Chapter 7. AC Equivalent Circuit Modeling
7.1. Introduction 7.2. The basic ac modeling approach 7.3. Example: A nonideal flyback converter 7.4. State-space averaging 7.5. Circuit averaging and averaged switch modeling 7.6. The canonical circuit model 7.7. Modeling the pulse-width modulator 7.8. Summary of key points
Fundamentals of Power Electronics2 Chapter 7: AC equivalent circuit modeling 7.1. Introduction
Objective: maintain v(t) A simple dc-dc regulator system, employing a equal to an accurate, buck converter constant value V. Power Switching converter Load There are input disturbances: +
v (t) + v(t) ¥ in vg(t) g – R feedback connection ¥ in R –
There are transistor
uncertainties: gate driver compensator
δ pulse-width vc – v (t) G (s) + ¥ in element modulator c
δ values (t) vc(t) voltage reference vref ¥ in Vg dTs Ts t t ¥ in R Controller
Fundamentals of Power Electronics3 Chapter 7: AC equivalent circuit modeling Applications of control in power electronics
Dc-dc converters Regulate dc output voltage. Control the duty cycle d(t) such that v(t) accurately follows a reference
signal vref. Dc-ac inverters Regulate an ac output voltage. Control the duty cycle d(t) such that v(t) accurately follows a reference
signal vref (t). Ac-dc rectifiers Regulate the dc output voltage. Regulate the ac input current waveform.
Control the duty cycle d(t) such that ig (t) accurately follows a reference signal iref (t), and v(t) accurately follows a reference signal vref. Fundamentals of Power Electronics4 Chapter 7: AC equivalent circuit modeling Objective of Part II
Develop tools for modeling, analysis, and design of converter control systems Need dynamic models of converters:
How do ac variations in vg(t), R, or d(t) affect the output voltage v(t)? What are the small-signal transfer functions of the converter? ¥ Extend the steady-state converter models of Chapters 2 and 3, to include CCM converter dynamics (Chapter 7) ¥ Construct converter small-signal transfer functions (Chapter 8) ¥ Design converter control systems (Chapter 9) ¥ Model converters operating in DCM (Chapter 10) ¥ Current-programmed control of converters (Chapter 11)
Fundamentals of Power Electronics5 Chapter 7: AC equivalent circuit modeling Modeling
¥ Representation of physical behavior by mathematical means ¥ Model dominant behavior of system, ignore other insignificant phenomena ¥ Simplified model yields physical insight, allowing engineer to design system to operate in specified manner ¥ Approximations neglect small but complicating phenomena ¥ After basic insight has been gained, model can be refined (if it is judged worthwhile to expend the engineering effort to do so), to account for some of the previously neglected phenomena
Fundamentals of Power Electronics6 Chapter 7: AC equivalent circuit modeling Neglecting the switching ripple
Suppose the duty cycle The resulting variations in transistor gate is modulated drive signal and converter output voltage: sinusoidally: gate ω drive d(t)=D+Dm cos mt
where D and Dm are constants, | Dm | << D , and the modulation t ω frequency m is much smaller than the actual waveform v(t) converter switching including ripple ω π frequency s = 2 fs. averaged waveform
t
Fundamentals of Power Electronics7 Chapter 7: AC equivalent circuit modeling Output voltage spectrum with sinusoidal modulation of duty cycle
modulation switching switching spectrum frequency and its frequency and of v(t) harmonics { { harmonics{ sidebands
ω ω m s ω
Contains frequency components at: With small switching ripple, high- ¥ Modulation frequency and its frequency components (switching harmonics harmonics and sidebands) are small. ¥ Switching frequency and its If ripple is neglected, then only low- harmonics frequency components (modulation ¥ Sidebands of switching frequency frequency and harmonics) remain.
Fundamentals of Power Electronics8 Chapter 7: AC equivalent circuit modeling Objective of ac converter modeling
¥ Predict how low-frequency variations in duty cycle induce low- frequency variations in the converter voltages and currents ¥ Ignore the switching ripple ¥ Ignore complicated switching harmonics and sidebands Approach: ¥ Remove switching harmonics by averaging all waveforms over one switching period
Fundamentals of Power Electronics9 Chapter 7: AC equivalent circuit modeling Averaging to remove switching ripple
Average over one switching Note that, in steady-state, period to remove switching ripple: vL(t) =0 Ts
diL(t)T s iC(t) T =0 L = vL(t) s dt Ts
dvC(t) Ts C = iC(t) by inductor volt-second dt Ts balance and capacitor charge where balance.
t + Ts x (t) = 1 x(τ) dτ L T s Ts t
Fundamentals of Power Electronics10 Chapter 7: AC equivalent circuit modeling Nonlinear averaged equations
The averaged voltages and currents are, in general, nonlinear functions of the converter duty cycle, voltages, and currents. Hence, the averaged equations
diL(t) Ts L = vL(t) dt Ts
dvC(t) Ts C = iC(t) dt Ts
constitute a system of nonlinear differential equations. Hence, must linearize by constructing a small-signal converter model.
Fundamentals of Power Electronics11 Chapter 7: AC equivalent circuit modeling Small-signal modeling of the BJT
Nonlinear Ebers-Moll model Linearized small-signal model, active region
C C
β β FiB FiB iB iB B B β RiB rE
E E
Fundamentals of Power Electronics12 Chapter 7: AC equivalent circuit modeling Buck-boost converter: nonlinear static control-to-output characteristic
0 0.5 1 D 0
quiescent operating point Example: linearization at the quiescent –V operating point g linearized function D = 0.5
actual nonlinear V characteristic
Fundamentals of Power Electronics13 Chapter 7: AC equivalent circuit modeling Result of averaged small-signal ac modeling
Small-signal ac equivalent circuit model
V ±V d(t) L g +
1 : D – D' : 1 +
v(t) + g – Id(t) Id(t) C v(t) R
±
buck-boost example
Fundamentals of Power Electronics14 Chapter 7: AC equivalent circuit modeling 7.2. The basic ac modeling approach
Buck-boost converter example
12 + i(t) + v(t) vg(t) – CR L –
Fundamentals of Power Electronics15 Chapter 7: AC equivalent circuit modeling Switch in position 1
+ Inductor voltage and capacitor i(t) current are: v (t) + LCRv(t) di(t) g – v (t)=L = v (t) L dt g – dv(t) v(t) i (t)=C =± C dt R
Small ripple approximation: replace waveforms with their low-frequency averaged values:
di(t) ≈ vL(t)=L vg(t) dt Ts
v(t) dv(t) T i (t)=C ≈ ± s C dt R
Fundamentals of Power Electronics16 Chapter 7: AC equivalent circuit modeling Switch in position 2
+ Inductor voltage and capacitor i(t) current are: v (t) + LCRv(t) di(t) g – v (t)=L = v(t) L dt – dv(t) v(t) i (t)=C =±i(t)± C dt R
Small ripple approximation: replace waveforms with their low-frequency averaged values:
di(t)≈ vL(t)=L v(t) dt Ts v(t) dv(t) ≈ Ts iC(t)=C ± i(t) ± dt Ts R
Fundamentals of Power Electronics17 Chapter 7: AC equivalent circuit modeling 7.2.1 Averaging the inductor waveforms
Inductor voltage waveform v (t) L vg(t) Ts Low-frequency average is v (t) v (t) v(t) L T = d g + d' T found by evaluation of s Ts s 0 t + Ts dT T t x (t) = 1 x(τ)dτ s s L T s Ts t v(t) T Average the inductor voltage s in this manner:
t + Ts v (t) = 1 v (τ)dτ ≈ d(t) v (t) + d'(t) v(t) L T L g T T s Ts t s s Insert into Eq. (7.2): di(t) This equation describes how Ts the low-frequency components L = d(t) vg(t) + d'(t) v(t) T dt Ts s of the inductor waveforms evolve in time. Fundamentals of Power Electronics18 Chapter 7: AC equivalent circuit modeling 7.2.2 Discussion of the averaging approximation
Use of the average inductor voltage vL(t) v (t) g T allows us to determine the net change s v (t) v (t) v(t) L T = d g + d' T in inductor current over one switching s Ts s period, while neglecting the switching 0 dTs Ts t ripple. v(t) In steady-state, the average inductor Ts i(t) voltage is zero (volt-second balance), i(dT ) and hence the inductor current s
waveform is periodic: i(t + T ) = i(t). i(0) vg v i(T ) s Ts Ts s There is no net change in inductor L L current over one switching period. t During transients or ac variations, the 0 dTs Ts average inductor voltage is not zero in Inductor voltage and current general, and this leads to net variations waveforms in inductor current.
Fundamentals of Power Electronics19 Chapter 7: AC equivalent circuit modeling Net change in inductor current is correctly predicted by the average inductor voltage
Inductor equation: di(t) L = v (t) dt L Divide by L and integrate over one switching period:
t + T t + T s 1 s τ τ di = vL( )d t L t Left-hand side is the change in inductor current. Right-hand side can
be related to average inductor voltage by multiplying and dividing by Ts as follows: 1 i(t + T s)±i(t)= Ts vL(t) L Ts
So the net change in inductor current over one switching period is 〈 〉 exactly equal to the period Ts multiplied by the average slope vL Ts /L.
Fundamentals of Power Electronics20 Chapter 7: AC equivalent circuit modeling Average inductor voltage correctly predicts average slope of iL(t)
Actual waveform, Averaged waveform including ripple i(t) i(t) Ts vg(t) v(t) L L
i(0) i(Ts)
dvg(t) +d'v(t) Ts Ts L
0 dTs Ts t
The net change in inductor current over one switching period is exactly 〈 〉 equal to the period Ts multiplied by the average slope vL Ts /L.
