Small-Signal Analysis of Closed-Loop PWM Boost
Converter in CCM with Complex Impedance Load
Dalvir K. Saini, Agasthya Ayachit,
and Marian K. Kazimierczuk
Hiroo Sekiya
Graduate School of Advanced Integration Science
Chiba University
Chiba, 263-8522, Japan
Email: sekiya@faculty.chiba-u.jp
Department of Electrical Engineering
Wright State University
Dayton, Ohio 45435, USA
Email: {saini.11, ayachit.2, marian.kazimierczuk}@wright.edu
Abstract—The following closed-loop transfer functions of the
boost converter operating in continuous-conduction mode (CCM)
supplying a complex impedance load are derived and analyzed:
input-to-output voltage Mvcl and reference-to-output Tcl . The
load of the boost dc-dc converter is composed of a seriesconnected resistance and inductance. The dynamic characteristics
of the closed-loop boost converter with a third-order doublelead integral compensator are evaluated for different load inductances. The theoretically predicted results are validated through
switching-circuit simulations using a suitable converter design
example.
I. I NTRODUCTION
A typical distributed power system involves a cascaded
connection of a main dc voltage supply (or a dc feeder)
linked to subsequent point-of-load (POL) dc-dc converters,
each functioning at different operating conditions. The input
impedance of the POL dc-dc converters exhibit characteristics
such as (a) negative resistance up to the control bandwidth and
(b) reactive impedance from low- to high-frequencies. Thus,
the effective load impedance to the main dc voltage supply can
be treated as a series combination of the load resistance RL
and load reactance XL [1]- [5]. The present work considers a
closed-loop boost converter with a series-connected resistance
and inductance load.
The consequences of adding a high load inductance on
the dynamics of the open-loop duty cycle-to-output transfer
function of the boost converter were discussed in [5]. It was
shown that the right-half plane zero zp moves closer to origin
and enters the left-half plane as the value of the inductance
was increased. The following conclusion was made:
⎧
⎨ right-half plane for LL < L/(1 − D)2
+∞
for LL = L/(1 − D)2
ωzp = zp =
⎩
left-half plane
for LL > L/(1 − D)2 ,
(1)
where LL is the load inductance, L is the inductance at the
input of the boost converter, and D is the duty cycle.
This paper extends the previous work to analyze the characteristics of the main dc voltage supply controlled by a doublelead integral controller. The expressions for the closed-loop
reference-to-output transfer function Tpcl and the closed-loop
978-1-4799-5341-7/16/$31.00 ©2016 IEEE
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Fig. 1. Complete small-signal model of the boost converter in CCM.
input-to-output transfer function Mvcl are derived and their
frequency and time responses are carefully studied.
The objectives of this paper are as follows:
• To adopt the load inductance model to an already existing
small-signal model of the closed-loop boost converter
with resistive load.
• To develop the expressions for the closed-loop controlto-output Tpcl and the closed-loop input-to-output voltage
Mvcl of the closed-loop boost converter in CCM.
• To analyze the effect of different values of the load
inductance on the location of poles and zeros of the
derived transfer functions.
II. O PEN -L OOP T RANSFER F UNCTIONS
The open-loop control-to-output transfer function Tp and
input-to-output voltage transfer function Mv for the boost
converter with complex impedance load has been derived in
[5]. This section presents a discussion of the transfer functions
in brief.
A. Open-Loop Control-to-Output Transfer Function Tp
Fig. 1 shows the small-signal model of the boost converter
with an inductive load. The impedances in the model can be
lumped and expressed as follows.
Z2 =
Z1 = r + sL,
1
rC +
||(RL + sLL ).
sC
(2)
(3)
The open-loop control-to-output voltage transfer function of
the boost converter with impedance load is obtained by setting
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Fig. 2. Circuit of boost DC-DC PWM converter with voltage-mode control
with inductive laod.
