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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN"><html> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> <!--This HTML is auto-generated from an M-file.To make changes, update the M-file and republish this document. --> <title>Configurable Simulink Model for DC-DC Converters</title> <meta name="generator" content="MATLAB 7.7"> <meta name="date" content="2009-03-31"> <meta name="m-file" content="DCConverterPWMPID"><style type="text/css">body { background-color: white; margin:10px;}h1 { color: #990000; font-size: x-large;}h2 { color: #990000; font-size: medium;}/* Make the text shrink to fit narrow windows, but not stretch too far in wide windows. */ p,h1,h2,div.content div { max-width: 600px; /* Hack for IE6 */ width: auto !important; width: 600px;}pre.codeinput { background: #EEEEEE; padding: 10px;}@media print { pre.codeinput {word-wrap:break-word; width:100%;}} span.keyword {color: #0000FF}span.comment {color: #228B22}span.string {color: #A020F0}span.untermstring {color: #B20000}span.syscmd {color: #B28C00}pre.codeoutput { color: #666666; padding: 10px;}pre.error { color: red;}p.footer { text-align: right; font-size: xx-small; font-weight: lighter; font-style: italic; color: gray;} </style></head> <body> <div class="content"> <h1>Configurable Simulink Model for DC-DC Converters</h1> <!--introduction--> <!--/introduction--> <h2>Contents</h2> <div> <ul> <li><a href="#1">DC-DC Converters</a></li> <li><a href="#4">Principles</a></li> <li><a href="#5">Model under ideal assumptions</a></li> <li><a href="#6">Buck converter:</a></li> <li><a href="#7">Boost converter:</a></li> <li><a href="#8">Buck-boost converter:</a></li> <li><a href="#9">Normalized buck model</a></li> <li><a href="#10">Normalized boost model</a></li> <li><a href="#11">Normalized buck-boost model</a></li> <li><a href="#12">Model with body resistors</a></li> <li><a href="#13">Buck model with <img src="DCConverterPWMPID_eq56918.png" alt="$R_\mathrm{L}$"> and <img src="DCConverterPWMPID_eq86081.png" alt="$R_\mathrm{c}$"></a></li> <li><a href="#14">Boost model with <img src="DCConverterPWMPID_eq56918.png" alt="$R_\mathrm{L}$"> and <img src="DCConverterPWMPID_eq86081.png" alt="$R_\mathrm{c}$"></a></li> <li><a href="#15">Buck-boost model with <img src="DCConverterPWMPID_eq56918.png" alt="$R_\mathrm{L}$"> and <img src="DCConverterPWMPID_eq86081.png" alt="$R_\mathrm{c}$"></a></li> <li><a href="#16">Simulink Model</a></li> </ul> </div> <h2>DC-DC Converters<a name="1"></a></h2> <p>There are three kinds of switching mode DC-DC converters, buck, boost and buck-boost. The buck mode is used to reduce output voltage, whilst the boost mode can increase the output voltage. In the buck-boost mode, the output voltage can be maintained either higher or lower than the source but in the opposite polarity. The simplest forms of these converters are schematically represented in Figure 1. </p> <p><html> <img src="buck.png" width="180"> <img src="boost.png" width="180"> <img src="buck_boost.png" width="180"> </html></p> <p>Figure 1. Buck (left), Boost (middle) and Buck-Boost (right) converters.