Fundamentals of Power Electronics21 Chapter 7: AC equivalent circuit modeling di(t) Ts dt
We have 1 i(t + T s)±i(t)= Ts vL(t) L Ts Rearrange:
i(t + Ts) ± i(t) L = vL(t) T Ts s 〈 〉 Define the derivative of i Ts as (Euler formula): di(t) T i(t + T )±i(t) s = s dt Ts
Hence, di(t) Ts L = vL(t) dt Ts
Fundamentals of Power Electronics22 Chapter 7: AC equivalent circuit modeling Computing how the inductor current changes over one switching period
i(t) i(dT ) Let’s compute the actual s
inductor current waveform, i(0) vg v i(T ) Ts Ts s using the linear ripple L L approximation.
0 dTs Ts t
vg(t) i(dT ) = i(0) + dT Ts With switch in s s L position 1: (final value) = (initial value) + (length of interval) (average slope)
v(t) With switch in Ts i(Ts) = i(dTs) + d'Ts position 2: L (final value) = (initial value) + (length of interval) (average slope)
Fundamentals of Power Electronics23 Chapter 7: AC equivalent circuit modeling Net change in inductor current over one switching period
Eliminate i(dTs), to express i(Ts) Ts directly as a function of i(0): i(Ts)=i(0) + d(t) vg(t) + d'(t) v(t) T L Ts s
vL(t) Ts
The intermediate step of Actual waveform, Averaged waveform computing i(dTs) is eliminated. including ripple i(t) i(t) Ts The final value i(Ts) is equal to vg(t) v(t) the initial value i(0), plus the L L switching period Ts multiplied 〈 〉 i(0) i(Ts) by the average slope vL Ts /L. dvg(t) +d'v(t) Ts Ts L
0 dTs Ts t
Fundamentals of Power Electronics24 Chapter 7: AC equivalent circuit modeling 7.2.3 Averaging the capacitor waveforms
Average capacitor current: v(t) T iC(t) ± s R v(t) v(t) Ts Ts iC(t) = d(t)± +d'(t)±i(t) ± iC(t) T Ts R Ts R s 0 dTs Ts t
Collect terms, and equate to C d〈 v 〉 /dt: v(t) Ts ± Ts ± i(t) R Ts v(t) dv(t) v(t) t C Ts =±d'(t) i(t) ± Ts 0 dt Ts R dTs Ts
v(dTs) v(t) Ts v(0) v(Ts) v(t) T v(t) T i(t) T ± s ± s ± s RC RC C
Capacitor voltage and current waveforms
Fundamentals of Power Electronics25 Chapter 7: AC equivalent circuit modeling 7.2.4 The average input current
We found in Chapter 3 that it was ig(t) i(t) Ts sometimes necessary to write an equation for the average converter input ig(t) current, to derive a complete dc Ts equivalent circuit model. It is likewise 0 necessary to do this for the ac model. 0 0 dTs Ts t Buck-boost input current waveform is Converter input current i(t) during subinterval 1 Ts waveform ig(t) = 0 during subinterval 2
Average value:
ig(t) = d(t) i(t) Ts Ts
Fundamentals of Power Electronics26 Chapter 7: AC equivalent circuit modeling 7.2.5. Perturbation and linearization
Converter averaged equations:
di(t) Ts L = d(t) vg(t) + d'(t) v(t) dt Ts Ts dv(t) v(t) C Ts =±d'(t) i(t) ± Ts dt Ts R
ig(t) =d(t) i(t) Ts Ts
—nonlinear because of multiplication of the time-varying quantity d(t) with other time-varying quantities such as i(t) and v(t).
Fundamentals of Power Electronics27 Chapter 7: AC equivalent circuit modeling Construct small-signal model: Linearize about quiescent operating point
If the converter is driven with some steady-state, or quiescent, inputs d(t)=D
vg(t) =Vg Ts then, from the analysis of Chapter 2, after transients have subsided the inductor current, capacitor voltage, and input current
i(t) , v(t) , ig(t) Ts Ts Ts
reach the quiescent values I, V, and Ig, given by the steady-state analysis as V =± D V D' g I=± V D'R
Ig =DI
Fundamentals of Power Electronics28 Chapter 7: AC equivalent circuit modeling Perturbation
So let us assume that the input voltage and duty cycle are equal to some given (dc) quiescent values, plus superimposed small ac variations:
vg(t) = Vg + vg(t) Ts d(t)=D+d(t)
In response, and after any transients have subsided, the converter dependent voltages and currents will be equal to the corresponding quiescent values, plus small ac variations:
i(t) = I + i(t) Ts v(t) = V + v(t) Ts
ig(t) = Ig + ig(t) Ts
Fundamentals of Power Electronics29 Chapter 7: AC equivalent circuit modeling The small-signal assumption
If the ac variations are much smaller in magnitude than the respective quiescent values,
vg(t) << Vg d(t) << D i(t) << I v(t) << V
ig(t) << Ig
then the nonlinear converter equations can be linearized.
Fundamentals of Power Electronics30 Chapter 7: AC equivalent circuit modeling Perturbation of inductor equation
Insert the perturbed expressions into the inductor differential equation:
dI+i(t) L = D + d(t) V + v (t) + D' ± d(t) V + v(t) dt g g note that d’(t) is given by
d'(t)= 1±d(t) =1± D+d(t) =D'±d(t) with D’ = 1 – D
Multiply out and collect terms:
➚0 di(t) L dI + = DV + D'V + Dv (t)+D'v(t)+ V ±V d(t) + d(t) v(t)±v(t) dt dt g g g g
Dc terms 1 st order ac terms 2 nd order ac terms (linear)(nonlinear)
Fundamentals of Power Electronics31 Chapter 7: AC equivalent circuit modeling The perturbed inductor equation
➚0 di(t) L dI + = DV + D'V + Dv (t)+D'v(t)+ V ±V d(t) + d(t) v(t)±v(t) dt dt g g g g
Dc terms 1st order ac terms 2nd order ac terms (linear)(nonlinear) Since I is a constant (dc) term, its derivative is zero The right-hand side contains three types of terms: ¥ Dc terms, containing only dc quantities ¥ First-order ac terms, containing a single ac quantity, usually multiplied by a constant coefficient such as a dc term. These are linear functions of the ac variations ¥ Second-order ac terms, containing products of ac quantities. These are nonlinear, because they involve multiplication of ac quantities
Fundamentals of Power Electronics32 Chapter 7: AC equivalent circuit modeling Neglect of second-order terms
➚0 di(t) L dI + = DV + D'V + Dv (t)+D'v(t)+ V ±V d(t) + d(t) v(t)±v(t) dt dt g g g g
Dc terms 1st order ac terms 2nd order ac terms (linear)(nonlinear)
Providedvg(t) << Vg then the second-order ac terms are much d(t) << D smaller than the first-order terms. For example, i(t) << I d(t) vg(t) << Dvg(t) when d(t) << D v(t) << V
ig(t) << Ig So neglect second-order terms. Also, dc terms on each side of equation are equal.
Fundamentals of Power Electronics33 Chapter 7: AC equivalent circuit modeling Linearized inductor equation
Upon discarding second-order terms, and removing dc terms (which add to zero), we are left with
di(t) L = Dv (t)+D'v(t)+ V ±V d(t) dt g g
This is the desired result: a linearized equation which describes small- signal ac variations.
Note that the quiescent values D, D’, V, Vg, are treated as given constants in the equation.
Fundamentals of Power Electronics34 Chapter 7: AC equivalent circuit modeling Capacitor equation
Perturbation leads to dV+v(t) V+v(t) C =± D'±d(t) I+i(t) ± dt R Collect terms:
➚0 dv(t) v(t) C dV + =±D'I±V +±D'i(t)± +Id(t) + d(t)i(t) dt dt R R
Dc terms 1st order ac terms 2nd order ac term (linear)(nonlinear) Neglect second-order terms. Dc terms on both sides of equation are equal. The following terms remain: dv(t) v(t) C =±D'i(t)± +Id(t) dt R This is the desired small-signal linearized capacitor equation.
Fundamentals of Power Electronics35 Chapter 7: AC equivalent circuit modeling Average input current
Perturbation leads to
Ig + i g(t)= D+d(t) I+i(t)
Collect terms:
Ig + ig(t) = DI + Di(t)+Id(t) + d(t)i(t)
Dc term 1st order ac term Dc term 1st order ac terms 2nd order ac term (linear)(nonlinear) Neglect second-order terms. Dc terms on both sides of equation are equal. The following first-order terms remain:
ig(t)=Di(t)+Id(t) This is the linearized small-signal equation which described the converter input port.
Fundamentals of Power Electronics36 Chapter 7: AC equivalent circuit modeling 7.2.6. Construction of small-signal equivalent circuit model
The linearized small-signal converter equations:
di(t) L = Dv (t)+D'v(t)+ V ±V d(t) dt g g dv(t) v(t) C =±D'i(t)± +Id(t) dt R
ig(t)=Di(t)+Id(t)
Reconstruct equivalent circuit corresponding to these equations, in manner similar to the process used in Chapter 3.