Fig. 3. Complete closed-loop small-signal model of the boost converter in
CCM.
vi and io to zero. Thus, the transfer function in terms of
impedances can be expressed as [5]
vo (s) Tp (s) =
d(s) vi =0, io =0
(4)
VO (1 − D)2 RL Z2 − Z1 Z2
=
,
1 − D (1 − D)2 RL Z2 + RL Z1
A. Design of Controller
The transfer function Tp has a strong dependence on the
value of the inductance LL . For smaller values of the load
inductance, one of the two zeros of Tp exists on the right-half
of the s-plane, while for higher values of the inductance LL ,
the zero shifts to the left half-plane altering the dynamics of
the boost converter. As LL increases, the ringing in its step
response also increases.
To obtain a wider closed-loop bandwidth and hence a faster
step response, an integral-double-lead (IDL) compensator is
used. The feedback path impedance Zf of the controller is [6]
Zf =
where the impedances Z1 and Z2 are as given in (2) and (3),
respectively. Including the load inductance LL into the transfer
function results in an additional zero, whose location is always
in the left-half of the s-plane.
where
h11 =
ωz(Zi) =
(6)
RA RB
,
RA + RB
(7)
(8)
R1 + h11
,
C3 [R3 (R1 + h11 ) + h11 R1 ]
(9)
1
.
C3 (R1 + R3 )
(10)
ωp(Zi) =
The voltage transfer function Tc of the integral double-lead
controller is given as [6]
Tc (s) =
[s + ωz1(T c) ][s + ωz2(T c) ]
vc (s)
= Tcx
,
ve (s)
s[s + ωp1(T c) ][s + ωp2(T c) ]
(11)
where the high-frequency gain is
Tcx =
III. C LOSED -L OOP T RANSFER F UNCTIONS
Fig. 2 shows the circuit of the closed-loop boost converter
with third-order integral double-lead controller. The controller
has a pole at the origin and two pole-zero pairs. The controller
achieves a low dc steady-state error and allows a wide closedloop bandwidth. Fig. 3 shows a small-signal model of closedloop boost converter with the transfer functions of the feedback
network, controller, and pulse-width modulator represented as
β, Tc , and Tm , respectively.
sC2 (s
and the input path impedance Zi of the controller is
s + ωz(Zi)
R1 R3
,
Zi = h11 +
R1 + R3 s + ωp(Zi)
B. Open-Loop Input-to-Output Voltage Transfer Function Mv
Using the analysis presented in [5], [6], [7], the expression
for the open-loop input-to-output voltage transfer function in
terms of the impedances is given by
vo (s) Z2 (1 − D)
=
,
Mv (s) =
vi (s) d=0, io =0
Z1 + Z2 (1 − D)2
(5)
1
R 2 C1
,
2
+ RC21C+C
)
1 C2
s+
R1 + R3
,
C2 [R1 R3 + h11 (R1 + R2 )]
(12)
and the poles and zeros are
ωz1(T c) = ωp(Zi) ,
ωp1(T c) =
C1 + C2
,
R2 C1 C2
ωz2(T c) =
1
,
R2 C1
ωp2(T c) = ωz(Zi) .
(13)
(14)
The following sections utilize the controller transfer function
Tc in deriving the closed-loop transfer functions.
434
B. Closed-Loop Control-to-Output Transfer Function Tpcl
The closed-loop control-to-output transfer function is expressed as [6], [7]
vo (s) Tc (s)Tm Tp (s)
, (15)
Tpcl (s) =
=
vr (s)
1 + βTc (s)Tm Tp (s)
vi =0, io =0
vr =0, io =0
where Mv is the open-loop audio susceptibility given in (5).
IV. R ESULTS AND D ISCUSSION
A. Design of Boost DC-DC Converter
A boost converter is designed for the following specifications: input dc voltage VI = 12 V, switching frequency
fs = 100 kHz, minimum output power POmax = 10 W,
and the dc output voltage is VO = 20 V. Using the design
equations presented in [6], the values of the boost inductor
and capacitors for a nominal duty ratio of D = 0.46 are
found to be: L = 156 μH, and C = 6.8 μF. The equivalent
average resistance (EAR) considered in the inductor branch
is r = 0.24 Ω. The equivalent series resistance of the filter
capacitor is rC = 0.111 Ω.