</p> <p>These converters consist of the same components, an inductor, <img src="DCConverterPWMPID_eq93576.png" alt="$L$">, a capacitor, <img src="DCConverterPWMPID_eq22407.png" alt="$C$"> and a switch, which has two states <img src="DCConverterPWMPID_eq77814.png" alt="$u=1$"> and <img src="DCConverterPWMPID_eq45348.png" alt="$u=0$">. All converters connect to a DC power source with a voltage (unregulated), <img src="DCConverterPWMPID_eq25938.png" alt="$V_\mathrm{in}$"> and provide a regulated voltage, <img src="DCConverterPWMPID_eq34436.png" alt="$v_\mathrm{o}$"> to the load resistor, <img src="DCConverterPWMPID_eq85930.png" alt="$R$"> by controlling the state of the switch. In some situations, the load also could be inductive, for example a DC motor, or approximately, a current load, for example in a cascade configuration. For simplicity, here, only current and resistive loads are to be considered. </p> <h2>Principles<a name="4"></a></h2> <p>The working principles of the these DC-DC converters can be explained as follows. In the buck mode, when the switch is on position 1, the DC source supplies power to the circuit which results an output voltage across the resistor. When the switch changes its position to 0, the energy stored in the inductor and capacitor will discharge through the resistor. Appropriately controlling the switching position can maintain the output voltage at a desired level lower than the source. </p> <p>In the boost mode, when the switch is on position 1, the circuit is separated into two parts: on the left, the source is charging the inductor, meanwhile the capacitor on the right maintains the output voltage using previously stored energy. When the switch changes its position to 0, both the DC source and energy stored in the inductor will supply power to the circuit on the right, hence boost the output voltage. Again, the output voltage can be maintain at desired level by controlling the switching sequence. </p> <p>Finally, for the buck-boost mode, switch positions 1 and 0 represents charging and discharging modes of the inductor. Appropriately controlling the switching sequence can result in output voltage higher or lower than the DC source. Since the inductor cannot change the direction of current, the output voltage is opposite to the DC source. </p> <h2>Model under ideal assumptions<a name="5"></a></h2> <p>Under ideal assumptions: ideal switch, ideal capacitor and ideal inductor, these converters can be described using ordinary differentiation equations as follows: </p> <h2>Buck converter:<a name="6"></a></h2> <p><img src="DCConverterPWMPID_eq22007.png" alt="$C{dv_{c} \over dt} = i_{L} - v_{c}/R - i_{o}$"></p> <p><img src="DCConverterPWMPID_eq20544.png" alt="$L{di_{L} \over dt} = uv_{in} - v_{c}$"></p> <p>where, <img src="DCConverterPWMPID_eq65970.png" alt="$i_\mathrm{o}$"> is the load current. </p> <h2>Boost converter:<a name="7"></a></h2> <p><img src="DCConverterPWMPID_eq30609.png" alt="$C{dv_\mathrm{c}\over dt}=(1-u)i_\mathrm{L}-v_\mathrm{c}/R-i_\mathrm{o}$"></p> <p><img src="DCConverterPWMPID_eq36756.png" alt="$L{di_\mathrm{L}\over dt}=v_\mathrm{in} - (1-u) v_\mathrm{c}$"></p> <h2>Buck-boost converter:<a name="8"></a></h2> <p><img src="DCConverterPWMPID_eq30609.png" alt="$C{dv_\mathrm{c}\over dt}=(1-u)i_\mathrm{L}-v_\mathrm{c}/R-i_\mathrm{o}$"></p> <p><img src="DCConverterPWMPID_eq83258.png" alt="$L{di_\mathrm{L}\over dt}=uv_\mathrm{in} - (1-u) v_\mathrm{c}$"></p> <p>Introduce the following state, time and load normalization:</p> <p><img