Fundamentals of Power Electronics37 Chapter 7: AC equivalent circuit modeling Inductor loop equation
di(t) L = Dv (t)+D'v(t)+ V ±V d(t) dt g g
V ±V d(t) L g + – +–di(t) L dt + D vg(t) + D'v(t) – – i(t)
Fundamentals of Power Electronics38 Chapter 7: AC equivalent circuit modeling Capacitor node equation
dv(t) v(t) C =±D'i(t)± +Id(t) dt R
dv(t) v(t) C + dt R D' i(t) Id(t) C v(t) R
±
Fundamentals of Power Electronics39 Chapter 7: AC equivalent circuit modeling Input port node equation
ig(t)=Di(t)+Id(t)
ig(t)
v(t) + g – Id(t) D i(t)
Fundamentals of Power Electronics40 Chapter 7: AC equivalent circuit modeling Complete equivalent circuit
Collect the three circuits: Vg±V d(t) i(t) L + – +
+ + vg(t) Id(t) D i(t) Dvg(t) D'v(t) + D'i(t) Id(t) C v(t) R – – – ±
Replace dependent sources with ideal dc transformers: V ±V d(t) L g +
1 : D – D' : 1 +
v(t) + g – Id(t) Id(t) C v(t) R
±
Small-signal ac equivalent circuit model of the buck-boost converter
Fundamentals of Power Electronics41 Chapter 7: AC equivalent circuit modeling 7.2.7. Results for several basic converters
V d(t) buck g L
1 : D + – i(t) +
v(t) + g – Id(t) C R v(t)
±
L Vd(t) boost D' : 1 + – i(t) +
v(t)+ Id(t) g– CRv(t)
±
Fundamentals of Power Electronics42 Chapter 7: AC equivalent circuit modeling Results for several basic converters
buck-boost
V ±V d(t) L g +
1 : D – D' : 1 i(t) +
v(t) + g – Id(t) Id(t) C v(t) R
±
Fundamentals of Power Electronics43 Chapter 7: AC equivalent circuit modeling 7.3. Example: a nonideal flyback converter
Flyback converter example
D 1 : n 1 +¥ MOSFET has on- ig(t) resistance Ron L CRv(t) ¥ Flyback transformer v (t) + has magnetizing g – – inductance L, referred to primary
Q1
Fundamentals of Power Electronics44 Chapter 7: AC equivalent circuit modeling Circuits during subintervals 1 and 2
Flyback converter, with
transformer equivalent transformer model circuit + ig i i Subinterval 1 + 1:n C i (t) D1 g L v C R v v + L + g – i(t) + 1 : n iC(t) – – CRv(t) L vL(t) + vg(t) – ideal – – Ron
Q 1 transformer model i/n + ig i i + 1:n C Subinterval=0 2– v/n C R v v + vL g – + – –
Fundamentals of Power Electronics45 Chapter 7: AC equivalent circuit modeling Subinterval 1
Circuit equations: transformer model + vL(t)=vg(t)±i(t)Ron i g i + 1:n iC v(t) i (t)=± C L v C R v R v + L g – ig(t)=i(t) – – Small ripple approximation:
Ron vL(t)= vg(t) ± i(t) Ron Ts Ts v(t) MOSFET conducts, diode is i (t)=± Ts C R reverse-biased
ig(t)= i(t) Ts
Fundamentals of Power Electronics46 Chapter 7: AC equivalent circuit modeling Subinterval 2
Circuit equations: transformer model i/n v(t) vL(t)=± + n ig i i + 1:n C i(t) v(t) =0 – iC(t)=± ± n R C R v + vL v/n vg ig(t)=0 – + – – Small ripple approximation: v(t) Ts vL(t)=± n MOSFET is off, diode i(t) v(t) i (t)=± Ts ± Ts conducts C n R
ig(t)=0
Fundamentals of Power Electronics47 Chapter 7: AC equivalent circuit modeling Inductor waveforms
vg(t) ± Ron i(t) i(t) T Ts vL(t) s vg – iRon L i(t) Ts vL(t) Ts ± v(t) 0 Ts t dTs Ts nL
– v/n 0 dTs Ts t Average inductor voltage: ± v(t) Ts vL(t) = d(t) vg(t) ± i(t) Ron + d'(t) Ts Ts Ts n Hence, we can write: di(t) v(t) Ts Ts L = d(t) vg(t) ± d(t) i(t) T Ron ± d'(t) dt Ts s n
Fundamentals of Power Electronics48 Chapter 7: AC equivalent circuit modeling Capacitor waveforms
i(t) v(t) i (t) v(t) T T C i ± v s ± s n R nC RC
iC(t) Ts v(t) T 0 v(t) s dT T t T s s ± s RC – v/R 0 dTs Ts t Average capacitor current: ± v(t) i(t) v(t) Ts Ts Ts iC(t) = d(t) + d'(t) ± Ts R n R Hence, we can write: dv(t) i(t) v(t) C Ts = d'(t) Ts ± Ts dt n R
Fundamentals of Power Electronics49 Chapter 7: AC equivalent circuit modeling Input current waveform
ig(t) i(t) Ts
ig(t) Ts
0 0 0 dTs Ts t
Average input current:
ig(t) = d(t) i(t) Ts Ts
Fundamentals of Power Electronics50 Chapter 7: AC equivalent circuit modeling The averaged converter equations
di(t) v(t) Ts Ts L = d(t) vg(t) ± d(t) i(t) T Ron ± d'(t) dt Ts s n dv(t) i(t) v(t) C Ts = d'(t) Ts ± Ts dt n R
ig(t) = d(t) i(t) Ts Ts — a system of nonlinear differential equations Next step: perturbation and linearization. Let
vg(t) = Vg + vg(t) i(t) T =I+i(t) Ts s d(t)=D+d(t) v(t) =V+v(t) Ts
ig(t) =Ig +ig(t) Ts
Fundamentals of Power Electronics51 Chapter 7: AC equivalent circuit modeling Perturbation of the averaged inductor equation
di(t) v(t) Ts Ts L = d(t) vg(t) ± d(t) i(t) T Ron ± d'(t) dt Ts s n
dI+i(t) V + v(t) L = D + d(t) V + v (t) ± D' ± d(t) ± D + d(t) I + i(t) R dt g g n on
0 ➚ di(t) v(t) L dI + = DV ± D'V ± DR I + Dv (t)±D' + V +V ±IR d(t)±DR i(t) dt dt g n on g n g n on on
Dc terms 1st order ac terms (linear) v(t) + d(t)vg(t)+d(t) n ±d(t)i(t)Ron
2nd order ac terms (nonlinear)
Fundamentals of Power Electronics52 Chapter 7: AC equivalent circuit modeling Linearization of averaged inductor equation
Dc terms: V 0=DVg± D' n ± DRonI Second-order terms are small when the small-signal assumption is satisfied. The remaining first-order terms are:
di(t) v(t) L = Dv (t)±D' + V +V ±IR d(t)±DR i(t) dt g n g n on on
This is the desired linearized inductor equation.
Fundamentals of Power Electronics53 Chapter 7: AC equivalent circuit modeling Perturbation of averaged capacitor equation
Original averaged equation: dv(t) i(t) v(t) C Ts = d'(t) Ts ± Ts dt n R
Perturb about quiescent operating point:
dV+v(t) I + i(t) V + v(t) C = D' ± d(t) ± dt n R
Collect terms:
➚0 dv(t) D'i(t) v(t) Id(t) d(t)i(t) C dV + = D'I ± V + ± ± ± dt dt n R n R n n
Dc terms 1st order ac terms 2 nd order ac term (linear)(nonlinear)
Fundamentals of Power Electronics54 Chapter 7: AC equivalent circuit modeling Linearization of averaged capacitor equation
Dc terms:
0= D'I ±V n R Second-order terms are small when the small-signal assumption is satisfied. The remaining first-order terms are:
dv(t) D'i(t) v(t) Id(t) C = ± ± dt n R n
This is the desired linearized capacitor equation.
Fundamentals of Power Electronics55 Chapter 7: AC equivalent circuit modeling Perturbation of averaged input current equation
Original averaged equation:
ig(t) = d(t) i(t) Ts Ts
Perturb about quiescent operating point:
Ig + ig(t)= D+d(t) I+i(t)
Collect terms:
Ig + ig(t) = DI + Di(t)+Id(t) + d(t)i(t)
Dc term 1 st order ac term Dc term 1 st order ac terms 2 nd order ac term (linear)(nonlinear)
Fundamentals of Power Electronics56 Chapter 7: AC equivalent circuit modeling Linearization of averaged input current equation
Dc terms:
Ig = DI
Second-order terms are small when the small-signal assumption is satisfied. The remaining first-order terms are:
ig(t)=Di(t)+Id(t)
This is the desired linearized input current equation.
Fundamentals of Power Electronics57 Chapter 7: AC equivalent circuit modeling Summary: dc and small-signal ac converter equations
Dc equations: V 0=DVg± D' n ± DRonI 0= D'I ±V n R
Ig =DI Small-signal ac equations:
di(t) v(t) L = Dv (t)±D' + V +V ±IR d(t)±DR i(t) dt g n g n on on dv(t) D'i(t) v(t) Id(t) C = ± ± dt n R n
ig(t)=Di(t)+Id(t)
Next step: construct equivalent circuit models.
