AC modeling of quasi-resonant converters Extension of State-Space Averaging to model non-PWM switches Use averaged switch modeling technique: apply averaged.
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Presentation on theme: "AC modeling of quasi-resonant converters Extension of State-Space Averaging to model non-PWM switches Use averaged switch modeling technique: apply averaged."— Presentation transcript:
1 AC modeling of quasi-resonant converters Extension of State-Space Averaging to model non-PWM switchesUse averaged switch modeling technique: apply averaged PWM model, with d replaced by µBuck example with full-wave ZCS quasi-resonant cell:µ = F
2 7.3. State Space AveragingA formal method for deriving the small-signal ac equations of a switching converterEquivalent to the modeling method of the previous sectionsUses the state-space matrix description of linear circuitsOften cited in the literatureA general approach: if the state equations of the converter can be written for each subinterval, then the small-signal averaged model can always be derivedComputer programs exist which utilize the state-space averaging method
3 7.3.1. The state equations of a network A canonical form for writing the differential equations of a systemIf the system is linear, then the derivatives of the state variables are expressed as linear combinations of the system independent inputs and state variables themselvesThe physical state variables of a system are usually associated with the storage of energyFor a typical converter circuit, the physical state variables are the inductor currents and capacitor voltagesOther typical physical state variables: position and velocity of a motor shaftAt a given point in time, the values of the state variables depend on the previous history of the system, rather than the present values of the system inputsTo solve the differential equations of a system, the initial values of the state variables must be specified
4 State equations of a linear system, in matrix form A canonical matrix form:State vector x(t) contains inductor currents, capacitor voltages, etc.:Input vector u(t) contains independent sources such as vg(t)Output vector y(t) contains other dependent quantities to be computed, such as ig(t)Matrix K contains values of capacitance, inductance, and mutual inductance, so that K dx/dt is a vector containing capacitor currents and inductor winding voltages. These quantities are expressed as linear combinations of the independent inputs and state variables. The matrices A, B, C, and E contain the constants of proportionality.
5 Example State vector Matrix K Input vector Choose output vector as To write the state equations of this circuit, we must express the inductor voltages and capacitor currents as linear combinations of the elements of the x(t) and u( t) vectors.
6 Circuit equations Find iC1 via node equation: Find vL via loop equation:
7 Equations in matrix form The same equations:Express in matrix form:
8 Output (dependent signal) equations Express elements of the vector y as linear combinations of elements of x and u:
9 Express in matrix formThe same equations:Express in matrix form:
10 7.3.2. The basic state-space averaged model Given: a PWM converter, operating in continuous conduction mode, with two subintervals during each switching period.During subinterval 1, when the switches are in position 1, the converter reduces to a linear circuit that can be described by the following state equations:During subinterval 2, when the switches are in position 2, the converter reduces to another linear circuit, that can be described by the following state equations:
11 Equilibrium (dc) state-space averaged model Provided that the natural frequencies of the converter, as well as the frequencies of variations of the converter inputs, are much slower than the switching frequency, then the state-space averaged model that describes the converter in equilibrium iswhere the averaged matrices areand the equilibrium dc components are
12 Solution of equilibrium averaged model Equilibrium state-space averaged model:Solution for X and Y:
13 Small-signal ac state-space averaged model whereSo if we can write the converter state equations during subintervals 1 and 2, then we can always find the averaged dc and small-signal ac models
14 Relevant background State-Space Averaging: see textbook section 7.3 Averaged Circuit Modeling and Circuit Averaging: see textbook section 7.4
15 Averaged Switch Modeling Separate switch elements from remainder of converterRemainder of converter consists of linear circuitThe converter applies signals xT to the switch networkThe switch network generates output signals xsWe have solved for how xs depends on xT
16 Block diagram of converter Switch network as a two-port circuit:
21 Finding µ: ZCS examplewhere, from previous slide,
22 Derivation of the averaged system equations of the resonant switch converter Equations of the linear network(previous Eq. 1):Substitute the averaged switchnetwork equation:Result:Next: try to manipulate into sameform as PWM state-space averaged result
23 Conventional state-space equations: PWM converter with switches in position 1 In the derivation of state-space averaging for subinterval 1:the converter equations can be written as a set of lineardifferential equations in the following standard form (Eq. 7.90):These equations must be equal:Solve for the relevant terms:But our Eq. 1 predicts that the circuit equations for this interval are:
24 Conventional state-space equations: PWM converter with switches in position 2 Same arguments yield the following result:and
25 Manipulation to standard state-space form Eliminate Xs1 and Xs2 from previous equations.Result is:Collect terms, and use the identity µ + µ’ = 1:—same as PWM result, but with d µ
26 Perturbation and Linearization The switch conversion ratio µ is generally a fairlycomplex function. Must use multivariable Taylor series,evaluating slopes at the operating point:
27 Small signal modelSubstitute and eliminate nonlinear terms, to obtain:Same form of equations as PWM small signal model. Hence same model applies, including the canonical model of Section 7.5.The dependence of µ on converter signals constitutes built-in feedback.
28 Salient features of small-signal transfer functions, for basic converters
29 Parameters for various resonant switch networks
30 Example 1: full-wave ZCS Small-signal ac model Averaged switch equations:Linearize:Resulting ac model:µ = F
31 Low-frequency modelTank dynamics occur only at frequency near or greater than switching frequency —discard tank elements—same as PWM buck, with d replaced by F
32 Example 2: Half-wave ZCS quasi-resonant buck Now, µ depends on js:
33 Small-signal modeling Perturbation and linearization of µ(v1r, i2r, fs):withLinearized terminal equations of switch network: