Schematic Diagram Of The Boost Converter

Published Date: 02 Nov 2017

CHAPTER 3

The schematic diagram of a Boost converter is shown in the Figure. 3.1. In this converter, the output voltage is always greater than the input voltage. It has a very simple structure, continuous input current, step up conversion ratio and it also has a clamped voltage stress and more over clamped switch voltage stress to the output voltage.

Figure 3.1. Schematic diagram of the Boost Converter

The output voltage is given by,

(3.1)

The inductor and capacitor values of the Boost converter are derived by having the same assumption as that of the Buck converter. Now the critical value of the inductor LC which decides the condition for the continuous current mode of the operation is given by,

(3.2)

Where fS is the switching frequency. The inductor value is determined as,

(3.3)

Similarly the capacitor value can be determined by assuming appropriate ripple voltage as,

(3.4)

In this type of converter, a very high peak current flows through the switch. It is very difficult to attain the stability of this converter due to high sensitivity of the output voltage to the duty cycle variations. When compared to buck converter the inductor and capacitor sizes are larger since high RMS current would flow through the filter capacitor. The following are the parameters considered for design: VS = 24V, VO = 50V, fS = 20 kHz, L = 72µH, C = 216.9X10-6 F and R = 23Ω.

The dynamic equation describing the Boost converter can be explained by assuming two modes of operation, which are discussed now. The inductor current iL and the capacitor voltage VO are the state variables. The semiconductor Switch is in on condition for the time interval 0 ≤ t ≤ Ton and hence the inductor L gets connected to the supply and stores the energy. Since the diode is in off condition, the output stage gets isolated from the supply. Here the inductor current flows through the inductor and completes its path through the source.The equivalent circuit for this mode is shown in the Figure. 3.2.

Figure.3.2. Equivalent circuit of Boost Converter for mode 1

Applying Kirchoff’s laws, the following equations describing mode1 are obtained as,

(3.6)

Now the coefficient matrices for this mode are obtained as,

(3.7)

(3.8)

During the time interval Ton ≤ t ≤ T, the diode is in on state and the switch is off state and hence the energy from the source as well as the energy stored in the inductor is fed to the load. The inductor current flows through the inductor L, the capacitor C, the diode and the load. The equivalent circuit for this mode is shown in Figure. 3.3

Figure. 3.3. Equivalent circuit of Boost Converter for mode 2

Applying Kirchoff’s the following equations describing mode 2 are obtained,

(3.10)

The coefficient matrices for this mode is defined as follows,

(3.11)

(3.12)

The output voltage VO (t) across the load is expressed as,

(3.13)

Derivation of state feedback matrix for boost converter

The root locus of the Boost converter under continuous time is drawn as shown in the Figure 3.4. The open loop poles of the Boost converter is shown by cross in the figure. The desired poles are arbitrarily placed in order to obtain the state feedback matrix. By substituting the values of L and C thus designed, the state coefficient matrices for the Boost converter is obtained as follows:

(3.13)

(3.14)

(3.15)

(3.16)

Figure 3.4. Root locus of Boost converter in s-domain

The derivation of the state feedback matrix for the Boost converter is carried out in the same manner as that for the Buck converter using pole placement technique which has already been explained in the second chapter. The state feedback matrix for the Boost converter is derived as follows:

Step1: The characteristic polynomial to find the unknown values of is formed as follows:

(3.17)

Step 2: The desired characteristic equation is formed by arbitrarily placing the poles as follows:

(3.18)

By equating the like powers of s in the equations (3.17) and (3.18), the state feedback matrices are obtained as k1 = 1.008 and k2 = 0.07206.

In order to check the robustness of the control law, the step input is used and the output response has been illustrated in the Figure 3.5. From the Figure 3.5, it is very well understood that the system settles down faster and the state feedback matrix is capable enough to realize the stability of the Boost converter.

Figure 3.4. Step response of the Buck converter in continuous time domain

The state equations for the Buck converter under discrete time domain is defined as,

(3.19)

(3.20)

(3.21)

(3.22)

As per the Nyquist criterion which states that sampling frequency of the analog signal should be atleast twice the maximum signal frequency, the sampling frequency is assumed as 1 MHz. The digital state feedback matrix can be derived in the same method as that for the continuous domain excepting the domain considered here is z. The root locus of this converter in z domain is drawn as shown in the Figure 3.5. The desired poles are chosen arbitrarily in order to find the digital state feedback matrix. The open loop poles are located as shown in the figure. The stability of the converter is obtained in such a way by moving the poles towards the left half of the z-plane. More the poles are moved towards the left hand side , more stability of the converter is obtained.

Figure 3.5. Root locus of the Boost Converter in z-domain

The digital state feedback matrix can be obtained by substitution method and is explained as follows:

Step1: The characteristic polynomial to find the unknown values of is formed as follows:

(3.23) Step 2: The desired characteristic equation is formed by arbitrarily placing the poles as follows:

(3.24)

By equating the like powers of s in the equations (3.23) and (3.24), the state feedback matrices are obtained as kd1 = 1.854 and kd2 = 1.