B. Compensator Design
An integral-double-lead controller is designed for the boost
converter with the specifications discussed earlier. With a
reference voltage VR = 2.5 V, the voltage transfer function of
R
B
= RAR+R
.
the feedback network is calculated to be β = VVO
B
The resistances in the feedback network are assumed as
RB = 620 Ω and RA = 4.3 Ω. The h parameters are
RB
= 542 Ω, and h112 >> RL
calculated as: h11 = RRAA+R
B
(can be neglected). The components of the integral-doublelead controller have values: R1 = 100 Ω, R2 = 100 Ω,
R3 = 12.2 kΩ, C1 = 4.7 nF, C2 = 150 pF, and C3 = 4.7 nF.
The loop gain transfer function T was evaluated and the
following results were found for the designed boost converter
with integral double-lead controller. The phase margin was
P M = 60.5◦ , gain margin GM = 11.5 dB, and a cross-over
frequency fc = 4 kHz was used. The theoretically obtained
plots of the transfer functions were validated using Saber
switching circuit simulations and the results are presented in
the following section.
LL = 5.3 mH
L = 0.53 mH
L
LL = 53 μH
3
10
4
10
5
10
5
10
10
6
f (Hz)
0
pcl
−100
−200
−300
C. Closed-Loop Input-to-Output Transfer Function Mvcl
The small-signal model of the closed-loop boost converter
required to determine the input-to-output transfer function
Mvcl is obtained by setting vr and io to zero in Fig. 3. The
expression for the closed-loop input-to-output transfer function
in terms of the impedances is given by [6], [7]
vo (s) Mv (s)
, (16)
Mvcl (s) =
=
vi (s) 1 + βTc (s)Tm Tp (s)
−100
−150
2
10
φ T (°)
where Tc (s) is the voltage transfer function of the controller
given in (11), Tp (s) is the open-loop control-to-output transfer
function given in the impedance form in (4), the β is the
B
, while the transfer
feedback factor given by β = RAR+R
B
function of the pulse-width modulator is Tm = VT1m , where
VT m is the amplitude of the sawtooth waveform.
−50
pcl
| T | (dB)
0
−400
2
10
3
10
4
10
10
6
f (Hz)
Fig. 4. Theoretically obtained Bode plots of the closed-loop control-to-output
transfer function Tpcl at the three different values of LL .
C. Analysis of Tpcl
Fig. 4 shows the Bode magnitude and phase plot of the
closed-loop reference-to-output transfer function Tpcl for the
different values of the load inductance. In [5], the authors
verified the movement of the inherent RHP zero to the LHP
as the load inductance was increased to LL ≥ L/(1 − D)2 .
However, in the closed-loop reference-to-output transfer function, although the movement of the RHP zero towards the
origin was observed, it crossed the origin only at extremely
high values of the load inductance. Thus, a closed-loop boost
converter with inductance load has characteristics similar to
a closed-loop boost converter with resistive load for a wide
range of load inductance values. In other words, the presence
of complex load does not affect the dynamics of the boost
converter. The theoretically obtained Bode plots were validated
through Saber switching circuit simulations and the result is
shown in Fig. 5. Fig. 6 shows the response of the output
voltage for step changes in the duty cycle obtained using Saber
circuit simulator. The rise time, overshoot, and the steady-state
error for the response plots for all the inductances are identical.
D. Analysis of Mvcl
Fig. 7 shows the Bode magnitude and phase plots of the
closed-loop input-to-output voltage transfer function for the
different values of the load inductance. The magnitude and
phase plots have identical characteristics for each of the three
selected inductances, indicating the insensitivity of the closedloop input-to-output transfer function to changes in the load
inductance. Fig. 8 shows the simulation results obtained using
Saber validating the theoretically predicted model of the boost
converter. Further, Fig. 9 shows the output voltage response
for small-signal changes in the input voltage obtained for the
different load inductance values.