Fundamentals of Power Electronics58 Chapter 7: AC equivalent circuit modeling Small-signal ac equivalent circuit: inductor loop
di(t) v(t) L = Dv (t)±D' + V +V ±IR d(t)±DR i(t) dt g n g n on on
V d(t) Vg ± IRon + L DRon n + – +–di(t) L dt D v (t) + + D' v(t) g – – n i(t)
Fundamentals of Power Electronics59 Chapter 7: AC equivalent circuit modeling Small-signal ac equivalent circuit: capacitor node
dv(t) D'i(t) v(t) Id(t) C = ± ± dt n R n
dv(t) v(t) C + dt R D' i(t) Id(t) n n C v(t) R
±
Fundamentals of Power Electronics60 Chapter 7: AC equivalent circuit modeling Small-signal ac equivalent circuit: converter input node
ig(t)=Di(t)+Id(t)
ig(t)
v(t) + g – Id(t) D i(t)
Fundamentals of Power Electronics61 Chapter 7: AC equivalent circuit modeling Small-signal ac model, nonideal flyback converter example
Combine circuits: d(t) V ±IR + V L DR g on n ig(t) on + – i(t) +
D'v(t) v(t) + Id(t) D i(t) + D v (t) + D' i(t) Id(t) C v(t) g – – g n – n n R
±
Replace dependent sources with ideal transformers:
d(t) V ±IR + V L g on n ig(t)
1 : D + – D' : n i(t) + DRon
v(t) + Id(t) Id(t) C v(t) g – n R
±
Fundamentals of Power Electronics62 Chapter 7: AC equivalent circuit modeling 7.4. State Space Averaging
¥ A formal method for deriving the small-signal ac equations of a switching converter ¥ Equivalent to the modeling method of the previous sections ¥ Uses the state-space matrix description of linear circuits ¥ Often cited in the literature ¥ A general approach: if the state equations of the converter can be written for each subinterval, then the small-signal averaged model can always be derived ¥ Computer programs exist which utilize the state-space averaging method
Fundamentals of Power Electronics63 Chapter 7: AC equivalent circuit modeling 7.4.1. The state equations of a network
¥ A canonical form for writing the differential equations of a system ¥ If the system is linear, then the derivatives of the state variables are expressed as linear combinations of the system independent inputs and state variables themselves ¥ The physical state variables of a system are usually associated with the storage of energy ¥ For a typical converter circuit, the physical state variables are the inductor currents and capacitor voltages ¥ Other typical physical state variables: position and velocity of a motor shaft ¥ At a given point in time, the values of the state variables depend on the previous history of the system, rather than the present values of the system inputs ¥ To solve the differential equations of a system, the initial values of the state variables must be specified
Fundamentals of Power Electronics64 Chapter 7: AC equivalent circuit modeling State equations of a linear system, in matrix form
dx(t) A canonical matrix form: K = Ax(t)+Bu(t) dt y(t)=Cx(t)+Eu(t) dx (t) State vector x(t) contains 1 x(t) dt inductor currents, capacitor 1 dx(t) dx (t) x(t)= x(t) , = 2 voltages, etc.: 2 dt dt
Input vector u(t) contains independent sources such as vg(t) Output vector y(t) contains other dependent quantities to be computed, such
as ig(t) Matrix K contains values of capacitance, inductance, and mutual inductance, so that K dx/dt is a vector containing capacitor currents and inductor winding voltages. These quantities are expressed as linear combinations of the independent inputs and state variables. The matrices A, B, C, and E contain the constants of proportionality. Fundamentals of Power Electronics65 Chapter 7: AC equivalent circuit modeling Example
i(t) L State vector + + v (t) – + iR1(t) iC1(t) L iC2(t) R2 v1(t) i (t) R C v (t) C v (t) in 1 1 1 2 2 + x(t)= v2(t) v (t) i(t) R3 out – – – Matrix K Input vector Choose output vector as
C1 00 v (t) K = 0C2 0 u(t)= iin(t) y(t)= out 00L iR1(t)
To write the state equations of this circuit, we must express the inductor voltages and capacitor currents as linear combinations of the elements of the x(t) and u( t) vectors.
Fundamentals of Power Electronics66 Chapter 7: AC equivalent circuit modeling Circuit equations
i(t) L + + v (t) – + iR1(t) iC1(t) L iC2(t) R2 i (t) R C v (t) C v (t) in 1 1 1 2 2 + v (t) R3 out – – –
dv (t) v(t) Find i via node equation: i (t)=C 1 = i (t)± 1 ±i(t) C1 C1 1 dt in R
dv2(t) v2(t) Find iC2 via node equation: iC2(t)=C2 = i(t)± dt R2 +R3
di(t) v(t)=L = v (t)±v(t) Find vL via loop equation: L dt 1 2
Fundamentals of Power Electronics67 Chapter 7: AC equivalent circuit modeling Equations in matrix form
dv (t) v(t) The same equations: i(t)=C 1 = i (t)± 1 ±i(t) C1 1 dt in R
dv2(t) v2(t) iC2(t)=C2 = i(t)± dt R2 +R3 di(t) v(t)=L = v (t)±v(t) L dt 1 2 Express in matrix form:
dv (t) 1 ± 1 0±1 dt R1 v(t) C1 00 1 1 dv2(t) 1 0C2 0 = 0± 1 v2(t)+0iin(t) dt R2+R3 0 00L di(t) i(t) 1±10 dt
dx(t) K = Ax(t)+Bu(t) dt
Fundamentals of Power Electronics68 Chapter 7: AC equivalent circuit modeling Output (dependent signal) equations
i(t) L vout(t) + + v (t) – + y(t)= iR1(t) iC1(t) L iC2(t) iR1(t) R2 i (t) R C v (t) C v (t) in 1 1 1 2 2 + v (t) R3 out – – – Express elements of the vector y as linear combinations of elements of x and u:
R3 vout(t)=v2(t) R2 +R3
v1(t) iR1(t)= R1
Fundamentals of Power Electronics69 Chapter 7: AC equivalent circuit modeling Express in matrix form
R3 The same equations: vout(t)=v2(t) R2 +R3
v1(t) iR1(t)= R1
Express in matrix form:
R v(t) 0 3 0 1 vout(t) 0 = R2 + R3 v(t)+i(t) i (t) 2 0in R1 1 00 i(t) R1
y(t)= Cx(t)+Eu(t)
Fundamentals of Power Electronics70 Chapter 7: AC equivalent circuit modeling 7.4.2. The basic state-space averaged model
Given: a PWM converter, operating in continuous conduction mode, with two subintervals during each switching period. During subinterval 1, when the switches are in position 1, the converter reduces to a linear circuit that can be described by the following state equations: dx(t) K = A x(t)+B u(t) dt 1 1
y(t)=C1 x(t)+E1 u(t) During subinterval 2, when the switches are in position 2, the converter reduces to another linear circuit, that can be described by the following state equations: dx(t) K = A x(t)+B u(t) dt 2 2
y(t)=C2 x(t)+E2 u(t)
Fundamentals of Power Electronics71 Chapter 7: AC equivalent circuit modeling Equilibrium (dc) state-space averaged model
Provided that the natural frequencies of the converter, as well as the frequencies of variations of the converter inputs, are much slower than the switching frequency, then the state-space averaged model that describes the converter in equilibrium is 0 = AX+BU Y=CX+EU
where the averaged matrices and the equilibrium dc are components are
A = D A 1 + D' A 2 X = equilibrium (dc) state vector
B = D B1 + D' B2 U = equilibrium (dc) input vector
C = D C1 + D' C2 Y = equilibrium (dc) output vector
E = D E1 + D' E2 D = equilibrium (dc) duty cycle
Fundamentals of Power Electronics72 Chapter 7: AC equivalent circuit modeling Solution of equilibrium averaged model
Equilibrium state-space averaged model:
0=AX+BU Y=CX+EU
Solution for X and Y: X =±A±1 BU Y=±CA±1 B+E U
Fundamentals of Power Electronics73 Chapter 7: AC equivalent circuit modeling Small-signal ac state-space averaged model
dx(t) K = Ax(t)+Bu(t)+ A ±A X+ B ±B U d(t) dt 1 2 1 2
y(t)=Cx(t)+Eu(t)+ C1 ±C2 X+ E1 ±E2 U d(t)
where x(t)=small ± signal (ac) perturbation in state vector u(t)=small ± signal (ac) perturbation in input vector y(t)=small ± signal (ac) perturbation in output vector d(t)=small ± signal (ac) perturbation in duty cycle
So if we can write the converter state equations during subintervals 1 and 2, then we can always find the averaged dc and small-signal ac models
Fundamentals of Power Electronics74 Chapter 7: AC equivalent circuit modeling 7.4.3. Discussion of the state-space averaging result
As in Sections 7.1 and 7.2, the low-frequency components of the inductor currents and capacitor voltages are modeled by averaging
over an interval of length Ts. Hence, we define the average of the state vector as:
t + T 1 s τ τ x(t) T = x( ) d s Ts t