src="DCConverterPWMPID_eq90066.png" alt="$x_1={v_\mathrm{c}\over v_\mathrm{in}}$">, </p> <p><img src="DCConverterPWMPID_eq12105.png" alt="$x_2={i_\mathrm{L}\over v_\mathrm{in}}\sqrt{{L}\over{C}}$">, </p> <p><img src="DCConverterPWMPID_eq57167.png" alt="$\tau = {t \over \sqrt{LC}}$">, </p> <p><img src="DCConverterPWMPID_eq91095.png" alt="$\gamma = {\sqrt{LC}\over R}$">, </p> <p><img src="DCConverterPWMPID_eq99797.png" alt="$d = {i_\mathrm o \over v_\mathrm{in}}\sqrt{{L}\over {C}}$"></p> <p>Then the normalized state equations of three converters are as follows:</p> <h2>Normalized buck model<a name="9"></a></h2> <p><img src="DCConverterPWMPID_eq55293.png" alt="$\dot{x_1} = -\gamma x_1 + x_2 - d$"></p> <p>$\dot{x_2}= -x_1 + u $</p> <p>where, with an abuse of notation, `.' represents the derivation with respect to the normalized time, <img src="DCConverterPWMPID_eq89224.png" alt="$\tau$">. </p> <h2>Normalized boost model<a name="10"></a></h2> <p><img src="DCConverterPWMPID_eq53325.png" alt="$\dot{x_1} = -\gamma x_1 + (1-u)x_2 - d$"></p> <p><img src="DCConverterPWMPID_eq06915.png" alt="$\dot{x_2} = -(1-u)x_1 + 1$"></p> <h2>Normalized buck-boost model<a name="11"></a></h2> <p><img src="DCConverterPWMPID_eq98147.png" alt="$\dot{x_1} = -\gamma x_1 + (1-u)x_2 -d$"></p> <p><img src="DCConverterPWMPID_eq01392.png" alt="$\dot{x_2} = -(1-u)x_1 + u$"></p> <h2>Model with body resistors<a name="12"></a></h2> <p>In more general cases, a body resistor of the inductor, <img src="DCConverterPWMPID_eq56918.png" alt="$R_\mathrm{L}$"> and an equivalent series resistor (ESR) of the capacitor, <img src="DCConverterPWMPID_eq86081.png" alt="$R_\mathrm{c}$"> can be added to the above models. </p> <h2>Buck model with <img src="DCConverterPWMPID_eq56918.png" alt="$R_\mathrm{L}$"> and <img src="DCConverterPWMPID_eq86081.png" alt="$R_\mathrm{c}$"><a name="13"></a></h2> <p>Since,</p> <p><img src="DCConverterPWMPID_eq05501.png" alt="$C{dv_\mathrm{c}\over dt} = i_\mathrm{L} - v_\mathrm{o}/R - i_\mathrm{o}$"></p> <p><img src="DCConverterPWMPID_eq97811.png" alt="$v_\mathrm{o} = v_\mathrm{c} + R_\mathrm{c}C{dv_\mathrm{c}\over dt}$"></p> <p><img src="DCConverterPWMPID_eq03813.png" alt="$L{di_\mathrm{L}\over dt} = uv_\mathrm{in} - v_\mathrm{o} - R_\mathrm{L}i_\mathrm{L}$"></p> <p>Inserting the second equation into the first leads to:</p> <p><img src="DCConverterPWMPID_eq85629.png" alt="$C{dv_\mathrm{c}\over dt} = i_\mathrm{L} - v_\mathrm{c}/R - {R_\mathrm{c}\over R}C{dv_\mathrm{c}\over dt}- i_\mathrm{o}$"></p> <p><img src="DCConverterPWMPID_eq70073.png" alt="$\left(1+{R_\mathrm{c}\over R}\right)C{dv_\mathrm{c}\over dt}=i_\mathrm{L} - v_\mathrm{c}/R - i_\mathrm{o}$"></p> <p>Hence,</p> <p><img src="DCConverterPWMPID_eq16026.png" alt="$v_\mathrm{o}={Rv_\mathrm{c}\over R+R_\mathrm{c}}+{RR_\mathrm{c}\over R+R_\mathrm{c}}(i_\mathrm{L}-i_\mathrm{o})$"></p> <p>and the overall model is</p> <p><img src="DCConverterPWMPID_eq11655.png" alt="$C{dv_\mathrm{c}\over dt}={R\over R+R_\mathrm{c}}\left(i_\mathrm{L} - {v_\mathrm{c}\over R} - i_\mathrm{o}\right)$"></p> <p><img src="DCConverterPWMPID_eq77863.png" alt="$L{di_\mathrm{L}\over dt} = uv_\mathrm{in} - {Rv_\mathrm{c}\over R+R_\mathrm{c}} - \left(R_\mathrm{L}+{RR_\mathrm{c}\over R+R_\mathrm{c}}\right) i_\mathrm{L} + {RR_\mathrm{c}i_\mathrm{o}\over R+R_\mathrm{c}}$"></p> <p><img