In order to check the robustness of the control law, the step input is used and the output response has been demonstrated in the Figure. 3.6. From the Figure. 3.6, it is very well understood that the system settles down faster and the digital state feedback matrix is efficient enough to realize the stability of the Boost converter.

Figure 3.6. Step response of the Boost converter in continuous time domain

Derivation of observer gain matrix for Boost converter

The derivation of full order state observer gain matrix has already been explained in the second chapter. Now, for the Boost converter this matrix can be derived by the substitution method by assuming appropriate natural frequency of oscillation and damping ratio as per the thumb rule. By assuming the damping ratio, ζ = 0.6 and the natural frequency of oscillation, ωn = 12.02788 x 103 rad/sec, the desired characteristic equation can be obtained as follows,

(3.25)

The polynomial equation with unknown values of observer poles is given by,

(3.26)

Comparing the equations (3.25) and (3.26), the observer gain matrix is obtained. The values are.

Since the observer poles are mainly designed to check the robustness of the control law, it is essential that the estimated state variables and error variables should converge at zero from any non-zero initial value. This ensures the asymptotic stability of the system under consideration with the desired pole locations. It is demonstrated in the figures 3.7, 3.8, 3.9 and 3.10 respectively. All the variables under consideration attains zero value from any non zero value thereby ensuring that the observer poles are dynamic and the state feedback control is very strong enough to achieve the stability of the system under consideration.

Figure 3.7. Estimation of state variable 1 Figure 3.8. Estimation of state variable 2

Figure 3.9. Estimation of error variable 1 Figure 3.10. Estimation of error variable 2

By using the Separation principle which has already been explained in chapter 2, the transfer function of the observer controller for the Boost converter under continuous time domain is given by,

(3.27)

Similar case can be derived for the discrete time system and it is discussed now. By assuming the damping ratio, ζ = 0.5, sampling time TS = 1µs and the natural frequency of oscillation, ωn = 192.4461 x 103 rad/sec, the desired characteristic equation can be obtained as follows,

(3.28)

The above equation can be simplified as,

(3.29)

The polynomial equation with unknown values of observer poles is given by,

(3.30)

Comparing the equations (3.29) and (3.30), the observer gain matrix is obtained. The values are.

By using the Separation principle which has already been explained in chapter 2, the transfer function of the prediction observer controller for the Boost converter under discrete time domain is given by,

(3.31)

Simulation results

Boost converter (continuous time domain)

The design and the performance of Boost converter is accomplished in continuous conduction mode and simulated using MATLAB/ Simulink. The ultimate aim is to achieve a robust controller inspite of uncertainty and large load disturbances. The performance parameters of the converter under consideration are rise time, settling time, maximum peak overshoot and steady state error, which are shown in the Table 3.1. It is evident that the converter settles down at 0.015s and the rise time of the converter is 0.01s. No overshoots or undershoots are evident and no steady state error is observed. The simulation of the Boost converter is also carried out by varying the load, not limiting it to R load and it is illustrated in the Table 3.2.

Table 3.1. Performance Parameters of Boost converter

Parameters

Values

Settling Time (s)

0.015

Peak Overshoot (%)

0

Steady State Error (V)

0

Rise Time (s)

0.01

Output Ripple Voltage (V)

0

Table 3.2. Output Response for Load Variations

Sl. No

Load

Reference

Voltage(V)

Output

Voltage (V)

R(Ω)

L(H)

E(V)

1

25

--

--

50

50.00

2

30

--

--

50

50.01

3

25

1x10-6

--

50

50.00

4

25

100 x 10-6

--

50

50.05

5

20

1x10-6

5

50

50.04

6

20

50x10-6

5

50

50.20

It is obviously understood that the Boost converter with observer controller is efficient enough to track the output voltage irrespective of the load variations. When the load resistance is varied as 25Ω and 30 Ω, the converter is able to track the output voltages as 50V and 50.01V respectively for the reference voltage of about 50V. Again when the inductance of 1µH and 100 µH are added to the resistances of 25Ω, the output thus obtained is of the order of 50V and 50.05V respectively. The steady state error observed is of very considerable order of about 0.05V. The simulation is also carried out again using RLE load with a resistance of 20Ω, two different inductances of 1µH and 50 µH and an ideal voltage source of about 50V. The response of the converter is such that the controller is capable to work under all the load transients thereby tracking the voltage as 50.04V and 50.20V respectively.

The simulation is also carried out by varying the input voltage and load resistance and the corresponding, input voltage, load resistance, output voltage, inductor current and load current are shown in the Figure. 3.11. The input voltage is first set as 24V until 0.4s and again varied from 24V to 22V upto 0.6s. Again at 0.6s it is varied to 24V and at 0.8s it has been varied to 26V respectively.

