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Sagot :
To find out which transformer Roberto should use to achieve an ending voltage of 4 V, let's go through the problem step by step.
### Step 1: Determine the Voltage After the First Transformer
Given:
- Primary coils of the first transformer [tex]\( (N_{p1}) = 300 \)[/tex]
- Secondary coils of the first transformer [tex]\( (N_{s1}) = 50 \)[/tex]
- Initial voltage [tex]\( V_{in} = 120 \text{ V} \)[/tex]
The voltage ratio of a transformer is determined by the ratio of the number of turns in the secondary winding to the number of turns in the primary winding:
[tex]\[ \frac{V_{out1}}{V_{in}} = \frac{N_{s1}}{N_{p1}} \][/tex]
We need to calculate the output voltage after the first transformer.
[tex]\[ V_{out1} = V_{in} \times \frac{N_{s1}}{N_{p1}} \][/tex]
[tex]\[ V_{out1} = 120 \text{ V} \times \frac{50}{300} \][/tex]
[tex]\[ V_{out1} = 120 \text{ V} \times \frac{1}{6} \][/tex]
[tex]\[ V_{out1} = 20 \text{ V} \][/tex]
After the first transformer, the voltage is reduced to 20 V.
### Step 2: Determine the Suitable Transformer from the List to Achieve 4 V
Roberto has three options for the second transformer:
1. W: Primary = 80 coils, Secondary = 20 coils
2. Y: Primary = 60 coils, Secondary = 12 coils
3. Z: Primary = 70 coils, Secondary = 35 coils
We need to check which transformer reduces the 20 V output from the first transformer to 4 V.
For each transformer:
[tex]\[ V_{out2} = V_{out1} \times \frac{N_{s2}}{N_{p2}} \][/tex]
#### Transformer W:
- Primary = 80 coils
- Secondary = 20 coils
[tex]\[ V_{out2} = 20 \text{ V} \times \frac{20}{80} \][/tex]
[tex]\[ V_{out2} = 20 \text{ V} \times \frac{1}{4} \][/tex]
[tex]\[ V_{out2} = 5 \text{ V} \][/tex]
#### Transformer Y:
- Primary = 60 coils
- Secondary = 12 coils
[tex]\[ V_{out2} = 20 \text{ V} \times \frac{12}{60} \][/tex]
[tex]\[ V_{out2} = 20 \text{ V} \times \frac{1}{5} \][/tex]
[tex]\[ V_{out2} = 4 \text{ V} \][/tex]
#### Transformer Z:
- Primary = 70 coils
- Secondary = 35 coils
[tex]\[ V_{out2} = 20 \text{ V} \times \frac{35}{70} \][/tex]
[tex]\[ V_{out2} = 20 \text{ V} \times \frac{1}{2} \][/tex]
[tex]\[ V_{out2} = 10 \text{ V} \][/tex]
### Conclusion
Only transformer Y outputs the desired voltage of 4 V after the second step. Therefore, Roberto should use transformer Y to achieve an ending voltage of 4 V.
### Step 1: Determine the Voltage After the First Transformer
Given:
- Primary coils of the first transformer [tex]\( (N_{p1}) = 300 \)[/tex]
- Secondary coils of the first transformer [tex]\( (N_{s1}) = 50 \)[/tex]
- Initial voltage [tex]\( V_{in} = 120 \text{ V} \)[/tex]
The voltage ratio of a transformer is determined by the ratio of the number of turns in the secondary winding to the number of turns in the primary winding:
[tex]\[ \frac{V_{out1}}{V_{in}} = \frac{N_{s1}}{N_{p1}} \][/tex]
We need to calculate the output voltage after the first transformer.
[tex]\[ V_{out1} = V_{in} \times \frac{N_{s1}}{N_{p1}} \][/tex]
[tex]\[ V_{out1} = 120 \text{ V} \times \frac{50}{300} \][/tex]
[tex]\[ V_{out1} = 120 \text{ V} \times \frac{1}{6} \][/tex]
[tex]\[ V_{out1} = 20 \text{ V} \][/tex]
After the first transformer, the voltage is reduced to 20 V.
### Step 2: Determine the Suitable Transformer from the List to Achieve 4 V
Roberto has three options for the second transformer:
1. W: Primary = 80 coils, Secondary = 20 coils
2. Y: Primary = 60 coils, Secondary = 12 coils
3. Z: Primary = 70 coils, Secondary = 35 coils
We need to check which transformer reduces the 20 V output from the first transformer to 4 V.
For each transformer:
[tex]\[ V_{out2} = V_{out1} \times \frac{N_{s2}}{N_{p2}} \][/tex]
#### Transformer W:
- Primary = 80 coils
- Secondary = 20 coils
[tex]\[ V_{out2} = 20 \text{ V} \times \frac{20}{80} \][/tex]
[tex]\[ V_{out2} = 20 \text{ V} \times \frac{1}{4} \][/tex]
[tex]\[ V_{out2} = 5 \text{ V} \][/tex]
#### Transformer Y:
- Primary = 60 coils
- Secondary = 12 coils
[tex]\[ V_{out2} = 20 \text{ V} \times \frac{12}{60} \][/tex]
[tex]\[ V_{out2} = 20 \text{ V} \times \frac{1}{5} \][/tex]
[tex]\[ V_{out2} = 4 \text{ V} \][/tex]
#### Transformer Z:
- Primary = 70 coils
- Secondary = 35 coils
[tex]\[ V_{out2} = 20 \text{ V} \times \frac{35}{70} \][/tex]
[tex]\[ V_{out2} = 20 \text{ V} \times \frac{1}{2} \][/tex]
[tex]\[ V_{out2} = 10 \text{ V} \][/tex]
### Conclusion
Only transformer Y outputs the desired voltage of 4 V after the second step. Therefore, Roberto should use transformer Y to achieve an ending voltage of 4 V.
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