( s) ds ˆ( ) vˆ con() s F -F () s m v v ˆ o () s Gvs () s 1 Ns () Ns = = with G () () vs s = vˆ s() s 1+ Gvd() s Fv() s Fm 1+ Gvd() s Fv() s Fm D() s D() s Characteristic equation: Ds ()( 1+ Gvd() sfv() sfm) = 0 When Ds () = 0 doesnothave any RHP roots, stability can be assessed by applying Nyquist criterion to 1+ Tm() s = 0 with Tm() s = Gvd() s Fv() s Fm. 21
Nyquist Analysis on 1+T m (s) = 0 Charateristic equation: 1+ Tm() s = 0 Polar plot of Tm() s Unit circle N = 2 (-1, 0) s=+ j N = 0 Nyquist criterion: Z = N + P Z: number of RHP roots in 1 + Tm() s = 0 N: number of encirclements of (-1, 0) point made by polar plot of Tm( s) P: number of RHP poles in Tm() s When P = 0, it becomes that Z = N s=+ j0 s=+ j0 Number of RHP roots in " 1+ Tm( s) = 0" = number of encirclements of (-1, 0) point For stability, the polar plot of Tm() s should not encircle (-1, 0) point. 22
Absolute Stability (-1, 0) (-1, 0) Stable system Unstable system (-1, 0) Marginally stable system (-1, 0) Conditionally stable system 23
Stability Analysis Using Bode Plot (-1,0) (- 1,0) (-1,0) T m 0dB T m 0dB T m 0dB T m T m Tm -180 180 180 - - Stable system Unstable system Marginally stable system 24
Marginally Stable Buck Converter 40 Magnitude (db) 20 0-20 f c = 2kHz -40 0 Phase (deg) -45-90 -135-180 0.1 1 10 Frequency (khz) Bode plot of loop gain Polar plot of loop gain T 2 2 2 103 m( jωc) = Tm( j πfc) = Tm( j π ) = 1-180 =-1 1+ T 2 2 103 m( j π ) = 0 25
Marginally Stable Buck Converter 5.0 4.5 vo ()(V) t 4.0 3.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 Time (ms) 1+ Tm( j2 2 103) = 0 s = j 3 12, 2 2 10 are theroots of the characteristic equation oscillation at the frequency c = 2 2 103 2 2 1 oscillation period: tos = = = = 05. ms c 2 2 103 2 103 26
Conditionally Stable System v O gion B gion A vi Smaller magnitude gion C Loop gain plot Operational region gion A: fast slope of vi-vo curve large gain stable gionb or C: slow slope of vi - vo curve small gain unstable 27
Effect of Gain Increase and Phase Delay lative Stability: Gain Margin and Phase Margin T ( jω) = T ( jω) T ( jω) m m m More phase delay Larger gain Gain increase or phase delay destablizes the system. 28