The low-frequency components of the input and output vectors are modeled in a similar manner. By averaging the inductor voltages and capacitor currents, one obtains:
d x(t) Ts K = d(t) A1 + d'(t) A2 x(t) + d(t) B1 + d'(t) B2 u(t) dt Ts Ts
Fundamentals of Power Electronics75 Chapter 7: AC equivalent circuit modeling Change in state vector during first subinterval
During subinterval 1, we have dx(t) K = A x(t)+B u(t) dt 1 1
y(t)=C1 x(t)+E1 u(t) So the elements of x(t) change with the slope dx(t) = K±1 A x(t)+B u(t) dt 1 1
Small ripple assumption: the elements of x(t) and u(t) do not change significantly during the subinterval. Hence the slopes are essentially constant and are equal to
dx(t) ±1 = K A1 x(t) +B1 u(t) dt Ts Ts
Fundamentals of Power Electronics76 Chapter 7: AC equivalent circuit modeling Change in state vector during first subinterval
±1 x(t) K A 1 x + B1 u dx(t) ±1 Ts Ts = K A1 x(t) +B1 u(t) dt Ts Ts
x(0) ±1 A + d'A x + dB K d 1 2 Ts
Net change in state vector over first 0 dT subinterval: s
±1 x(dTs) = x(0) + dTs K A1 x(t) +B1 u(t) Ts Ts
final initial interval slope value value length
Fundamentals of Power Electronics77 Chapter 7: AC equivalent circuit modeling Change in state vector during second subinterval
Use similar arguments. State vector now changes with the essentially constant slope
dx(t) ±1 = K A2 x(t) +B2 u(t) dt Ts Ts
The value of the state vector at the end of the second subinterval is therefore
±1 x(Ts) = x(dTs) + d'Ts K A2 x(t) +B2 u(t) Ts Ts
final initial interval slope value value length
Fundamentals of Power Electronics78 Chapter 7: AC equivalent circuit modeling Net change in state vector over one switching period
We have:
±1 x(dTs)=x(0) + dTs K A1 x(t) +B1 u(t) Ts Ts
±1 x(Ts)=x(dTs)+ d'Ts K A2 x(t) +B2 u(t) Ts Ts
Eliminate x(dTs), to express x(Ts) directly in terms of x(0) :
±1 ±1 x(Ts)=x(0) + dTsK A1 x(t) +B1 u(t) +d'TsK A2 x(t) +B2 u(t) Ts Ts Ts Ts
Collect terms:
±1 ±1 x(Ts)=x(0) + TsK d(t)A1 +d'(t)A2 x(t) +TsK d(t)B1 +d'(t)B2 u(t) Ts Ts
Fundamentals of Power Electronics79 Chapter 7: AC equivalent circuit modeling Approximate derivative of state vector
±1 ±1 x(t) K A 1 x + B1 u K A 2 x + B2 u Ts Ts Ts Ts x(t) Ts x(T ) x(0) s
±1 A + d'A x + dB1 + d'B2 u T K d 1 2 Ts s
0 dTs Ts t Use Euler approximation: d x(t) Ts ≈ x(Ts)±x(0) dt Ts We obtain: d x(t) Ts K = d(t) A1 + d'(t) A2 x(t) + d(t) B1 + d'(t) B2 u(t) dt Ts Ts
Fundamentals of Power Electronics80 Chapter 7: AC equivalent circuit modeling Low-frequency components of output vector
y(t) C1 x(t) + E1 u(t) Ts Ts
y(t) Ts
C2 x(t) T + E2 u(t) T 0 s s 0 dTs Ts t
Remove switching harmonics by averaging over one switching period:
y(t) = d(t) C1 x(t) + E1 u(t) + d'(t) C2 x(t) + E2 u(t) Ts Ts Ts Ts Ts Collect terms:
y(t) = d(t) C1 + d'(t) C2 x(t) + d(t) E1 + d'(t) E2 u(t) Ts Ts Ts
Fundamentals of Power Electronics81 Chapter 7: AC equivalent circuit modeling Averaged state equations: quiescent operating point
The averaged (nonlinear) state equations:
d x(t) Ts K = d(t) A1 + d'(t) A2 x(t) + d(t) B1 + d'(t) B2 u(t) dt Ts Ts
y(t) = d(t) C1 + d'(t) C2 x(t) + d(t) E1 + d'(t) E2 u(t) Ts Ts Ts
The converter operates in equilibrium when the derivatives of all elements of are zero. Hence, the converter quiescent < x(t) >Ts operating point is the solution of 0 = AX+BU Y=CX+EU
where A = D A 1 + D' A 2 and X = equilibrium (dc) state vector
B = D B1 + D' B2 U = equilibrium (dc) input vector
C = D C1 + D' C2 Y = equilibrium (dc) output vector
E = D E1 + D' E2 D = equilibrium (dc) duty cycle Fundamentals of Power Electronics82 Chapter 7: AC equivalent circuit modeling Averaged state equations: perturbation and linearization
Let x(t) = X + x(t) with U >> u(t) Ts u(t) = U + u(t) D >> d(t) Ts y(t) = Y + y(t) X >> x(t) Ts d(t)=D+d(t) ⇒ d'(t)=D'±d(t) Y >> y(t)
Substitute into averaged state equations: d X+x(t) K = D+d(t) A + D'±d(t) A X+x(t) dt 1 2
+ D+d(t) B1 + D'±d(t) B2 U+u(t)
Y+y(t) = D+d(t) C1 + D'±d(t) C2 X+x(t)
+ D+d(t) E1 + D'±d(t) E2 U+u(t)
Fundamentals of Power Electronics83 Chapter 7: AC equivalent circuit modeling Averaged state equations: perturbation and linearization
dx(t) K = AX + BU + Ax(t)+Bu(t)+ A ±A X+ B ±B U d(t) dt 1 2 1 2
first±order ac dc terms first±order ac terms
+ A 1 ± A 2 x(t)d(t)+ B1 ±B2 u(t)d(t)
second±order nonlinear terms
Y+y(t) = CX + EU + Cx(t)+Eu(t)+ C1 ±C2 X+ E1 ±E2 U d(t)
dc +1st order ac dc terms first±order ac terms
+ C1 ± C2 x(t)d(t)+ E1 ±E2 u(t)d(t)
second±order nonlinear terms
Fundamentals of Power Electronics84 Chapter 7: AC equivalent circuit modeling Linearized small-signal state equations
Dc terms drop out of equations. Second-order (nonlinear) terms are small when the small-signal assumption is satisfied. We are left with:
dx(t) K = Ax(t)+Bu(t)+ A ±A X+ B ±B U d(t) dt 1 2 1 2
y(t)=Cx(t)+Eu(t)+ C1 ±C2 X+ E1 ±E2 U d(t)
This is the desired result.
Fundamentals of Power Electronics85 Chapter 7: AC equivalent circuit modeling 7.4.4. Example: State-space averaging of a nonideal buck-boost converter
Q1 D1 ig(t) Model nonidealities: + i(t) ¥ MOSFET on- resistance R v (t) + LCRv(t) on g – ¥ Diode forward voltage
– drop VD
state vector input vector output vector
i(t) vg(t) x(t)= u(t)= y(t)= ig(t) v(t) VD
Fundamentals of Power Electronics86 Chapter 7: AC equivalent circuit modeling Subinterval 1
i (t) Ron di(t) g L = vg(t)±i(t)Ron + dt i(t) dv(t) v(t) C =± + LCRv(t) dt R vg(t) – ig(t)=i(t) –
± R 0 i(t) on i(t) vg(t) L 0 d = +10 0C dt v(t) 0±1 v(t) 00 VD R
dx(t) K A x(t) B u(t) dt 1 1
i(t) vg(t) ig(t) =10 +00 v(t) VD
y(t) C1 x(t) E1 u(t) Fundamentals of Power Electronics87 Chapter 7: AC equivalent circuit modeling Subinterval 2
V di(t) D + L = v(t)±VD – dt i (t) + dv(t) v(t) g C =± ±i(t) dt R + LCRv(t) vg(t) – ig(t)=0 i(t) –
i(t) 01 i(t) vg(t) L 0 d = + 0±1 v(t) v(t) 00 V 0C dt ±1 ± 1 D R dx(t) K A x(t) B u(t) dt 2 2
i(t) vg(t) ig(t) =00 +00 v(t) VD
y(t) C2 x(t) E2 u(t)
Fundamentals of Power Electronics88 Chapter 7: AC equivalent circuit modeling Evaluate averaged matrices
± Ron 0 01 ±DRon D' A = DA 1 + D'A 2 = D +D' = 0±1 ±1 ± 1 ± D' ± 1 R R R
In a similar manner,
B = DB + D'B = D ± D' 1 2 00
C=DC1+D'C2=D0
E=DE1+D'E2=00
Fundamentals of Power Electronics89 Chapter 7: AC equivalent circuit modeling DC state equations
± DRon D' V 0 = I + D ± D' g 0=AX+BU 0 1 V 00 V or, ± D' ± D Y=CX+EU R
I Vg Ig= D0 +00 V VD
D 1 DC solution: 2 D' R V I = 1 D' R g V R D V 1+ D on ± 1 D D'2 R D' 2 V I = 1 D D g g R 2 D'R V 1+ D on D' R D D'2 R
Fundamentals of Power Electronics90 Chapter 7: AC equivalent circuit modeling Steady-state equivalent circuit
DC state equations: ± DRon D' V 0 = I + D ± D' g 0 ± D' ± 1 V 00 VD R
I Vg Ig= D0 +00 V VD
Corresponding equivalent circuit:
DRon D'VD 1 : D D' : 1 – + + Ig I
+ V Vg – R
–
Fundamentals of Power Electronics91 Chapter 7: AC equivalent circuit modeling Small-signal ac model
Evaluate matrices in small-signal model: V ± IR + V V ± V ± IR + V A ± A X + B ± B U = ± V + g on D = g on D 1 2 1 2 I 0 I
C1 ± C2 X + E1 ± E2 U = I
Small-signal ac state equations:
± DR D' L 0 d i(t) on i(t) D ± D' vg(t) Vg±V±IRon + VD = + 0 + d(t) 0 C dt v(t) ± D' ± 1 v(t) 00 v➚(t) I R D
i(t) v(t) i (t) = D 0 + 00 g + 0 d(t) g v(t) 00 ➚0 I vD(t)
Fundamentals of Power Electronics92 Chapter 7: AC equivalent circuit modeling Construction of ac equivalent circuit
Small-signal ac di(t) L = D' v(t)±DRon i(t)+Dvg(t)+ Vg ±V±IRon + VD d(t) equations, in dt dv(t) v(t) scalar form: C =±D'i(t)± +Id(t) dt R
ig(t)=Di(t)+Id(t)
Corresponding equivalent circuits: ig(t)
inductor equation + d(t) Vg ± V + VD ± IRon vg(t) Id(t) Di(t) L DRon – + – input +–di(t) L dt eqn + D vg(t) + D' v(t) – – i(t) dv(t) v(t) C + dt R D' i(t) Id(t) C v(t) R capacitor eqn ±
Fundamentals of Power Electronics93 Chapter 7: AC equivalent circuit modeling Complete small-signal ac equivalent circuit
Combine individual circuits to obtain
d(t) V ±V+V ±IR L g D on ig(t)
1 : D + – D' : 1 i(t) + DRon
v(t) + Id(t) Id(t) C g – v(t) R
±
