On the Extraction of Input and Output Impedance of PWM DC-DC Converters

Copyright © BAJECE ISSN: 2147-284X http://dergipark.gov.tr/bajece Abstract— The buck, buck-boost and boost converter, are the most popular types of DC-DC converters. The input/output characteristics of these converters operating in Continuous Current Mode (CCM) is the object of considerations in this paper. The input impedance of converter helps the designer to select a suitable energy source for the converter. The internal impedance of selected input energy source must be quite lower than the input impedance of converter in order to avoid any voltage drop. The output impedance of DC-DC converters must be quite small in order to supply the load with high current demand. So, extraction of input/output characteristics of DC-DC converters is an important task and helps the designer to decide about the performance of system. Extraction of input/output characteristics is using pencil-and-paper analysis is quite tedious and error prone. This paper show how input/output impedance of DC-DC converters can extracted with the aid of MATLAB programming. This paper can be used as a tutorial on the extraction of input/output impedance of DC-DC converters.


I. INTRODUCTION
he input impedance of a DC-DC converter is the impedance seen from the input DC source. The output impedance is defined as the output voltage response of converter for the excitation of current at constant input voltage and duty ratio . In some descriptions, the output impedance includes the load conductance , in other it does not.  The input impedance of converter helps the designer to select the suitable input DC source. The input impedance of the converter must be much larger than the output impedance of the input DC source. The output impedance of the converter is even more important than the input impedance. Output impedance must be as low as possible. Output impedance of the converter is especially important if the converter supplies a low-voltage, high-current load, with large values of output current slew rate. The most representative example of such a load is a processor in modern computer systems. The processor requires about (or even less) and drawn current is typically over . Current slew rates may approach [1,2]. According to the given numbers, the processor can be modelled as a resistor (or lower). The output resistance of converter should be substantially lower than , to ensure a good efficiency. Usually a buck converter is used to supply the processor. The output impedance of the buck converter supplying the processor (or other type of DC-DC converters) can be reduced with the aid of negative feedback. The relation between the open-loop output impedance ( ) and closed-loop output impedance ( ) is: where is the loop gain [3,4,5].
The buck, buck-boost and boost converter, are the most popular types of converters. Their input/output characteristics are the object of considerations in this paper. A well-known reference such as [3]   When the MOSFET switch is closed, the diode is reversebiased and the equivalent circuit of Fig. 3 applies. shows the MOSFET drain-source resistance.
is a fictitious current source added to the circuit in order to measure the output impedance ( ).  According to Fig. 4, the circuit differential equations can be written as: State Space Averaging (SSA) is one of the most important tools to study the dynamics of converters operating in CCM. Foundation of SSA was laid down in [7] and later extended in [8,9,10], as well as many other publications. Theory of SSA has been studied in many text books for instance see [10] and [11]. SSA has two important steps: averaging and linearization. The SSA procedure can be summarized as follows [11]: Step 1-Circuit differential equations are written for different working modes (i.e on/off state of semiconductor switches).
Step 2-Equations are time averaged over one period.
Step 3-Steady state operating points are calculated by equating the derivative terms to zero.
Step 4-The averaged equations are linearized around the steady state operating point found in the third step.
Applying the SSA to the Equations (2)-(9) leads to 6 different transfer functions: and . Openloop input and output impedance of the converter is extracted with the aid of and , respectively.
Applying the aforementioned steps manually is tedious and error prone (especially if the converter order is high). MATLAB ® can be very helpful to do the mathematical machinery of SSA. The program shown in appendix (program 1) extracts the small signal transfer functions of a buck converter with component values as shown in Table 1.    The block diagram shown in Fig. 8 can be drawn for the studied buck converter. We want to study the effect of feedback on output impedance.  Assume that the controller is a simple I-type controller ( ). Fig. 10, shows the step response of the closed loop. Fig. 10.
Step response of closed-loop control system shown in Fig. 9 with .
According to Fig. 9, the closed loop output impedance ( ) is :   Fig. 11, is a comparison between the open-loop output impedance ( , Equation (12)) and closed-loop output impedance ( , Equation (14)). The closed loop output impedance is reduced at low frequency portion of the graph. Reduction of output impedance is one of the desired properties of feedback control. There are many efforts presented in the literature to achieve satisfactory output impedance of PWM DC-DC converters, especially buck type. The methods can be categorized into two groups:  Sophisticated design of control loops in the converter [13,14,15,16]  Modifications of the basic structure of the power stage [17][18]. The starting point of the first method is the precise description of the converter, in particular the use of accurate formulas for open-loop output impedance. The program given in appendix can be helpful for this purpose [19].
III. BUCK-BOOST CONVERTER Schematic of the buck-boost converter is shown in Fig. 12.  MOSFET. According to Fig. 13, the circuit differential equations can be written as: When the MOSFET switch is opened, the diode becomes forward-biased. According to Fig. 14, the circuit differential equations can be written as: The program shown in appendix (program 2) extracts the small signal transfer functions of a buck-boost converter with component values as shown in Table 2.    IV. BOOST CONVERTER Schematic of boost converter is shown in Fig. 18. When the MOSFET is closed, the diode is reverse biased. Fig. 19. Equivalent circuit of boost converter with closed MOSFET.
According to Fig. 19, the circuit differential equations can be written as: When the MOSFET switch is opened, the diode becomes forward-biased. Fig. 20, shows the equivalent circuit for this case. According to Fig. 20, the circuit differential equations can be written as: The program shown in appendix (program 3) extracts the small signal transfer functions of a boost converter with component values as shown in Table 3. Switching Frequencyis 25 kHz.  The programs given in appendix calculates the steady state operating point of the converters as well. The steady state operating point for studied boost converter is and . and show the average inductor current and average capacitor voltage, respectively. The average current drawn from the input DC source is the same as the average current of inductor. So, the input DC source sees the converter as a load. If we substitute in the Equation (35), we obtain the which is quite close to the expected value. The DC gain of obtained input impedance (at ) can be checked in a similar way for other type of converters.
V. CONCLUSION Input/output characteristics of DC-DC converters are important parameters. The input impedance helps the designer to select the suitable input source. The output impedance of the converter shows whether the converter can supply the output load successfully or not. This paper studied the input/output characteristics of buck, buck-boost and boost converters. MATLAB programming is used to do the mathematical machinery. Input/output characteristics of other types of converters can be extracted in a similar way shown in the paper. The control to output transfer function of power electronics converters is used to design the control loop of converter. The Buck converter has a minimum phase control to output transfer function while the Boost and Buck-Boost converters have non-minimum phase control to output transfer functions. The feedback control of power converters affect the output impedance of converter. The output impedance of converter decreases with the aid of feedback control as shown in the second sectiom of paper. [