g
Transfer Function Modeling
for the Buck converter
Sanda Lefteriu
Cécile Labarre
Ecole des Mines de Douai
May 10, 2016
Sanda Lefteriu (Ecole des Mines de Douai)
Transfer Function Modeling for the Buck converter
May 10, 2016
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Motivation
Outline
1
Modeling Systems using Frequency Domain Measurements
2
Transfer Function Modeling for the Buck converter
Average model of the Buck converter
Generalized Transfer Function (GTF)
Augmented MNA for PSL circuits
Results
3
Conclusion
Sanda Lefteriu (Ecole des Mines de Douai)
Transfer Function Modeling for the Buck converter
May 10, 2016
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Motivation
Obtaining measurements in the frequency domain
u(t) = a0 sin(2πfi t + φ0 ) −→
Σ
−→ y (t) = a1 sin(2πfi t + φ1 )
Problem statement: H(s) models a data set with N samples obtained from an
unknown underlying system, if, for i=1, · · · , N,
H(si = jωi ) ≈ Hi , where ωi = 2πfi , Hi :=
Sanda Lefteriu (Ecole des Mines de Douai)
a1
φ1 − φ0 .
a0
Transfer Function Modeling for the Buck converter
May 10, 2016
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Motivation
Obtaining measurements in the frequency domain
u(t) = a0 sin(2πfi t + φ0 ) −→
Σ
−→ y (t) = a1 sin(2πfi t + φ1 )
Problem statement: H(s) models a data set with N samples obtained from an
unknown underlying system, if, for i=1, · · · , N,
H(si = jωi ) ≈ Hi , where ωi = 2πfi , Hi :=
a1
φ1 − φ0 .
a0
For LTI systems, H(s) is a rational scalar/matrix function ⇒
rational approximation problem.
Sanda Lefteriu (Ecole des Mines de Douai)
Transfer Function Modeling for the Buck converter
May 10, 2016
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Modeling the Buck converter
Outline
1
Modeling Systems using Frequency Domain Measurements
2
Transfer Function Modeling for the Buck converter
Average model of the Buck converter
Generalized Transfer Function (GTF)
Augmented MNA for PSL circuits
Results
3
Conclusion
Sanda Lefteriu (Ecole des Mines de Douai)
Transfer Function Modeling for the Buck converter
May 10, 2016
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Modeling the Buck converter
DC-DC switching power converters
They consist of LTI components (resistors, capacitors, inductors) together with
switches (transistors and diodes), whose operation is controlled to yield the
desired voltage conversion.
Sanda Lefteriu (Ecole des Mines de Douai)
Transfer Function Modeling for the Buck converter
May 10, 2016
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Modeling the Buck converter
DC-DC switching power converters
They consist of LTI components (resistors, capacitors, inductors) together with
switches (transistors and diodes), whose operation is controlled to yield the
desired voltage conversion.
Frequency domain analysis via the Bode plot is not possible.
Sanda Lefteriu (Ecole des Mines de Douai)
Transfer Function Modeling for the Buck converter
May 10, 2016
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g
Modeling the Buck converter
DC-DC switching power converters
They consist of LTI components (resistors, capacitors, inductors) together with
switches (transistors and diodes), whose operation is controlled to yield the
desired voltage conversion.
Frequency domain analysis via the Bode plot is not possible.
Ultimate goal: use the measurements to derive a model for the converter, valid
from low up to high frequency.
Sanda Lefteriu (Ecole des Mines de Douai)
Transfer Function Modeling for the Buck converter
May 10, 2016
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Modeling the Buck converter
State-of-the-art
Average models describe dynamics using average values of currents and voltages,
neglecting ripple effects due to switching ⇒ state-space averaging (SSA)
[Middlebrook & Cuk, 1976] employs the weighted combination of state equations
of switching phases in pulse-width modulated (PWM) converters. They are only
valid until half of the switching frequency.
Sanda Lefteriu (Ecole des Mines de Douai)
Transfer Function Modeling for the Buck converter
May 10, 2016
6 / 20
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Modeling the Buck converter
State-of-the-art
Average models describe dynamics using average values of currents and voltages,