Fundamentals of Power Electronics94 Chapter 7: AC equivalent circuit modeling 7.5. Circuit Averaging and Averaged Switch Modeling
● Historically, circuit averaging was the first method known for modeling the small-signal ac behavior of CCM PWM converters
● It was originally thought to be difficult to apply in some cases
● There has been renewed interest in circuit averaging and its corrolary, averaged switch modeling, in the last decade
● Can be applied to a wide variety of converters
● We will use it to model DCM, CPM, and resonant converters
● Also useful for incorporating switching loss into ac model of CCM converters
● Applicable to 3¿ PWM inverters and rectifiers
● Can be applied to phase-controlled rectifiers
● Rather than averaging and linearizing the converter state equations, the averaging and linearization operations are performed directly on the converter circuit
Fundamentals of Power Electronics95 Chapter 7: AC equivalent circuit modeling Separate switch network from remainder of converter
Power input Load Time-invariant network containing converter reactive elements +
v (t) + C L v(t) g – R
iL(t) + vC(t) – –
i1(t) i2(t)
+ +
Switch network port 1
v1(t) v2(t) 2 port – –
Control d(t) input
Fundamentals of Power Electronics96 Chapter 7: AC equivalent circuit modeling Boost converter example
L Ideal boost converter + example i(t) v (t) + CRv(t) g –
–
Two ways to i (t) i (t) i (t) i (t) define the 1 2 1 2 a (b) (a) switch network + + + +
v1(t) v2(t) v1(t) v2(t)
– – – –
Fundamentals of Power Electronics97 Chapter 7: AC equivalent circuit modeling Discussion
● The number of ports in the switch network is less than or equal to the number of SPST switches
● Simple dc-dc case, in which converter contains two SPST switches: switch network contains two ports The switch network terminal waveforms are then the port voltages and
currents: v1(t), i1(t), v2(t), and i2(t). Two of these waveforms can be taken as independent inputs to the switch network; the remaining two waveforms are then viewed as dependent outputs of the switch network.
● Definition of the switch network terminal quantities is not unique. Different definitions lead equivalent results having different forms
Fundamentals of Power Electronics98 Chapter 7: AC equivalent circuit modeling Boost converter example
Let’s use definition (a):
i1(t) i2(t) L + + + i(t)
v (t) v (t) v (t) + CRv(t) 1 2 g –
– – –
Since i1(t) and v2(t) coincide with the converter inductor current and output voltage, it is convenient to define these waveforms as the independent inputs to the switch network. The switch network
dependent outputs are then v1(t) and i2(t).
Fundamentals of Power Electronics99 Chapter 7: AC equivalent circuit modeling Obtaining a time-invariant network: Modeling the terminal behavior of the switch network
Replace the switch network with dependent sources, which correctly represent the dependent output waveforms of the switch network
L + + i(t) i1(t)
v (t) + v (t) + i (t) v (t) CRv(t) g – 1 – 2 2
– – Switch network
Boost converter example
Fundamentals of Power Electronics100 Chapter 7: AC equivalent circuit modeling Definition of dependent generator waveforms
L v1(t) v2(t) + + i(t) i1(t)
v (t) + v (t) + i (t) v (t) CRv(t) 〈v (t)〉 g – 1 – 2 2 1 Ts – – 0 Switch network 0 0 dT T t s s The waveforms of the dependent i2(t) i1(t) generators are defined to be identical to the actual terminal waveforms of the 〈i (t)〉 switch network. 2 Ts The circuit is therefore electrical 0 0 identical to the original converter. 0 dT T t s s So far, no approximations have been made.
Fundamentals of Power Electronics101 Chapter 7: AC equivalent circuit modeling The circuit averaging step
Now average all waveforms over one switching period:
Power input Load Averaged time-invariant network containing converter reactive elements +
〈v (t)〉 + C 〈 〉 g Ts L R v(t) – Ts 〈i (t)〉 + 〈v (t)〉 – L Ts – C Ts
〈i (t)〉 〈i (t)〉 1 Ts 2 Ts
+ +
Averaged port 〈 〉 1 switch network 〈 〉 v1(t) T v2(t) T s 2 s port – –
Control d(t) input
Fundamentals of Power Electronics102 Chapter 7: AC equivalent circuit modeling The averaging step
The basic assumption is made that the natural time constants of the converter are much longer than the switching period, so that the converter contains low-pass filtering of the switching harmonics. One may average the waveforms over an interval that is short compared to the system natural time constants, without significantly altering the
system response. In particular, averaging over the switching period Ts removes the switching harmonics, while preserving the low-frequency components of the waveforms. In practice, the only work needed for this step is to average the switch dependent waveforms.
Fundamentals of Power Electronics103 Chapter 7: AC equivalent circuit modeling Averaging step: boost converter example
L + + i(t) i1(t)
v (t) + v (t) + i (t) v (t) CRv(t) g – 1 – 2 2
– – Switch network
L
〈i(t)〉 〈i (t)〉 + + Ts 1 Ts
〈v (t)〉 + 〈v (t)〉 + 〈i (t)〉 〈v (t)〉 C R 〈v(t)〉 g Ts – 1 Ts – 2 Ts 2 Ts Ts
– – Averaged switch network
Fundamentals of Power Electronics104 Chapter 7: AC equivalent circuit modeling Compute average values of dependent sources
v1(t) v2(t) Average the waveforms of the dependent sources: 〈v (t)〉 1 Ts
0 0 v1(t) T = d'(t) v2(t) T 0 dTs Ts t s s i (t) 2 i (t) 1 i2(t) = d'(t) i1(t) Ts Ts 〈i (t)〉 2 Ts
0 0 0 dTs Ts t L
〈i(t)〉 〈i (t)〉 + + Ts 1 Ts
〈v (t)〉 + d'(t) 〈v (t)〉 + d'(t) 〈i (t)〉 〈v (t)〉 C R 〈v(t)〉 g Ts – 2 Ts – 1 Ts 2 Ts Ts
– – Averaged switch model
Fundamentals of Power Electronics105 Chapter 7: AC equivalent circuit modeling Perturb and linearize
vg(t) = Vg + vg(t) As usual, let: Ts d(t)=D+d(t) ⇒d'(t)=D'±d(t)
i(t) = i1(t) =I+i(t) Ts Ts
v(t) = v2(t) =V+v(t) Ts Ts
v1(t) =V1 +v1(t) Ts
i2(t) =I2 +i2(t) Ts The circuit becomes: L
I+i(t) +
V+v(t)+ D'±d(t)V+v(t)+ D'±d(t)I+i(t)CRV+v(t) gg– –
–
Fundamentals of Power Electronics106 Chapter 7: AC equivalent circuit modeling Dependent voltage source
D'±d(t) V+v(t) =D' V+v(t) ±Vd(t)±v(t)d(t) nonlinear, 2nd order Vd(t) + –
+ D' V+v(t) –
Fundamentals of Power Electronics107 Chapter 7: AC equivalent circuit modeling Dependent current source
D' ± d(t) I + i(t) = D' I + i (t) ± Id(t)±i(t)d(t) nonlinear, 2nd order
D' I + i(t) Id(t)
Fundamentals of Power Electronics108 Chapter 7: AC equivalent circuit modeling Linearized circuit-averaged model
L Vd(t) + – I+i(t) +
V+v(t)+ D' V+v(t) + D' I+i(t) Id(t) CRV+v(t) gg– –
–
L Vd(t) D' : 1 + – I+i(t) +
V+v(t)+ Id(t) V+v(t) gg– CR
–
Fundamentals of Power Electronics109 Chapter 7: AC equivalent circuit modeling Summary: Circuit averaging method
Model the switch network with equivalent voltage and current sources, such that an equivalent time-invariant network is obtained Average converter waveforms over one switching period, to remove the switching harmonics Perturb and linearize the resulting low-frequency model, to obtain a small-signal equivalent circuit
Fundamentals of Power Electronics110 Chapter 7: AC equivalent circuit modeling Averaged switch modeling: CCM
Circuit averaging of the boost converter: in essence, the switch network was replaced with an effective ideal transformer and generators:
2 D' : 1 + – i(t) + I + i(t) + 1 Vd(t) v(t) Id(t) V + v(t)
– – Switch network
Fundamentals of Power Electronics111 Chapter 7: AC equivalent circuit modeling Basic functions performed by switch network
2 D' : 1 + – i(t) + I + i(t) + 1 Vd(t) v(t) Id(t) V + v(t)
– – Switch network
For the boost example, we can conclude that the switch network performs two basic functions: ¥ Transformation of dc and small-signal ac voltage and current levels, according to the D’:1 conversion ratio ¥ Introduction of ac voltage and current variations, drive by the control input duty cycle variations Circuit averaging modifies only the switch network. Hence, to obtain a small- signal converter model, we need only replace the switch network with its averaged model. Such a procedure is called averaged switch modeling.
