Texas Instruments | AC Modeling of Power Stage in Flyback Converter | Application notes | Texas Instruments AC Modeling of Power Stage in Flyback Converter Application notes

Texas Instruments AC Modeling of Power Stage in Flyback Converter Application notes
Application Report
SLVA589 – July 2013
AC Modeling of Power Stage in Flyback Converter
Rushil K K
PWR-DCDC Controllers
ABSTRACT
This application report provides a step-by-step procedure for constructing the AC model for the
power stage of a Flyback DC/DC Converter in Continuous Conduction Mode. The model is used in
the TIna simulator to plot and verify the control-to-output transfer function.
1
2
3
4
5
Contents
Introduction .................................................................................................................................. 1
1.1 How it Works .......................................................................................................................... 2
Circuit Manipulations ................................................................................................................... 3
Construction of the Mathematical Model .................................................................................... 5
Simulation and Results ................................................................................................................ 8
Conclusion: ................................................................................................................................ 11
Figure 1.
Figure 2.
Figure 3.
Figure 4.
Figure 5.
Figure 6.
Figure 7.
Figure 8.
1
Figures
Basic Topology of Flyback Converter ............................................................................ 2
Separating Out the Non-Linear Switch Elements .......................................................... 3
Voltage Waveforms for the 2-Port Network.................................................................... 4
Current Waveforms for the 2-Port Network.................................................................... 4
Replacing the Non-Linear Switch Network with Equivalent Linear Network ............... 7
Complete Linear Circuit for Flyback Converter ............................................................. 8
Implementation of Example Case in Tina Simulator ................................................... 10
Control-to-Output Transfer Function Plotted using TIna ............................................ 11
Introduction
The flyback DC-DC converter is derived from the basic buck-boost converter. The conversion
ratio for this converter is given by,
𝐷
π‘‰π‘œπ‘’π‘‘
= 𝑛.
1βˆ’π·
𝑉𝑖𝑛
As seen from this equation, the conversion ratio is similar to that of the buck-boost converter
with an included factor of n, which is the turns ratio of the flyback transformer.
1
SLVA589
1.1
How it Works
In a flyback converter, when the current ramps up in the primary, energy is built up in the
transformer core. When the FET switches off, this energy is dumped into the secondary,
allowing current to flow in the secondary, so the primary current collapses immediately to zero.
This causes an immediate rise in current in the secondary, after which, the current in the
secondary ramps down linearly as it discharges, supplying energy to the load. The current in the
secondary when the FET switches off, is proportional to the peak in the primary, and is
determined by the turns ratio.
The circuit topology of the flyback converter is as shown in Figure 1.
Figure 1.
2
Basic Topology of a Flyback Converter
AC Modeling of Power Stage in Flyback Converter
SLVA589
2
Circuit Manipulations
The circuit can be re-drawn, to separate out the reactive components and the switching components, as shown in Figure 2.
Figure 2.
Separating Out the Non-Linear Switch Elements
The switching elements (MOSFET and Diode) are shown in the dashed rectangular box. These
are non-linear switching elements. The box is treated as a 2-port network with input voltage
𝑣1 (𝑑), input current 𝑖1 (𝑑), output voltage 𝑣2 (𝑑), and output current 𝑖2 (𝑑).
The waveforms for 𝑣1 (𝑑) and 𝑣2 (𝑑) over one switching period, 𝑇𝑠 , are shown in Figure 3. A
simplifying assumption of zero MOSFET and diode conduction drop has been made for
obtaining the waveform shown below.
AC Modeling of Power Stage in Flyback Converter
3
SLVA589
Note: The slope of the voltage waveform in the figure is exaggerated for the sake of clarity.
Figure 3.
Voltage Waveforms for the 2-Port Network
Similarly, the waveforms for 𝑖1 (𝑑) and 𝑖2 (𝑑) are shown in Figure 4.
Figure 4.