neglecting ripple effects due to switching ⇒ state-space averaging (SSA)
[Middlebrook & Cuk, 1976] employs the weighted combination of state equations
of switching phases in pulse-width modulated (PWM) converters. They are only
valid until half of the switching frequency.
In continuous current mode (CCM), the Buck converter has two working modes,
so it can be modeled by the Generalized Transfer Function (GTF)
[Biolek, 1997, Biolek et al., 2006], which combines the continuous and the
discrete behavior. The resulting function produces a Bode plot which is valid
past half the switching frequency.
Sanda Lefteriu (Ecole des Mines de Douai)
Transfer Function Modeling for the Buck converter
May 10, 2016
6 / 20
g
Modeling the Buck converter
State-of-the-art
Average models describe dynamics using average values of currents and voltages,
neglecting ripple effects due to switching ⇒ state-space averaging (SSA)
[Middlebrook & Cuk, 1976] employs the weighted combination of state equations
of switching phases in pulse-width modulated (PWM) converters. They are only
valid until half of the switching frequency.
In continuous current mode (CCM), the Buck converter has two working modes,
so it can be modeled by the Generalized Transfer Function (GTF)
[Biolek, 1997, Biolek et al., 2006], which combines the continuous and the
discrete behavior. The resulting function produces a Bode plot which is valid
past half the switching frequency.
DC-DC power converters are periodic switched linear (PSL) circuits characterized
by time-varying transfer function or alternatively, by bi-frequency transfer
function [Zadeh, 1950]. The steady-state response found as solution of
augmented MNA equation in the frequency domain [Trinchero, 2015].
Sanda Lefteriu (Ecole des Mines de Douai)
Transfer Function Modeling for the Buck converter
May 10, 2016
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Modeling the Buck converter
Average model of the Buck converter
Average model of the Buck converter
Figure : Topology of the Buck converter
1
2
In CCM, 2 linear working modes.
1st mode: switch is closed (S = 1, S = 0) for t ∈ [kT , (k + D)T ], where D is
the duty cycle, T is the switching period and k = ..., −1, 0, 1, ...
2nd mode: switch is open (S = 0, S = 1) for t ∈ [(k + D)T , (k + 1)T ].
Figure : Switching diagram
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Transfer Function Modeling for the Buck converter
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Modeling the Buck converter
Average model of the Buck converter
State-space equations
ẋ(t) = Ax(t) + Bi u(t), i = 1, 2, where
1
0
VC (t)
0
− RC C1
, B1 = 1 , B2 =
x(t) =
,A =
,
IL (t)
0
0
− L1
L
(1)
with u(t) = Vin (t). Using the average values of VC and IL
˙x̃(t) = Ax̃(t) + Bu(t), where B = D0 .
(2)
L
The output is y (t) = Vo (t) = VC (t) = Cx̃(t) + Du(t) with C = 1
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Transfer Function Modeling for the Buck converter
0 , D = 0.
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Modeling the Buck converter
Average model of the Buck converter
State-space equations
ẋ(t) = Ax(t) + Bi u(t), i = 1, 2, where
1
0
VC (t)
0
− RC C1
, B1 = 1 , B2 =
x(t) =
,A =
,
IL (t)
0
0
− L1
L
(1)
with u(t) = Vin (t). Using the average values of VC and IL
˙x̃(t) = Ax̃(t) + Bu(t), where B = D0 .
(2)
L
The output is y (t) = Vo (t) = VC (t) = Cx̃(t) + Du(t) with C = 1 0 , D = 0.
The line-to-output transfer function is H(s) = VVino (s)
(s) . Its expression based on
average model is
D
Havg (s) =
.
(3)
1 + s RL + s 2 LC
Sanda Lefteriu (Ecole des Mines de Douai)
Transfer Function Modeling for the Buck converter
May 10, 2016
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g
Modeling the Buck converter
Generalized Transfer Function (GTF)
Deriving the GTF
We follow the derivation in [Biolek, 1997, Biolek et al., 2006]. The solution of (1):
x(t) = e
A(t−t0 )
Z
t
e A(t−τ ) Bi u(τ )dτ, t ≥ t0 .
x0 +
(4)
t0
Phase 1: t ∈ [kT , kT + DT ), write (4) at the end of it
x(kT + DT ) = e ADT x(kT ) +
Z
kT +DT
e A(kT +DT −τ ) B1 u(τ )dτ.
(5)
kT
Phase 2: t ∈ [kT + DT , (k + 1)T ), write (4)
x(t) = e A(t−kT −DT ) x(kT + DT ).
(6)
We replace (5) into (6) and evaluate at the end of the period, for t = (k + 1)T :
Z
x((k + 1)T )=e AT x(kT ) + e A(k+1)T
kT+DT
