To Study about the Synchronous DC - DC Buck Converter.

AIM : To Study about the Synchronous DC - DC Buck Converter.

OBJECTIVES :

1. Explain the terms and Tune a compensator for DC Motor Using Bode Diagram Graphical Tuning.

2. Can a system with a pole on the right half of the s-plane be stable ?

3. Use the following link of Synchronous Buck Converter in order to solve the following questions :

   https://www.mathworks.com/help/physmod/sps/ug/synchronous-buck-converter.html;jsessionid=d329b9b7f786706ee82e2b490282

   • Run the simulation for the default values and comment on the results with respect to stability of the system.
   • Change the reference voltage and comment on whether it is tracking the reference voltage or not.
   • Change the PI gain values and comment on the stability of the system w.r.t. new values.
   • Change the switching frequency and comment on the stability of the system w.r.t. new values.

STUDY - 1 :

a.) Explanation of the following Terms :

• Rise time

• Settling time

• Peak overshoot

• Steady state error

Rise Time :

It is the time required for the response to rise from 0% to 100% of its final value. This is applicable for the under-damped systems. For the over-damped systems, consider the duration from 10% to 90% of the final value. Rise time is denoted by t_r .

Settling time :

It is the time required for the response to reach the steady state and stay within the specified tolerance bands around the final value. In general, the tolerance bands are 2% and 5%. The settling time is denoted by T_s .

Peak Overshoot :

Peak overshoot M_p is defined as the deviation of the response at peak time from the final value of response. It is also called the maximum overshoot.

Mathematically, we can write it as

M_p = c (t_p) - c (infty)

Where,

c(t_p) is the peak value of the response.

c(∞) is the final (steady state) value of the response.

Steady State Error :

Steady-state error is defined as the difference between the desired value and the actual value of a system output in the limit as time goes to infinity (i.e. when the response of the control system has reached steady-state).

It is represented as e_{ss}. We can find steady state error using the final value theorem as follows.

e_{ss} = lim_{t to infty}e(t) = lim_{s to 0}sE(s)

Where,

E(s) is the Laplace transform of the error signal, e(t).

b.) Tune Compensator for DC motor using Bode Diagram Graphical Tuning :

G(s) = 1.5 / (s² + 14s + 40.02)

Meet the following specifications -

Rise time of less than 0.5 seconds

Steady state error of less than 5%

Overshoot of less than 10%

Gain Margin is greater than 20db

Phase Margin is greater than 40 deg

By seeing the transfer function, one can understand it is second order LTI system. The Poles and zeros determine the characteristics of the system. To find poles of system, the roots of characteristic equation must be equated to zero -

Given Equation -

G(s) = 1.5 / (s² + 14s + 40.02)

Denominator -

s² + 14s + 40.02 = 0

s(s + 10) + 4(s + 10) = 0

(s + 10) (s + 4) = 0

s = -4 or s = -10

The poles are at -10 and -4 on the left side of s-plane. There is no Zero. The system is stable second order system.

Matlab Code For Control System Designer window :

• The compensator is designed to meet the shortcomings of desired performance of the plant. To design the compensator, define the plant transfer function in MATLAB window.

• To Open the Control System Designer window for the defined transfer function we uses controlSystemDesigner command.

Control System Designer window :

• The Control system designer window displays the BODE plot editor and Root locus editor side by side along with Step response.

• One must tune the compensator to satisfy the condition of design requirements.
• On bode plot editor, the magnitude plot needs to be adjusted to around 4 rad/sec to get rise time and overshoot as per the condition.

• The Rise time and Overshoot have got satisfied now.

• To achieve 5% of steady state error, the integrator must get added by right clicking the Magnitude plot of Bode plot editor as shown below.

• The gain of compensator is increased directly by selecting Edit compensator. We can calculate the gain value from

• The gain value can be increased to 100 .The lead compensator can be added with pole at -25 and zero at -5. Lead compensator will reshape the frequency-domain plot by providing sufficient phase lead angle over some appropriate frequency range to offset the excessive phase lag associated with the plant.The transfer function of lead compensator is
• Now the design requirements have been satisfied. Hence, this is one of the ways of designing compensator to meet design requirements. One can try their own method to design the compensator.

STUDY - 2 :

Close Loop Control System :

• From above image we can see that poles in the right half of the S plane are unstable.
• A system is said to be stable if for bounded input there exists the bounded output.
• The stability of a linear system may be determined directly from its transfer function.

