III. DC/DC and DC/AC Converters with Spread Spectrum Technique
3.3 Performance enhancement of output voltage regulation for DC/DC converters using
3.3.2 Power stage analysis of IBI LLC resonant converter
The IBI LLC resonant converter is numerically expressed using a first harmonic approximation (FHA). Fig. 3-17 (a) and (b) show the key operating waveforms of the IBI LLC resonant converter to analyze the input-to-output voltage gain. Since the IBI LLC resonant converter can be expressed as the cascaded boost converter and LLC resonant converter equivalently as shown in Fig. 3-17 (c), the link voltage can be expressed by the voltage gain of the boost converter (V =V /D) [40]. The inverter
D
1C
rL
rL
mV
inS
1V
linkL
1L
2V
inv+ -
D
2C
oR
oV
oS
2S
3S
4voltage changes according to the link voltage and the duty cycle. From the Fourier series, the first harmonic of the inverter voltage can be obtained which is expressed in (3-11) [41].
(a)
(b)
(c)
Fig. 3-17. Theoretical operation of IBI LLC resonant converter: (a) Key operating waveform when D is lower than 0.5, (b) Key operating waveform when D is higher than 0.5, (c) Equivalent circuit diagram of IBI LLC resonant converter.
D1
Cr
Lr
Lm
D2
Co Ro Vo
Vin
L1
L2
S3
S4
S1
S2
Vinv +
-
S3
S4
S1
S2
Vlink
Interleaved boost converters
Full-Bridge inverter
( ) ( )
1
2 2
2 1 cos 2 sin
2
link inv
s
V t
v t D
T
= − +
(3-11) where vinv1 is the first harmonic of the inverter voltage, Vlink is the link voltage, D is the duty cycle, and Ts is the switching period. Using the FHA, the voltage gain of the resonant network can be expressed as follows:
( ) ( )
( )
, 2
2 2
1 1 1 1
o FHA n n
n
inv n n n n n
V L f
M f = V = L f j f f QL
+ − + −
(3-12)
2 2
1 8
, , , r r ,
m s
n n r oe o
r r r r oe
L f L C
L f f Q R n R
L f L C R
= = = = =
(3-13) where Vo,FHA is the first harmonic of the secondary side voltage, M is the voltage gain of the resonant network, Ln is the inductor ratio, fn is the normalized switching frequency, fs is the switching frequency, fr and Q are the series resonant frequency and the quality factor of the resonant network, respectively, Lr and Cr is the resonant inductor and capacitor of the resonant network, respectively, Roe is primary referred effective load resistance, n is the turn ratio of the transformer, and Ro is the load resistance.
From (3-11)-(3-13), the input-to-output voltage gain of the IBI LLC resonant converter according to the duty and switching frequency can be derived as follows:
( , n) o ( ) ( )n
in
G D f V H D M f
=V =
(3-14)
( )
2 1 cos 2( )
2
H D n D
D
= −
(3-15) where H is the voltage gain according to the duty cycle variation.
Fig. 3-18 shows the voltage gain curves of IBI LLC resonant converter according to the duty cycle variation. By modulating duty cycle of the inverter, it can obtain wide input-to-output voltage gain.
Besides, the theoretical voltage gain curves are well-matched with the experimental results. Fig. 3-19 shows the voltage gain curves according to the normalized switching frequency with different resonant network designs in terms of the quality factor (Q) and inductor ratio (Ln). The voltage gain of the IBI LLC resonant converter is dependent on both the duty ratio and the switching frequency. Since the switching frequency changes periodically according to the SST, the major control variable of the IBI LLC resonant converter is the duty cycle. To implement the SST with the low output voltage fluctuation, the voltage gain can be insensitive according to the switching frequency variation by the resonant network design. The resonant network design with a high inductor ratio and low-quality factor can reduce the output voltage fluctuation of the IBI LLC resonant converter employing the SST.
Fig. 3-18. Input-to-output voltage gain of IBI LLC resonant converter according to duty cycle.
(a)
(b)
Fig. 3-19. Input-to-output voltage gain of IBI LLC resonant converter: (a) Voltage gain curve according to quality factor, (b) Voltage gain curve according to inductor ratio.
On the other hand, the soft-switching region of the IBI LLC resonant converter is closely related to the magnetizing inductance, which reduces the practical inductor ratio. The soft-switching region according to the resonant network design can be analyzed using a phaser analysis. The zero-voltage switching (ZVS) region of the inverter is determined by the turn-on current of the lower side switches (S2 and S4) regardless of the operating condition for the IBI LLC resonant converter. The turn-on current of the S2 is expressed in (3-16).
( ) ( ) ( )
2 2, 1 2, 2,
S S on L S on pri S on
I t =I t −I t
(3-16) where IL1 is the current passing through the input inductor L1, and Ipri is the primary side current. The input inductor current when S2 turns on can be obtained as follow:
( )
( )1 2, 1,min 1
1
1
L S on L L 2 in s
I t I I D DV T
L
= = + −
(3-17) Where IL1 is the average current passing through the input inductor L1 (=IL1=Pout Vin2). In addition, the primary side current can be obtained using the simple phaser analysis as follows:
1
pri inv pri pri
in
I V I I
= Z =
(3-18)
1 m oe
in r
r m oe
j L R
Z j L
j C j L R
= + +
+ (3-19)
( )
2,sin 2
pri s on pri 2 pri
s
I t I D I
T
= +
(3-20) where Zin is the input impedance of the resonant network.
Fig. 3-20 shows the turn-on current of the power switch S2 according to the duty cycle and inductor ratio. The duty cycle, which controls the output voltage, is designed to obtain the required input-to- output voltage gain with the PV panel voltage variation. In addition, the input inductors are designed to suppress the input current ripple by 10% compared to the input current rms value since the higher current ripple applied to the PV panel reduces the average power generation from the PV panel. With given power stage specifications, the inductor ratio changes the ZVS region. As shown in Fig. 3-20, a high inductor ratio cannot provide the ZVS condition at low duty cycle condition. Therefore, the inductor ratio of the resonant network is limited to provide the ZVS condition for the entire duty cycle range. Consequently, the output voltage fluctuation caused by the SST increases due to the limited inductor ratio.
Fig. 3-20. Turn-on current of the power switch S2 according to the duty cycle and inductor ratio.
Fig. 3-21. Expected output voltage fluctuation according to inductor ratio.
Fig. 3-21 shows the output voltage fluctuation induced by the switching frequency varies [fc -Δf, fc
+Δf] according to the inductor ratio at the nominal duty cycle (D=0.5). When the inductor ratio is designed as 6 to provide the soft-switching condition for entire duty cycle conditions as shown in Fig.
3-19, the output voltage fluctuation is expected as 8% of the nominal output voltage rms value without considering the practical output voltage ripple due to the switching operation. Therefore, the resonant network design with the high inductor ratio and low-quality factor cannot be an effective solution to suppress output voltage fluctuation induced by the SST considering the soft-switching capability of the IBI LLC resonant converter.