−Aτ
e
B1 u(τ )dτ.
kT
Sanda Lefteriu (Ecole des Mines de Douai)
Transfer Function Modeling for the Buck converter
May 10, 2016
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Modeling the Buck converter
Generalized Transfer Function (GTF)
Deriving the GTF
Buck converter is a DC-DC converter ⇒ use small signal analysis:
u(t) = u0 + ũe jΩt , with u0 >> ũ and Ω, perturbation frequency of choice.
Sanda Lefteriu (Ecole des Mines de Douai)
Transfer Function Modeling for the Buck converter
May 10, 2016
10 / 20
g
Modeling the Buck converter
Generalized Transfer Function (GTF)
Deriving the GTF
Buck converter is a DC-DC converter ⇒ use small signal analysis:
u(t) = u0 + ũe jΩt , with u0 >> ũ and Ω, perturbation frequency of choice.
The output at steady-state contains a DC component and an AC of the same
frequency Ω, together with other frequencies. Using the analogy with LTI, only
the component with frequency Ω is of interest.
Sanda Lefteriu (Ecole des Mines de Douai)
Transfer Function Modeling for the Buck converter
May 10, 2016
10 / 20
g
Modeling the Buck converter
Generalized Transfer Function (GTF)
Deriving the GTF
Buck converter is a DC-DC converter ⇒ use small signal analysis:
u(t) = u0 + ũe jΩt , with u0 >> ũ and Ω, perturbation frequency of choice.
The output at steady-state contains a DC component and an AC of the same
frequency Ω, together with other frequencies. Using the analogy with LTI, only
the component with frequency Ω is of interest.
The generalized transfer function for the line-to-output response is given by
HGTF (jΩ) = C e
jΩT
Z
I−e
AT −1 AT
DT
−Aτ
e
e
B1 e jΩτ dτ,
(7)
0
where I is the identity matrix.
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Transfer Function Modeling for the Buck converter
May 10, 2016
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Modeling the Buck converter
Augmented MNA for PSL circuits
Time-varying systems
Following [Zadeh, 1950], impulse response and transfer function can be extended
to time-varying systems. The output is
Z ∞
y (t) =
h(t, τ )u(τ )dτ, h(t, τ ) = R[δ(t − τ )],
−∞
with h(t, τ ), the generalized impulse response, R, the operator describing system
behavior, t, the observation time and τ , the excitation time.
The bi-frequency transfer function is
Z ∞Z ∞
h(t, τ )e −j(ωt−Ωτ ) dtdτ.
(8)
H(ω, Ω) =
−∞
−∞
where Ω and ω are the input and output frequencies.
The time-varying transfer function
Z ∞
H(t, Ω) =
h(t, τ )e −jΩ(t−τ ) dτ.
(9)
−∞
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Transfer Function Modeling for the Buck converter
May 10, 2016
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Modeling the Buck converter
Augmented MNA for PSL circuits
Periodical linear time-varying (PLTV) systems
H(t, Ω) is a periodic function of t ⇒ Fourier series w.r.t. ωs =
n=∞
X
H(t, Ω) =
2π
T :
Hn (Ω)e jnωs t .
(10)
n=−∞
Frequency-dependent Fourier coefficients Hn (Ω) called aliasing transfer functions:
Hn (Ω) =
1
T
Z
T
H(t, Ω)e −jnωs t dt.
0
The output in the frequency domain:
Y (ω) =
n=∞
X
Hn (ω − nωs )U(ω − nωs ).
(11)
n=−∞
The LTI relationship can be recovered for n = 0, namely Y (ω) = Hn (ω)U(ω).
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Transfer Function Modeling for the Buck converter
May 10, 2016
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Modeling the Buck converter
Augmented MNA for PSL circuits
Periodical linear time-varying (PLTV) systems
Figure : Block-diagram representation with infinitely many LTI systems [Trinchero, 2015]
Figure : Output for single tone ω0 contains new harmonics at multiples of ωs
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Transfer Function Modeling for the Buck converter
May 10, 2016
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Modeling the Buck converter
Augmented MNA for PSL circuits
Modified Nodal Analysis (MNA) for PSL circuits
Approach from [Trinchero, 2015] for computing steady-state of periodic switched
linear circuits via augmented time-invariant MNA equation in frequency-domain
adapted for small-signal transfer function computation.
Figure : Frequency domain representation of the Buck converter [Trinchero, 2015]
Substitute S, S with periodic switching elements Y1 , Y2 and replace voltage
source by its Norton equivalent ⇒ the augmented MNA equation is