Fundamentals of Power Electronics112 Chapter 7: AC equivalent circuit modeling Averaged switch modeling: Procedure
1. Define a switch network and its i1(t) i2(t) terminal waveforms. For a simple + + transistor-diode switch network as in
the buck, boost, etc., there are two v1(t) v2(t)
ports and four terminal quantities: v1, – – i1, v2, i2.The switch network also contains a control input d. Buck example: 2. To derive an averaged switch model, express the average values of 〈 〉 〈 〉 two of the terminal quantities, for example v2 Ts and i1 Ts, as 〈 〉 〈 〉 functions of the other average terminal quantities v1 Ts and i1 Ts . 〈 〉 〈 〉 v2 Ts and i1 Ts may also be functions of the control input d, but they should not be expressed in terms of other converter signals.
Fundamentals of Power Electronics113 Chapter 7: AC equivalent circuit modeling The basic buck-type CCM switch cell
i (t) i i (t) L 1 + vCE – C 2 i(t) i1(t) i2 + + + T2
+ i2 vg(t) v1(t) v2(t) CRv(t) i1(t) – T2
– – – 0 Switch network 0 0 dTs Ts t
v2(t) v1 i1(t) = d(t) i2(t) Ts Ts v2(t) T v2(t) = d(t) v1(t) 2 Ts Ts 0 0 0 dTs Ts t
Fundamentals of Power Electronics114 Chapter 7: AC equivalent circuit modeling Replacement of switch network by dependent sources, CCM buck example
Circuit-averaged model Perturbation and linearization of switch
L network: i(t) I 1 + i 1(t)=DI2+i2(t)+I2d(t) + + i2(t) V2+v2(t)=DV1+v1(t)+V1d(t) v (t) + v (t) i (t) + v (t) CRv(t) g – 1 1 – 2 I1 + i 1 1 : D I2 + i 2 + – – – + + Switch network V1 d V1 + v1 V + v I2 d 2 2
– –
L I + i 1 : D I2 + i 2 1 1 + – + + Resulting + I + i V1 d + averaged switchV + v V1 + v1 V + v CRV + v g g – I2 d 2 2 model: CCM buck converter – – – Switch network
Fundamentals of Power Electronics115 Chapter 7: AC equivalent circuit modeling Three basic switch networks, and their CCM dc and small-signal ac averaged switch models
i (t) i (t) I1 + i 1 1 : D I2 + i 2 +
1 2 – + + + + see also V1 d v (t) v (t) V1 + v1 V + v 1 2 I2 d 2 2 Appendix 3 Averaged switch – – – – modeling of a CCM SEPIC i (t) i (t) I1 + i 1 D' : 1 I2 + i 2 +
1 2 – + + + +
V2 d
v (t) V1 + v1 V + v 1 v2(t) I1 d 2 2
– – – –
i (t) i (t) I1 + i 1 D' : D I2 + i 2 +
1 2 – + + + + V 1 d V + v DD' I2 v (t) v (t) 1 1 d V2 + v2 1 2 DD'
– – – –
Fundamentals of Power Electronics116 Chapter 7: AC equivalent circuit modeling Example: Averaged switch modeling of CCM buck converter, including switching loss
i (t) i i (t) L 1 + vCE – C 2 i(t) + + + i1(t)=iC(t)
v (t) + v (t) (t) CRv(t) v2(t)=v1(t)±vCE(t) g – 1 v2
– – – Switch network Switch network terminal
waveforms: v1, i1, v2, i2. To v (t) i (t) derive averaged switch CE C 〈 〉 v model, express v2 Ts 1 〈 〉 and i1 Ts as functions of i 2 〈 〉 〈 〉 〈 〉 v1 Ts and i1 Ts . v2 Ts and 〈 i 〉 may also be 0 1 Ts 0 functions of the control t t t t t 1 2 input d, but they should ir vf tvr tif
Ts not be expressed in terms of other converter signals.
Fundamentals of Power Electronics117 Chapter 7: AC equivalent circuit modeling Averaging i1(t)
vCE(t) iC(t)
v1
i2
0 0 t t t t t 1 2 ir vf tvr tif
Ts
1 Ts i1(t) T = i1(t) dt s Ts 0 t + t + t + 1 t + 1 t 1 vf vr 2 ir 2 if = i2(t) T s Ts
Fundamentals of Power Electronics118 Chapter 7: AC equivalent circuit modeling 〈 〉 Expression for i1(t)
Given 1 Ts i1(t) T = i1(t) dt s Ts 0 t + t + t + 1 t + 1 t 1 vf vr 2 ir 2 if = i2(t) T s Ts Let Then we can write t + 1 t + 1 t + 1 t + 1 t 1 2 vf 2 vr 2 ir 2 if 1 d = i1(t) = i2(t) d + dv Ts Ts Ts 2
tvf + tvr dv = Ts
tir + tif di = Ts
Fundamentals of Power Electronics119 Chapter 7: AC equivalent circuit modeling Averaging the switch network output voltage v2(t)
vCE(t) iC(t)
v1
i2
0 0 t t t t t 1 2 ir vf tvr tif
Ts
Ts v (t) = v (t)±v (t) = 1 ± v (t) dt + v (t) 2 T 1 CE T CE 1 T s s Ts 0 s t + 1 t + 1 t 1 2 vf 2 vr v2(t) T = v1(t) T s s Ts
1 v2(t) = v1(t) d ± di Ts Ts 2
Fundamentals of Power Electronics120 Chapter 7: AC equivalent circuit modeling Construction of large-signal averaged-switch model
1 1 i1(t) = i2(t) d + dv v2(t) = v1(t) d ± di Ts Ts 2 Ts Ts 2
1 〈 〉 2 di(t) v1(t) T 〈i (t)〉 s 〈i (t)〉 1 Ts – + 2 Ts + +
〈v (t)〉 1 d (t) 〈i (t)〉 d(t) 〈i (t)〉 + d(t) 〈v (t)〉 〈v (t)〉 1 Ts 2 v 2 Ts 2 Ts – 1 Ts 2 Ts
– –
1 〈 〉 2 di(t) v1(t) T 〈i (t)〉 1 : d(t) s 〈i (t)〉 1 Ts – + 2 Ts + +
〈v (t)〉 1 d (t) 〈i (t)〉 〈v (t)〉 1 Ts 2 v 2 Ts 2 Ts
– –
Fundamentals of Power Electronics121 Chapter 7: AC equivalent circuit modeling Switching loss predicted by averaged switch model
1 〈 〉 2 di(t) v1(t) T 〈i (t)〉 1 : d(t) s 〈i (t)〉 1 Ts – + 2 Ts + +
〈v (t)〉 1 d (t) 〈i (t)〉 〈v (t)〉 1 Ts 2 v 2 Ts 2 Ts
– –
1 Psw = dv + d i i2(t) v1(t) 2 Ts Ts
Fundamentals of Power Electronics122 Chapter 7: AC equivalent circuit modeling Solution of averaged converter model in steady state
I 1 : D I L I
1 – + 2 + + + 1 2 Di V1 V+ V 1 D I V CRV g– 1 2 v 2 2
– – – Averaged switch network model
Output voltage: Efficiency calcuation: 1 D Pin = VgI 1 = V1I 2 D + Dv V = D ± 1 D V = DV 1± i 2 2 i g g 2D 1 Pout = VI2 = V1I2 D ± 2Di D 1 1± i P D ± 2 Di 2D η = out = = Pin 1 D D + 2 Dv 1+ v 2D Fundamentals of Power Electronics123 Chapter 7: AC equivalent circuit modeling 7.6. The canonical circuit model
All PWM CCM dc-dc converters perform the same basic functions: ¥ Transformation of voltage and current levels, ideally with 100% efficiency ¥ Low-pass filtering of waveforms ¥ Control of waveforms by variation of duty cycle Hence, we expect their equivalent circuit models to be qualitatively similar. Canonical model: ¥ A standard form of equivalent circuit model, which represents the above physical properties ¥ Plug in parameter values for a given specific converter
Fundamentals of Power Electronics124 Chapter 7: AC equivalent circuit modeling 7.6.1. Development of the canonical circuit model
1. Transformation of dc Converter model voltage and current levels 1 : M(D) + ¥ modeled as in Chapter 3 with ideal V + V R dc transformer g –
¥ effective turns ratio – M(D) ¥ can refine dc model by addition of effective loss D elements, as in Power Control Load Chapter 3 input input
Fundamentals of Power Electronics125 Chapter 7: AC equivalent circuit modeling Steps in the development of the canonical circuit model
2. Ac variations in
vg(t) induce ac variations in v(t) 1 : M(D) + ¥ these variations are also V + v (s) + V + v(s) R g g – transformed by the conversion ratio – M(D)
D Power Control Load input input
Fundamentals of Power Electronics126 Chapter 7: AC equivalent circuit modeling Steps in the development of the canonical circuit model
3. Converter H (s) must contain an e 1 : M(D) effective low- + pass filter Effective Zei(s) Zeo(s) characteristic V + v (s) + V + v(s) R g g – low-pass ¥ necessary to filter filter switching – ripple ¥ also filters ac variations D Power Control Load ¥ effective filter input input elements may not coincide with actual element values, but can also depend on operating point
Fundamentals of Power Electronics127 Chapter 7: AC equivalent circuit modeling Steps in the development of the canonical circuit model
H (s) e(s) d(s) e 1 : M(D) + – + Effective Zei(s) Zeo(s) V + v (s) + j(s) d(s) V + v(s) R g g – low-pass filter –
D + d(s) Power Control Load input input
4. Control input variations also induce ac variations in converter waveforms ¥ Independent sources represent effects of variations in duty cycle ¥ Can push all sources to input side as shown. Sources may then become frequency-dependent
Fundamentals of Power Electronics128 Chapter 7: AC equivalent circuit modeling Transfer functions predicted by canonical model
H (s) e(s) d(s) e 1 : M(D) + – + Effective Zei(s) Zeo(s) V + v (s) + j(s) d(s) V + v(s) R g g – low-pass filter –
D + d(s) Power Control Load input input
v(s) Line-to-output transfer function: Gvg(s)= =M(D)He(s) vg(s) v(s) Control-to-output transfer function: G (s)= =e(s)M(D)H(s) vd d(s) e
Fundamentals of Power Electronics129 Chapter 7: AC equivalent circuit modeling 7.6.2. Example: manipulation of the buck-boost converter model into canonical form
Small-signal ac model of the buck-boost converter
V ±V d L g +
1 : D – D' : 1 +
V +v(s) + C V + v(s) g g – Id Id R
–
¥ Push independent sources to input side of transformers ¥ Push inductor to output side of transformers ¥ Combine transformers
Fundamentals of Power Electronics130 Chapter 7: AC equivalent circuit modeling Step 1
Move current source through Push voltage source through 1:D transformer D’:1 transformer
V ± V g d D L +
– 1 : D D' : 1 +
V +v(s) + Id I d C V + v(s) g g – D' R
–
Fundamentals of Power Electronics131 Chapter 7: AC equivalent circuit modeling Step 2
How to move the current source past the inductor: Break ground connection of current source, and connect to node A instead. Connect an identical current source from node A to ground, so that the node equations are unchanged.