Figure 3.11. Output Response of Boost Converter with Observer Controller

(Vs – Input Voltage, Ro – Load resistance, Vo – Output Voltage, IL – Inductor current, Io – Load Current)

Simultaneously the load resistance is also varied from 20Ω to 19Ω, 10Ω and again to 18.5Ω respectively for 0.3s and 0.6s respectively. The corresponding output response of the Boost converter shows fixed output voltage regulation. Undershoots and Overshoots are not observed and the steady state error is also not apparent. The inductor current and load current are also shown in the Figure. 3.11, which shows no evidence of current ripples. In order to check the dynamic performance of the system, the L and C parameters of the Boost converter are varied and the output response of the system is shown in the Table 3.3.

Table 3.3. Output Response with Variable Converter Parameters

Sl.No

Inductance, L

Capacitance, C

Reference

Voltage(V)

Output

Voltage (V)

1

100µH

220µF

50

49.85

2

150µH

200µF

50

50.34

3

200µH

150µF

50

49.96

4

250 µH

100µF

50

50.10

5

250 µH

250µF

50

50.24

It is understood from the table 3.3 that the system is very much dynamic in tracking the reference voltages inspite of the variations in the inductance and capacitance values. The system does not show any overshoots or undershoots and it settles down fast with a settling time of about 0.015s for all the values. The steady state error thus noticeable ranges between 0.3% to 0.6% which is considered as within the tolerable limits. The load current, output power, losses and efficiency of the Boost converter is determined and is illustrated in table 3.4.

Table 3.4. Efficiency of the Boost Converter with Analog Observer Controller

Sl. No.

IO(A)

PO(W)

Losses(W)

(%)

1

1.5455

77.3368

1.9098

97.590

2

1.9723

98.7136

2.3662

97.659

3

2.1755

108.8185

3.5067

96.878

4

2.3750

118.6313

4.2135

96.570

5

2.5170

126.1017

5.6936

95.680

6

4.4590

222.8608

12.7787

94.577

The efficiency of the Boost converter remains more or less same with the increase in the load current. It is very well understood that the Boost converter with observer controller is highly efficient and the highest efficiency is obtained as 97.659% at a load current of about 1.9723A and the corresponding output power is 98.7136W

3.4.2 Boost converter (discrete time domain)

Simulation has been carried out for the Boost converter under discrete time domain and the performance parameters are tabulated in table 3.5. The system settles down much faster at 0.0075s and the rise time is only 0.005s as shown in the table 3.5. The overshoots and undershoots are not seen and there is no peak overshoot.

Table 3.5. Performance Parameters of Buck converter

Parameters

Values

Settling Time (s)

0.0075

Peak Overshoot (%)

0

Steady State Error (V)

0

Rise Time (s)

0.004

Output Ripple Voltage (V)

0

The output response with load variations is shown in the table.3.6. When the load resistance is varied as 25Ω and 30 Ω, the Boost converter with discrete controller is capable to track the output voltages as 50.02V and 49.95V respectively for the reference voltage of about 50V. Again when the inductance of 1µH and 100 µH are added to the resistances of 25Ω, the output thus obtained is of the order of 49.99V and 49.79V respectively. The steady state error observed is of very considerable order of about 0.01V and 0.05V respectively. The simulation is also carried out again using RLE load with a resistance of 20Ω, two inductances of 1µH and 50 µH and an ideal voltage source of about 5V. The response of the converter is such that the controller is proficient to work under all the load transients thereby tracking the voltage as 50.03V and 49.89V respectively.

Table 3.6 Output voltage of Boost converter for load variations (discrete)

Sl.No

Load

Reference

Voltage(V)

Output

Voltage (v)

R(Ω)

L(H)

E(V)

1

25

--

--

50

50.02

2

30

--

--

50

49.95

3

25

1x10-6

--

50

49.99

4

25

100x 10-6

--

50

49.79

5

20

1x10-6

5

50

50.03

6

20

50x10-6

5

50

49.89

The simulation is also carried out by varying the input voltage and load resistance and the corresponding, input voltage, load resistance, output voltage, inductor current and load current are shown in the Figure 3.12. The input voltage is first set as 24V until 0.01s and again varied to 22V upto 0.02s. Again at 0.02s it is varied to 24V and at 0.03s it has been varied to 26V up to 0.04s and again varied to 24V respectively till 0.05s. Simultaneously the load resistance is also varied as 22Ω, 20Ω and 10Ω respectively and the corresponding output response of the Boost converter shows tight output voltage regulation. Undershoots and Overshoots are not observed and the steady state error is also not evident. The inductor current and load current are also shown in the Figure 3.12, which shows no evidence of current ripples. In order to check the dynamic performance of the system, the L and C parameters of the Boost converter are varied and the output response of the system is shown in the Table 3.7.

Table 3.7. Output Response with Variable Converter Parameters

Sl.No

Inductance, L

Capacitance, C