V. C ONCLUSION
Small-signal analysis of the closed-loop boost dc-dc converter with an impedance load has been presented in this paper.
435
Mag_Mvcl (dB) : f(Hz)
Mag_Tpcl (dB) : f(Hz)
Tpcl@LL = 5.3 mH
10.0
Tpcl@LL = 0.53 mH
0.0
−10.0
Tpcl@LL = 53 uH
−20.0
−30.0
Mvcl@LL = 0.53 mH
−20.0
−40.0
Mvcl@LL = 53 uH
−60.0
−80.0
−40.0
Ph_Tpcl (deg) : f(Hz)
Ph_Mvcl (deg) : f(Hz)
Tpcl@LL = 5.3 mH
Tpcl@LL = 0.53 mH
−100.0
Mvcl@LL = 5.3 mH
100.0
Ph_Mvcl (deg)
0.0
Ph_Tpcl (deg)
Mvcl@LL = 5.3 mH
0.0
Mag_Mvcl (dB)
Mag_Tpcl (dB)
20.0
Tpcl@LL = 53 uH
−200.0
Mvcl@LL = 0.53 mH
0.0
Mvcl@LL = 53 uH
−100.0
−200.0
−300.0
10.0
100.0
1.0k
10.0k
f(Hz)
100.0k
1meg
10.0
Fig. 5. Simulated magnitude and phase Bode plots of the closed-loop controlto-output transfer function Tpcl at the three different values of LL .
100.0
1.0k
10.0k 100.0k 1meg
f(Hz)
Fig. 8. Simulated magnitude and phase Bode plots of the closed-loop inputto-output voltage transfer function Mvcl at the different values of LL .
vo (V) : t(s)
vo (V) : t(s)
Tpcl@LL = 5.3 mH
Mvcl@LL = 5.3 mH
1.0
10.0
Tpcl@LL = 0.53 mH
Mvcl@LL = 0.53 mH
0.8
Tpcl@LL = 53 uH
5.0
vo (V)
vo (V)
Mvcl@LL = 53 uH
0.6
0.4
0.2
0.0
0.0
500u
1.0m
t(s)
1.5m
0.0
2.0m
0.0
Fig. 6. Simulated results of the output voltage to a step change in the reference
voltage (Δvr = 1) at different values of LL .
−20
vcl
| M | (dB)
0
−40
−60
−80
2
3
10
4
10
5
(°)
vcl
M
10
L
L = 53 μH
L
−50
−100
−150
−200
2
10
3
10
4
10
5
10
2.0m
2.5m
3.0m
R EFERENCES
L = 0.53 mH
0
1.5m
t(s)
Fig. 9. Simulated magnitude and phase Bode plots of the closed-loop inputto-output voltage transfer function Mv at the different values of LL .
LL = 5.3 mH
50
φ
6
10
f (Hz)
1.0m
open-loop transfer functions derived by the authors in [5]. In
conclusion, as the load inductance of the closed-loop boost
converter increases (a) the location of the inherent RHP zero
in the control-to-output transfer function Tpcl moves closer to
the origin, and (b) the dynamics of the input-to-output transfer
function remains unchanged.
20
10
500u
6
10
f (Hz)
Fig. 7. Theoretically obtained Bode plots of the closed-loop input-to-output
voltage transfer function Mvcl at the different values of LL .
The feedback and control network consists of a third-order
double-lead integral compensator. The small-signal transfer
functions: the closed-loop control-to-output transfer function
Tpcl and the input-to-output voltage transfer function Mvcl
have been derived. Frequency-domain and time-domain characteristics of these expressions have been analyzed. It has
been shown that the addition of the load inductance does
not effect the closed-loop transfer functions as opposed to the
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