  

Y1 + Y2
0
I V1
Y1 Vin

ωC + G I
0 jω
−I V2=
0,
(12)
ωL
I
−I −jω
IL
0
where G =
1
R
and each block is a matrix of dimension 2N + 1.
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Transfer Function Modeling for the Buck converter
May 10, 2016
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Modeling the Buck converter
Augmented MNA for PSL circuits
MNA for PSL circuits
IL contains coefficients of truncated frequency domain representation
IL (ω) =
N
X
In δ(ω − nωs − ω0 ),
(13)
n=−N
with ω0 , the excitation frequency. Similarly for V1 , V2 .

Yi,0
Yi,−1 . . .
Yi,−2N
 Yi,1
Y
.
.
.
Y
i,0
i,−2N+1

Yi = 
..
.. . .
..

.
.
.
.
Yi,2N
Yi,2N−1
...
Y1 and Y2 are Toeplitz



 , i = 1, 2,

(14)
Yi,0
Y1,n are Fourier coefficients of window function with amplitude 1 when S is on:
Z DT
e −jnωs DT − 1
e −jnωs t dt =
, n = −2N, . . . , 2N,
Y1,n = fs
−j2πn
0
in Y1 , and Y2,n = Y1n e −jωs nDT , m = −2N, . . . , 2N in Y2 , for S due to delay DT .
ω0 − Nωs . . . ω0 + Nωs
ω = diag
.
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Transfer Function Modeling for the Buck converter
May 10, 2016
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Modeling the Buck converter
Augmented MNA for PSL circuits
MNA for PSL circuits
Small-signal analysis: u(t) = u0 + ũe jΩt , with u0 >> ũ and Ω, perturbation
frequency ⇒ U(ω) = u0 δ(ω) + ũδ(ω − Ω). Linear system solved for each input




0
0
UDC =  u0  , UAC =  ũ  , 0 ∈ Rn×1 .
0
0
We have V2 = V2,DC + V2,AC , where
V2,DC (ω)
=
N
X
Vn,DC δ(ω − nωs ),
(15)
Vn,AC δ(ω − nωs − Ω),
(16)
n=−N
V2,AC (ω)
=
N
X
n=−N
with coefficients Vn,DC , Vn,AC found by solving (12) for ω0 = 0, ω0 = Ω. The
transfer function at perturbation frequency Ω is
(
V0,AC
if Ω 6= nωs
ũ ,
.
(17)
HMNA (jΩ) = Vn,DC
+Vn,AC
,
if Ω = nωs
ũ
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Transfer Function Modeling for the Buck converter
May 10, 2016
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Modeling the Buck converter
Results
Simulation and Experimental Results
Buck converter with parameters: L = 1mH, C = 500µF, R = 12Ω,
fs = 20kHz= T1 , duty cycle D = 12 .
Figure : Simulation results and experimental validation
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Transfer Function Modeling for the Buck converter
May 10, 2016
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Conclusion
Outline
1
Modeling Systems using Frequency Domain Measurements
2
Transfer Function Modeling for the Buck converter
Average model of the Buck converter
Generalized Transfer Function (GTF)
Augmented MNA for PSL circuits
Results
3
Conclusion
Sanda Lefteriu (Ecole des Mines de Douai)
Transfer Function Modeling for the Buck converter
May 10, 2016
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Conclusion
Conclusion & Future work
Presented:
The line-to-output transfer function was simulated using the average model, the
GTF and the MNA for PSL circuits.
Its magnitude was measured with a spectrum analyzer.
Next steps:
Consider other modeling techniques (e.g., take into account parasitics).
Measure the phase.
Model, simulate and measure the control-to-output transfer function.
Thank you for your attention!
Sanda Lefteriu (Ecole des Mines de Douai)
Transfer Function Modeling for the Buck converter
May 10, 2016
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Conclusion
References
Biolek, D. (1997).
Modeling of periodically switched networks by mixed s-z description.
IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, 44(8), 750–758.
Biolek, D., Biolkova, V., & Dobes, J. (2006).
Modeling of switched DC-DC converters by mixed s-z description.
In Proceedings of the IEEE International Symposium on Circuits and Systems (pp. 831–834).
Middlebrook, R. & Cuk, S. (1976).
A general unified approach to modelling switching-converter power stages.
In IEEE Power Electronics Specialists Conference (pp. 18–34).
Trinchero, R. (2015).
EMI Analysis and Modeling of Switching Circuits.
PhD thesis, Politecnico di Torino.
Zadeh, L. A. (1950).
Frequency analysis of variable networks.
Proceedings of the IRE, 38(3), 291–299.
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Transfer Function Modeling for the Buck converter
May 10, 2016
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