4
Current Waveforms for the 2-Port Network
AC Modeling of Power Stage in Flyback Converter
SLVA589
3
Construction of the Mathematical Model
The waveforms in Figure 3 and Figure 4 are averaged over one switching period, which
effectively removes the switching harmonics. The switching harmonics have no role of
consequence in the development of the AC model. Observing the waveform, it can be seen that:
< π‘½πŸ (𝒕) >𝑻𝒔 = 𝒅′ (𝒕) βˆ— {π‘½π’Šπ’ + 𝑽𝒐�𝒏 }
< π‘½πŸ (𝒕) >𝑻𝒔 = 𝒅(𝒕) βˆ— {π’π‘½π’Šπ’ + 𝑽𝒐 }
Dividing the above equations, we get the following relation:
Hence, we have:
𝟏 𝒅′ (𝒕)
< π‘½πŸ (𝒕) >𝑻𝒔
οΏ½< 𝑽 (𝒕) > = βˆ—
𝒏 𝒅(𝒕)
𝟐
𝑻𝒔
< π‘½πŸ (𝒕) >𝑻𝒔 =
𝟏
𝒏
βˆ—
𝒅′ (𝒕)
𝒅(𝒕)
βˆ—< π‘½πŸ (𝒕) >𝑻𝒔
(1)
Equation 1 is a non-linear one, since it contains products of time-dependent variables. To obtain
a linear relation, the variables are perturbed about the quiescent operating point.
𝒅(𝒕) = 𝐷 + 𝑑̂
< π‘½πŸ (𝒕) >𝑻𝒔 = 𝑉1 + 𝑣
οΏ½1
Using the preceding equations in (1):
οΏ½2
< π‘½πŸ (𝒕) >𝑻𝒔 = 𝑉2 + 𝑣
𝟏 𝐷′ βˆ’ 𝑑̂
βˆ—
βˆ— (π‘½πŸ + π’—οΏ½πŸ )
𝒏 𝐷 + 𝑑̂
οΏ½ +𝒗
οΏ½ = 𝟏 βˆ— {𝑽 βˆ— 𝑫′ βˆ’ 𝑽 βˆ— 𝒅
οΏ½ +𝒗
οΏ½}
�𝟏 βˆ— 𝑫 + 𝒗
�𝟏 βˆ— 𝒅
�𝟐 βˆ— 𝑫′ βˆ’ 𝒗
�𝟐 βˆ— 𝒅
π‘½πŸ βˆ— 𝑫 + π‘½πŸ βˆ— 𝒅
𝟐
𝟐
𝒏
�𝟏 =
π‘½πŸ + 𝒗
οΏ½ and 𝒗
οΏ½ are small enough to
�𝟏 βˆ— 𝒅
�𝟐 βˆ— 𝒅
Since the perturbations are small, the non-linear terms 𝒗
be ignored. Upon rearranging and grouping terms appropriately:
(2)
The equation is re-arranged into the below form,
(3)
𝟏
Note: Equating the quiescent terms in Equation 2, we have π‘½πŸ βˆ— 𝐷 = 𝒏 π‘½πŸ βˆ— 𝑫′.
Substituting this relation for π‘½πŸ in Equation 3,
AC Modeling of Power Stage in Flyback Converter
5
SLVA589
Now, D+D’=1.
So the above equation finally reduces to:
(A)
Using the same procedure for the 2-port network currents 𝑖1 (𝑑) and 𝑖2 (𝑑), Equation B is obtained:
(B)
Equations (A) and (B) are the relations for the input voltage and output current of the 2-port
network and are used to obtain the linear model for the averaged switch network.
These relations can be implemented by a network consisting of an ideal transformer and an
independent voltage and current source, both driven by duty cycle variations, d(t), as shown in
Figure 5.
6
AC Modeling of Power Stage in Flyback Converter
SLVA589
Figure 5.
Replacing the Non-Linear Switch Network with an Equivalent Linear Network
The linear model for the switching element thus obtained is used in place of the MOSFET and
diode in the rectangular dashed box of Figure 2 to obtain the circuit shown in Figure 6.
