Ece 534 derive an equivalent circuit model for this

Question

Question 1. For Fig 1, the input voltage is dc and positive with the polarity shown. Specify how to implement the switches using a minimal number of diodes and transistors, such that the converter operates over the entire range of duty cycles. The switch states should vary as shown in Fig. 2. You may assume that the inductor current ripples and capacitor voltage ripples are small.

a) Realize the switches using SPST ideal switches, and explicitly define the voltage and current of each switch.

b) Express the on­state current and off­state voltage of each SPST switch in terms of the converter inductor currents, capacitor voltages, and/or input source voltage.

c) Solve the converter to determine the inductor currents and capacitor voltages.

d) Determine the polarities of the switch on­state currents and off­state voltages. Do the polarities vary with duty cycle?

e) State how each switch can be realized using transistors and/or diodes, and whether the realization requires single­quadrant, current­bidirectional two­quadrant, voltage­bidirectional two quadrant,or four­quadrant switches.

Question 2. A switching converter operates with a switching frequency of 150 kHz. The converter waveforms exhibit damped sinusoidal ringing, initiated by the transistor turn­off transition, which decays slowly but eventually reaches zero before the end of the switching period. This ringing occurs in a series resonant circuit formed by parasitic inductances and capacitances in the circuit. The frequency of the ringing is 10 MHz. During the first period of sinusoidal ringing, the ac inductor current reaches a peak magnitude of 0.5 A, and the ac capacitor voltage reaches a peak magnitude of 150 V. Determine the following quantities:

(a) The value of the total parasitic inductance,
(b) The value of the total parasitic capacitance,
(c) The energy lost per switching period, associated with this ringing, and
(d) The switching loss associated with this ringing.
(e) Derive a general expression for the switching loss, as a function of the switching frequency, ringing frequency, and the ringing voltage and current peak magnitudes during the first period of ringing. (15 points)

Question 3. In Fig 3 , the switching transition for a switch are shown with following switching characteristics

t_ri = 80ns

t_fv = 80ns

t_rv = 120ns

t_fi = 200ns

V_d = 300V

I₀ = 4A

Figure 3. Switching Circuit & switching transition

a. Using Fig 3, sketch the instantaneous switching loss for the switch as a function of time.

b. Calculate the switching power loss as a function of the switching frequency in

a range 25 ­ 100 kHz, assuming V_d =300V and I₀ = 4A.

c. Calculate and plot the total losses as a function of the switching frequency in a range 25 ­ 100 kHz in the switch when the on­state voltage is V_on =1.5V at 4A and the duty ratio of the switch is D = 0.4 (assume t_don=t_doff ).

Assumption for Q.4 :

A transistor having an on­resistance of 0.5 ohm is used. To simplify the problem, you may neglect all losses other than the transistor conduction loss. You may also neglect the dependence of MOSFET on­resistance on rated blocking voltage. These simplifying assumptions reduce the differences between converters, but do not change the conclusions regarding which converter performs best in the given situations.

Question 4. It is desired to interface a 500 V dc source to a 400 V, 10 A load using a dc­dc converter. Two possible approaches, using buck and buck­boost converters, are illustrated in Fig. 4 Use the assumptions described above to:

(a) Derive equivalent circuit models for both converters, which model the converter input and output ports as well as the transistor conduction loss.

(b) Determine the duty cycles that cause the converters to operate with the specified conditions.

(c) Compare the transistor conduction losses and efficiencies of the two approaches, and conclude which converter is better suited to the specified application.

Question 5. To reduce the switching harmonics present in the input current of a certain buck converter, an input filter is added as shown in Figure 5. Inductors L₁ and L₂ contain winding resistances R_L1 and R_L2 respectively. The MOSFET has on­resistance R_on and the diode forward voltage drop can be modeled by a constant voltage V_D plus a resistor R_D. All other losses can be ignored.

Figure 5. Buck converter with input filter

(a) Derive a complete equivalent circuit model for this circuit.
(b) Solve your model to find the dc output voltage V.
(c) Derive an expression for the efficiency. Manipulate your expression into a form similar to the below equation.

Question 6. In the Cuk converter shown in Figure 6, the MOSFET has on­resistance Ron and the diode has a constant forward voltage drop VD. All other losses can be neglected.

Figure 6. Cuk converter

(a) Derive an equivalent circuit model for this converter. Suggestion: if you don't know how to handle some of the terms in your dc equations, then temporarily leave them as dependent sources. A more physical representation of these terms may become apparent once dc transformers are incorporated into the model.

(b) Derive analytical expressions for the converter output voltage and for the efficiency.

(c) For V_D=0, plot V₂/V_g vs. D over the range 0≤D≤1, for (i) R_on/R = 0.01, (ii) R_on/R = 0.1, (iii) R_on/R = 1. Conclude from the result.

(d) For V_D=0, plot converter efficiency vs. D over the range 0≤D≤1, for (i) R_on/R = 0.01,

(ii) R_on/R = 0.1, (iii) R_on/R = 1. Conclude from the result.