Consider the buck converter R = 2 ohm L = 3H, C = 1/6 H The Power Source v_g = 1
ID: 1996427 • Letter: C
Question
Consider the buck converter R = 2 ohm L = 3H, C = 1/6 H The Power Source v_g = 1 V The nominal duty cycle is D = 0.5. The feedback controller is a proportional controller with gain K = 1 What is the transfer function (of -the linearized system) from the duty cycle d(t) to the output circuit model to the output voltage ? You can use the equivalent model to derive the modal (so that you do not have to start from differential equations and state space models, ) show essential derivation steps What are the closed loop systems poles If the load is changed from R = 2 ohm to R = 4 ohm but the Controller is the same what are the new closed -loop systems poles Compare the maximum overshoot the settling time and the peak time of the closed loop system in two cases.Explanation / Answer
1)The transfer function from the duty cycle to the output voltage of the open-loop Buck converter in CCM operation is essential in voltage mode control and vital for describing the voltage loop for current mode control. The transfer function from the duty cycle to the inductor current of the system isalso important for describing the current loop for current modecontrol. These two transfer functions can be obtained from thesmall-signal equivalent circuit model under the condition that the perturbation of the input voltage is equal to zero . Accordingly, based on the Kirchhoff’scurrent and voltage laws
A duty cycle estimator for determining a nominal duty cycle of an output regulator. The duty cycle estimator having at least two modes and including at least a mode one estimator and a mode two estimator.
The mode one estimator to determine the nominal duty cycle as a function of prior duty cycles. The mode two estimator to determine the nominal duty cycle as a function of accumulated error.
A mode selector, based on a mode selection criteria, to select a one of the at least two modes to generate the nominal duty cycle.
A useful procedure in network analysis is to simplify the network by reducing the number of components.
This can be done by replacing the actual components with other notional components that have the same effect.
A particular technique might directly reduce the number of components, for instance by combining impedances in series.
On the other hand, it might merely change the form into one in which the components can be reduced in a later operation.
For instance, one might transform a voltage generator into a current generator using Norton's theorem in order to be able to later combine the internal resistance of the generator with a parallel impedance load.
A resistive circuit is a circuit containing only resistors, ideal current sources, and ideal voltage sources.
If the sources are constant (DC) sources, the result is a DC circuit. Analysis of a circuit consists of solving for the voltages and currents present in the circuit. The solution principles outlined here also apply to phasor analysis of AC circuits.
Two circuits are said to be equivalent with respect to a pair of terminals if the voltage across the terminals and current through the terminals for one network have the same relationship as the voltage and current at the terminals of the other network.
If V_{2}=V_{1}} V_2=V_1 implies I_{2}=I_{1}} I_2=I_1 for all (real) values of V_{1}} V_{1}, then with respect to terminals ab and xy, circuit 1 and circuit 2 are equivalent.
The above is a sufficient definition for a one-port network. For more than one port, then it must be defined that the currents and voltages between all pairs of corresponding ports must bear the same relationship. For instance, star and delta networks are effectively three port networks and hence require three simultaneous equations to fully specify their equivalence.
2)
The response of a linear and time invariant system to any input can be derived from its impulse response and step response.
The eigenvalues of the system determine completely the natural response (unforced response).
In control theory, the response to any input is a combination of a transient response and steady-state response.
Therefore, a crucial design parameter is the location of the eigenvalues, or closed-loop poles.
In root-locus design, the gain K is usually parameterized.
Each point on the locus satisfies the angle condition and magnitude condition and corresponds to a different value of K.
For negative feedback systems, the closed-loop poles move along the root-locus from the open-loop poles to the open-loop zeroes as the gain is increased.
For this reason, the root-locus is often used for design of proportional control, i.e. those for which Gc=K.
Consider a simple feedback system with controller Gc and transfer function Hs in the feedback path. Note that a unity feedback system has
Hs=1 and the block is omitted. For this system, the open-loop transfer function is the product of the blocks in the forward path,
GcG=KG The product of the blocks around the entire closed loop is GcGH=KGH
Ts=KG/(1+KGH)
The closed-loop poles, or eigenvalues, are obtained by solving the characteristic equation
1+KGH=0. In general, the solution will be n complex numbers where n is the order of the characteristic polynomial.
The preceding is valid for single-input-single-output systems (SISO). An extension is possible for multiple input multiple output systems,
that is for systems where Gs are matrices whose elements are made of transfer functions. In this case the poles are the solution of the equation
det(I+G(s)K(s))=0
3)
Effect of addition of zero to closed loop transfer function
1) Makes the system overall response faster.
2) Rise time, peak time, decreases but overshoot increases.
3) Addition of right half zeros means system response slower and system exhibits inverse response. Such systems are said to be non-minimum phase systems.
Minimum phase system: The system which doesn’t have zeros in right half of s plane is said to be minimum phase system.
Non-minimum phase system: If a transfer function has poles or zeros in right half of s plane then the system shows the non-minimum phase behavior.
4)
The maximum overshoot is the maximum peak value of the response curve measured from unity. If the final steady-state value of the response differs from unity, then it is common to use the maximum percent overshoot.
Peak time (tp) is simply the time required by response to reach its first peak i.e. the peak of first cycle of oscillation, or first overshoot.
Settling time (ts) is the time required for a response to become steady. It is defined as the time required by the response to reach and steady within specified range of 2 % to 5 % of its final value.
Expression of Peak Time
As per definition at the peak time, the response curve reaches to its maximum value. Hence at that point,
The maximum overshoot occurs at n = 1.
Expression of Settling Time
It is already defined that settling time of a response is that time after which the response reaches to its steady-state condition with value above nearly 98 % of its final value. It is also observed that this duration is approximately 4 times of time constant of a signal. At the time constant of a second-order control system is 1/ n, the expiration of settling time can be given as