V ± V node g d D A L +
– 1 : D D' : 1 + I I d V +v(s) + d D' C V + v(s) g g – Id D' R
–
Fundamentals of Power Electronics132 Chapter 7: AC equivalent circuit modeling Step 3
The parallel-connected current source and inductor can now be replaced by a Thevenin-equivalent network:
V ± V sLI g d d D D' L – + +
– 1 : D D' : 1 +
I V +v(s) + d C V + v(s) g g – Id D' R
–
Fundamentals of Power Electronics133 Chapter 7: AC equivalent circuit modeling Step 4
Push current source past voltage source, again by: Now pushBreaking current ground source connection through 1:D of current transformer. source, and connecting to node B instead. Connecting an identical current source from node B to ground, so that the node equations are unchanged. Note that the resulting parallel-connected voltage and current sources are equivalent to a single voltage source.
node V ± V sLI g d d B D D' L – + +
– 1 : D D' : 1 +
DI d V +v(s) + Id DI d D' C V + v(s) g g – D' R
–
Fundamentals of Power Electronics134 Chapter 7: AC equivalent circuit modeling Step 5: final result
Push voltage source through 1:D transformer, and combine with existing input-side transformer. Combine series-connected transformers.
V ± V g ± s LI d(s) D DD' L D' : D D' 2 + – +
V + v (s) + I d(s) V + v(s) R g g – D' C
– Effective low-pass filter
Fundamentals of Power Electronics135 Chapter 7: AC equivalent circuit modeling Coefficient of control-input voltage generator
Voltage source coefficient is:
V +V e(s)= g ± sLI D DD'
Simplification, using dc relations, leads to
e(s)=± V 1± sDL D2 D'2 R
Pushing the sources past the inductor causes the generator to become frequency-dependent.
Fundamentals of Power Electronics136 Chapter 7: AC equivalent circuit modeling 7.6.3. Canonical circuit parameters for some common converters
e(s) d(s) L 1 : M(D) e + – +
V + v (s) + j(s) d(s) C V + v(s) R g g –
–
Table 7.1. Canonical model parameters for the ideal buck, boost, and buck-boost converters
Converter M(D) Le e(s) j(s) V V
Buck DL D2 R 1 L sL V 2 V 1± 2 Boost D' D' D'2 R D' R D L V sDL V Buck-boost ± 2 ± 1± ± 2 D' D' D2 D2'R D' R
Fundamentals of Power Electronics137 Chapter 7: AC equivalent circuit modeling 7.7. Modeling the pulse-width modulator
Pulse-width Power Switching converter Load modulator input converts voltage +
signal vc(t) into v (t) + v(t) duty cycle signal g – R feedback d(t). connection – What is the relation between transistor
gate driver compensator
vc(t) and d(t)? δ pulse-width vc – v (t) G (s) + modulator c
δ (t) vc(t) voltage reference vref
dTs Ts t t Controller
Fundamentals of Power Electronics138 Chapter 7: AC equivalent circuit modeling A simple pulse-width modulator
VM vsaw(t) Sawtooth v (t) v (t) c wave saw generator
comparator 0 t δ(t) δ(t) analog input + PWM waveform vc(t) –
02TdTs Ts s
Fundamentals of Power Electronics139 Chapter 7: AC equivalent circuit modeling Equation of pulse-width modulator
For a linear sawtooth waveform: VM vsaw(t) vc(t) ≤ ≤ v (t) d(t)= for 0 vc(t) VM c VM
So d(t) is a linear function of v (t). 0 c t δ(t)
02TdTs Ts s
Fundamentals of Power Electronics140 Chapter 7: AC equivalent circuit modeling Perturbed equation of pulse-width modulator
PWM equation: Block diagram:
v(t) d(t)= c for 0 ≤ v (t) ≤ V V c M M V + v (s) D + d(s) c c 1 V Perturb: M
vc(t)=Vc +vc(t) d(t)=D+d(t) pulse-width modulator Result: Dc and ac relations: V +v(t) D + d(t)= c c Vc V D = M VM v(t) d(t)= c VM
Fundamentals of Power Electronics141 Chapter 7: AC equivalent circuit modeling Sampling in the pulse-width modulator
The input voltage is a continuous function of time, but there sampler can be only one vc 1 d discrete value of the VM f duty cycle for each s switching period. Therefore, the pulse- pulse-width modulator width modulator samples the control waveform, with sampling rate equal to the switching frequency. In practice, this limits the useful frequencies of ac variations to values much less than the switching frequency. Control system bandwidth must
be sufficiently less than the Nyquist rate fs/2. Models that do not account for sampling are accurate only at frequencies much less than fs/2.
Fundamentals of Power Electronics142 Chapter 7: AC equivalent circuit modeling 7.8. Summary of key points
1. The CCM converter analytical techniques of Chapters 2 and 3 can be extended to predict converter ac behavior. The key step is to average the converter waveforms over one switching period. This removes the switching harmonics, thereby exposing directly the desired dc and low-frequency ac components of the waveforms. In particular, expressions for the averaged inductor voltages, capacitor currents, and converter input current are usually found. 2. Since switching converters are nonlinear systems, it is desirable to construct small-signal linearized models. This is accomplished by perturbing and linearizing the averaged model about a quiescent operating point. 3. Ac equivalent circuits can be constructed, in the same manner used in Chapter 3 to construct dc equivalent circuits. If desired, the ac equivalent circuits may be refined to account for the effects of converter losses and other nonidealities.
Fundamentals of Power Electronics143 Chapter 7: AC equivalent circuit modeling Summary of key points
4. The state-space averaging method of section 7.4 is essentially the same as the basic approach of section 7.2, except that the formality of the state-space network description is used. The general results are listed in section 7.4.2. 5. The circuit averaging technique also yields equivalent results, but the derivation involves manipulation of circuits rather than equations. Switching elements are replaced by dependent voltage and current sources, whose waveforms are defined to be identical to the switch waveforms of the actual circuit. This leads to a circuit having a time-invariant topology. The waveforms are then averaged to remove the switching ripple, and perturbed and linearized about a quiescent operating point to obtain a small-signal model.
Fundamentals of Power Electronics144 Chapter 7: AC equivalent circuit modeling Summary of key points
6. When the switches are the only time-varying elements in the converter, then circuit averaging affects only the switch network. The converter model can then be derived by simply replacing the switch network with its averaged model. Dc and small-signal ac models of several common CCM switch networks are listed in section 7.5.4. Switching losses can also be modeled using this approach. 7. The canonical circuit describes the basic properties shared by all dc-dc PWM converters operating in the continuous conduction mode. At the heart of the model is the ideal 1:M(D) transformer, introduced in Chapter 3 to represent the basic dc-dc conversion function, and generalized here to include ac variations. The converter reactive elements introduce an effective low-pass filter into the network. The model also includes independent sources which represent the effect of duty cycle variations. The parameter values in the canonical models of several basic converters are tabulated for easy reference.
Fundamentals of Power Electronics145 Chapter 7: AC equivalent circuit modeling Summary of key points
8. The conventional pulse-width modulator circuit has linear gain, dependent on the slope of the sawtooth waveform, or equivalently on its peak-to-peak magnitude.
Fundamentals of Power Electronics146 Chapter 7: AC equivalent circuit modeling