• An order linear system is asymptotically stable only if all of the components in the homogeneous response from a finite set of initial conditions decay to zero as time increases, or where the p are the system poles.
• In a stable system all components of the homogeneous response must decay to zero as time increases. If any pole has a positive real part there is a component in the output that increases without bound, causing the system to be unstable.
• In order for a linear system to be stable, all of its poles must have negative real parts, that is they must all lie within the left-half of the s-plane. An “unstable” pole, lying in the right half of the s-plane, generates a component in the system homogeneous response that increases without bound from any finite initial conditions.
• A system having one or more poles lying on the imaginary axis of the s-plane has non-decaying oscillatory components in its homogeneous response, and is defined to be marginally stable.

Stability :

The total response of a system is the sum of the forced and natural responses -

• A linear, time-invariant system is stable if the natural response approached zero as time approaches infinity.
• A linear, time-invariant system is unstable is the natural response grows without bound as time approaches infinity.
• A linear, time-invariant system is marginally stable if the natural response neither decays or grow, but remains constant or oscillates as time approaches infinity.

By looking at the placement of the poles of a system in the s-plane we can draw conclusions about the stability of a system.

• If the poles of the closed loop are in the left half of the s-plane (negative and real), the system is stable.
• If the poles of the closed loop are in the right half of the s-place (positive and real), the system is unstable.
• If the poles appear on the imaginary axis and none appear in the Right Hand Plane, the system is marginally stable.

Example -

We will do Stability analysis using Matlab Root Locus Commands :

Undefined Function -

Root Locus Command for a Transfer Function using Matlab :

Root Locus Outputs :

We will do Stability analysis using MATLAB Root Locus Plot –

Now we will look into the Simulink System –

By Increas in Controller Value to 13 –

By Increas in Controller Value to 150 –

• From the above plot we can see that the system is unstable and lies right half on the S – plane.

To make the system stable we will revaluate the value on the controller panel –

By Increas in Controller Value to 100 –

STUDY - 3 :

Simulink Model of Synchronous DC - DC Buck Converter :

• Here we see the model of Synchronous DC - DC Buck Converter. In that have a different types of passive components.
• This type of converter is used to operate low-level components. MOSFET acts as a switch and PI controller control that switching action.

Output Plots :

Current Measurment Plot :

• Here we see the max 7A current flow true the circuit but it varies from 4 to 7.

PI Control Plot :

Output Voltage Plot :

• Here we see the output plot of the Synchronous DC - DC Buck Converter. in that there are two signals one is the output voltage and others cyclic load current.
• The reference voltage gives to the feedback controller is 15V. Also, the controller track that reference voltage and try to reach.
• There is a small ripple see in the output voltage at the time period of m/s.

After Change the reference voltage :

• Here we changed the reference voltage from 15 to 24. that we do with help of reference voltage block.
• The model refers to that voltage and tries to get the output voltage as 24V.

Current Measurement Plot :

• Here we see the current plot when the reference voltage changed also the miner change observed in the current plot.

PI Control Plot :

Output Voltage Plot :

• In the output voltage plot, we observe that when we changed the reference voltage on the output side also get that same voltage.
• This means we say that the feedback controller and PI follow the reference voltage conditions.

After Change the PI gain values :

• The above model shows the gain block inside the PI controller action. We changed the value of gain from 200 to 260 and 0.5 to 1.
• Kp is Proportional and Ki Integral. the addition of that both gain gives PI output.

PI Control Plot :

• Here we see the PI controller changed the control frequency action as compared to the first model.

After Change the switching frequency :

• The above figure shows an inside view of the cyclic load. here we applied the controlled frequency to the final output to minimize reppletion.
• We increased the frequency of Pulse Generator and observed the output.

Output Voltage Plot :

• Here we observed the two signals first is voltage out and second is cyclic load out. the changed in frequency we see also the changed induce in signal period time.
• The voltage output ripples change with change in cyclic load current.

CONCLUSION :

• Successfully Explained the terms and Tuned a compensator for DC Motor Using Bode Diagram Graphical Tuning.

• Discussed about the Synchronous Buck Converter Simulink Model and observe all types of output waveforms.

REFERENCES :

https://skill-lync.freshdesk.com/support/solutions/articles/43000585187-tune-compensator-using-control-system-designer-tool

https://www.tutorialspoint.com/control_systems/control_systems_time_domain_specifications.htm

http://www.personal.reading.ac.uk/~sis01xh/teaching/CY2C2/splane1.pdf