AC Modeling of Power Stage in Flyback Converter
7
SLVA589
Figure 6.
Complete Linear Circuit for Flyback Converter
The inductance 𝐿𝑀 indicates the magnetizing inductance, and is used with an ideal transformer
of turns ratio 1:n.
4
Simulation and Results
Example:
Consider a flyback converter designed for the following specifications: 𝑉𝑖𝑛 = 5𝑉, π‘‰π‘œπ‘’π‘‘ = 10𝑉. A
load resistance of 10 Ξ© is connected at the output, so the load current drawn is 1 A. The
magnetizing inductance of the flyback transformer is 6 µH, and load capacitance used is 500 µF.
The turns ratio n=4, hence, the duty cycle, D, required to produce a 10-V output is D = 0.33.
The quiescent value for 𝑣1 (t), is 𝑉1 = 𝐷 β€² . (𝑉𝑖𝑛 + π‘‰π‘œ /𝑛), as seen from waveform.
Similarly, 𝐼2 = πΌπ‘™π‘œπ‘Žπ‘‘
Hence, 𝑉1 = 4.95 V.
Also, 𝐼2 = 1 A.
Now, for modeling in TIna, the voltage and current sources controlled by d(t) are implemented
using the VCVS and VCCS sources respectively, as shown in Figure 7.
The gain of the VCVS source is set to be
ac model is
𝑉1
𝐷.𝐷′
βˆ— 𝑑̂ )
𝑉1
𝐷.𝐷′
(since the voltage source obtained in the derived
so, the gain of VCVS source is 22.31.
8
AC Modeling of Power Stage in Flyback Converter
SLVA589
The gain of VCCS source must be
𝐼2
𝐷.𝐷′
(again, according to derived model).
The gain of VCCS source is calculated to be 4.5.
The turns ratio of the transformer (of the switch model) is nD/D’ = 2.
Verification:
For a flyback converter, the location of RHPZ and LC resonant pole, is at(1):
𝑓𝑅𝐻𝑃𝑍 =
Also, the gain at DC is,
(π‘…π‘œ βˆ— 𝐷′2 )
οΏ½ 2
(𝑛 βˆ— 2πœ‹ βˆ— 𝐿𝑀 βˆ— 𝐷)
𝑓𝐿𝐢 = 𝐷′�
(𝑛 βˆ— 2πœ‹ βˆ— �𝐿𝑀 𝐢)
𝐺𝑑0 = 𝑉0 /𝐷𝐷′
Using the example values in the above equations, we should have the following:
𝑓𝑅𝐻𝑃𝑍 = 21.88 kHz
𝐺𝑑0 = 45.09
𝑓𝐿𝐢 = 883.3𝐻𝑧
𝐺𝑑0 |𝑑𝐡 = 33.2𝑑𝐡
AC Modeling of Power Stage in Flyback Converter
9
SLVA589
The circuit is constructed in TIna Simulator as shown in Figure 7.
Figure 7.
Implementation of the Example Case in TIna Simulator
Running the AC analysis, the bode plot for the control-to-output transfer function is obtained as
shown in Figure 8.
10
AC Modeling of Power Stage in Flyback Converter
SLVA589
Figure 8.
Control-to-Output Transfer Function Plotted using TIna
As seen from Figure 8, the LC resonant pole appears at around 862 Hz.
The RHPZ is located at around 22.13 kHz.
Gain at DC is 33 dB. Hence, it is seen that the response obtained from the model matches
closely with the theoretical calculations.
5
Conclusion
The AC model of the power stage for a flyback converter can be constructed following the steps
shown in this application report. The response of the model, implemented in a TIna Simulator, is
found to be in alignment with what is theoretically expected.
6
References
1. Basso, C., β€œHow to Keep a FLYBACK Switch Mode Supply Stable with a Critical-Mode
Controller,” ON Semiconductorβ„’, AN1681/D Application Note, September 2000 – Rev. 0
AC Modeling of Power Stage in Flyback Converter
11
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