src="DCConverterPWMPID_eq16026.png" alt="$v_\mathrm{o}={Rv_\mathrm{c}\over R+R_\mathrm{c}}+{RR_\mathrm{c}\over R+R_\mathrm{c}}(i_\mathrm{L}-i_\mathrm{o})$"></p> <h2>Boost model with <img src="DCConverterPWMPID_eq56918.png" alt="$R_\mathrm{L}$"> and <img src="DCConverterPWMPID_eq86081.png" alt="$R_\mathrm{c}$"><a name="14"></a></h2> <p><img src="DCConverterPWMPID_eq51974.png" alt="$C{dv_\mathrm{c}\over dt} = (1-u)i_\mathrm{L} - v_\mathrm{o}/R - i_\mathrm{o}$"></p> <p><img src="DCConverterPWMPID_eq68771.png" alt="$L{di_\mathrm{L}\over dt} = v_\mathrm{in} - (1-u) v_\mathrm{o} - R_\mathrm{L}i_\mathrm{L}$"></p> <p><img src="DCConverterPWMPID_eq19857.png" alt="$v_\mathrm{o} = {Rv_\mathrm{c}\over R+R_\mathrm{c}}+{RR_\mathrm{c}\over R+R_\mathrm{c}}((1-u)i_\mathrm{L}-i_\mathrm{o})$"></p> <h2>Buck-boost model with <img src="DCConverterPWMPID_eq56918.png" alt="$R_\mathrm{L}$"> and <img src="DCConverterPWMPID_eq86081.png" alt="$R_\mathrm{c}$"><a name="15"></a></h2> <p><img src="DCConverterPWMPID_eq60873.png" alt="$C{dv_\mathrm{c}\over dt} = (1-u)i_\mathrm{L} - v_\mathrm{o}/R-i_\mathrm{o}$"></p> <p><img src="DCConverterPWMPID_eq50326.png" alt="$L{di_\mathrm{L}\over dt}=uv_\mathrm{in}-(1-u) v_\mathrm{o}-R_\mathrm{L}i_\mathrm{L}$"></p> <p><img src="DCConverterPWMPID_eq94809.png" alt="$v_\mathrm{o}={Rv_\mathrm{c}\over R+R_\mathrm{c}}+{RR_\mathrm{c}\over R+R_\mathrm{c}}((1-u)i_\mathrm{L}-i_\mathrm{o})$"></p> <h2>Simulink Model<a name="16"></a></h2> <p>These three modes of DC-DC converters have been uniformly implemented in the MATLAB/Simulink as show in Figure~\ref{fig:simulinkmodel}.</p> <p><img vspace="5" hspace="5" src="DCDCModelInside.png" alt=""> </p> <p>Figure 2. A uniform Simulink model of DC-DC converters.</p> <p>The input-output connections of the model is shown in Figure 3.</p> <p><img vspace="5" hspace="5" src="DCDCModel.png" alt=""> </p> <p>Figure 3. Input and output connections of the DC-DC converter model.</p> <p>The first input to the model is the switch signal eight 1 or 0. The second one defines the DC source voltage and internal resistance. The third input is used to define the output current. The model has two outputs, the output voltage and the inductor current, which are the states of the system. </p> <p>The model can be configure with a number of parameters as shown in Figure~\ref{fig:simulinkmodelparameters}. These parameters are: the capacitance, <img src="DCConverterPWMPID_eq22407.png" alt="$C$">, inductance, <img src="DCConverterPWMPID_eq93576.png" alt="$L$">, the internal resistance of the capacitor and the inductor, <img src="DCConverterPWMPID_eq31468.png" alt="$R_C$"> and <img src="DCConverterPWMPID_eq11714.png" alt="$R_L$"> respectively. Three converter modes can be selected through the pull-down menu. One can also define either zero or non-zero value to the initial capacitor voltage by selecting or de-selecting the ``zero capacitor voltage'' option. Finally, the option ``Positive Inductor Current'' defines whether the condition <img src="DCConverterPWMPID_eq01057.png" alt="$i_L\ge 0$"> should be enforced or not. </p> <p><img vspace="5" hspace="5" src="DCDCModelParameters.png" alt=""> </p> <p>Figure 4. Parameters of the DC-DC converter model.</p> <p class="footer"><br> Published with MATLAB® 7.7<br></p>⌨️ 快捷键说明
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