Answer :
To solve the given problem, we'll compute the transistor on time [tex]t_{on}[/tex] and the average voltage across the load [tex]V_t[/tex] for a buck-boost converter with specified parameters.
a) Transistor On Time, [tex]t_{on}[/tex]
Definition: The "on time" of the transistor in a switching converter is the duration for which the transistor is conducting during one cycle of the switching frequency.
Given Data:
- Switching frequency [tex]f = 6.5 \text{ kHz}[/tex]
- Duty ratio [tex]D = 45\% = 0.45[/tex]
Formula: The switching period [tex]T[/tex] is given by:
[tex]T = \frac{1}{f}[/tex]
Subsequently, the on time [tex]t_{on}[/tex] can be expressed by:
[tex]t_{on} = D \times T[/tex]Calculations:
- First, compute the switching period:
[tex]T = \frac{1}{6.5 \times 10^3} \approx 1.538 \times 10^{-4} \text{ seconds}[/tex] - Then, compute the on time:
[tex]t_{on} = 0.45 \times 1.538 \times 10^{-4} \approx 6.921 \times 10^{-5} \text{ seconds}[/tex] - Therefore, the transistor on time [tex]t_{on} \approx 69.21 \text{ microseconds}[/tex].
- First, compute the switching period:
b) Average Voltage Across the Load, [tex]V_t[/tex]
Definition: The average voltage across the load in a buck-boost converter is influenced by the duty cycle and the input voltage.
Given Data:
- Input voltage [tex]V_{in} = 10 \text{ V}[/tex]
- Duty ratio [tex]D = 0.45[/tex]
Formula: The average output voltage [tex]V_t[/tex] for a buck-boost converter is:
[tex]V_t = \frac{-D}{1-D} \times V_{in}[/tex]
(Note: The negative sign indicates a reversal of polarity typical in buck-boost configurations.)Calculations:
- Plug the values into the formula:
[tex]V_t = \frac{-0.45}{1-0.45} \times 10 = \frac{-0.45}{0.55} \times 10 \approx -8.182 \text{ V}[/tex] - Thus, the average voltage across the load [tex]V_t \approx -8.18 \text{ V}[/tex].
- Plug the values into the formula:
Summary:
- The transistor on time [tex]t_{on}[/tex] is approximately 69.21 microseconds.
- The average voltage across the load [tex]V_t[/tex] is approximately -8.18 V, indicating an inversion of polarity from the input voltage.