184_notes:examples:week14_step_down_transformer

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184_notes:examples:week14_step_down_transformer [2017/11/29 16:56] – tallpaul184_notes:examples:week14_step_down_transformer [2021/07/22 13:56] (current) – [Solution] schram45
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 +[[184_notes:ac|Return to Changing Flux from an Alternating Current notes]]
 +
 ===== Designing a Step-down Transformer ===== ===== Designing a Step-down Transformer =====
 Recall the [[184_notes:ac#Voltage_Transformer|discussion on voltage transformers]]. We designed a step-up transformer in the notes, which is used to convert small voltages from a generator into high voltages, which get carried long distances to residential areas. High-voltage power lines are dangerous, though, because the potential difference between the power lines and the ground is so enormous. Before the lines enter a residential area, they are sent to a transformer, from which low-voltage lines carry power to the residential area. How can you design a step-down transformer to convert a line that has a 240 kV potential difference to ground, into a power line that has a 120 V potential difference to ground? Recall the [[184_notes:ac#Voltage_Transformer|discussion on voltage transformers]]. We designed a step-up transformer in the notes, which is used to convert small voltages from a generator into high voltages, which get carried long distances to residential areas. High-voltage power lines are dangerous, though, because the potential difference between the power lines and the ground is so enormous. Before the lines enter a residential area, they are sent to a transformer, from which low-voltage lines carry power to the residential area. How can you design a step-down transformer to convert a line that has a 240 kV potential difference to ground, into a power line that has a 120 V potential difference to ground?
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   * The high-voltage line carries 240 kV.   * The high-voltage line carries 240 kV.
   * We want the low-voltage line to carry 120 V.   * We want the low-voltage line to carry 120 V.
-  * We know the step-up transformer was created by wrapping two solenoids around an iron torus, and connected the high-voltage line to one solenoid, and the low-voltage line to the other. The low-voltage solenoid had fewer turns. A more detailed representation is shown below.+  * We know the step-up transformer was created by wrapping two solenoids around an iron ring, and connected the high-voltage line to one solenoid, and the low-voltage line to the other. The low-voltage solenoid had fewer turns. A more detailed representation is shown below.
  
 ===Lacking=== ===Lacking===
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 ===Approximations & Assumptions=== ===Approximations & Assumptions===
-  * We have access to the same materials as we did for the step-up transformer.+  * We have access to the same materials as we did for the step-up transformer: This allows us to use some of the same relationships from the step-up transformer solution.
   * The step-down transformer we are building will have a similar design to the step-up transformer.   * The step-down transformer we are building will have a similar design to the step-up transformer.
  
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   * We represent the step-up transformer, as described in the Facts, with the following visual:   * We represent the step-up transformer, as described in the Facts, with the following visual:
  
-{{ 184_notes:week14_7.png?600 |Step-up Transformer}}+[{{ 184_notes:week14_7.png?600 |Step-up Transformer}}]
 ====Solution==== ====Solution====
-It makes sense intuitively to just flip the step-up transformer for our design. If the step-up transformer brings voltage down, we could just reverse it to bring the voltage up. Our design is shown below, where the number of turns in the primary solenoid is much greater than the number of turns in the secondary solenoid -- it's just the flipped step-up transformer, as described.+It makes sense intuitively to just flip the step-up transformer for our design. If the step-up transformer brings voltage down, we could just reverse it to bring the voltage up. Our design is shown below, where **now the number of turns in the primary solenoid is much greater than the number of turns in the secondary solenoid** -- it's just the flipped step-up transformer, as described.
  
-{{ 184_notes:14_step_down_design.png?600 |Step-down Transformer}}+[{{ 184_notes:14_step_down_design.png?600 |Step-down Transformer}}]
  
-As with the step-up transformer, the two solenoids are wrapped around the same iron torus. When current exists in the primary solenoid, it creates a magnetic field inside the solenoid, aligned along the torus. The magnetic field from the solenoid will cause all of the atoms within the iron to align with the magnetic field from the primary solenoid. Because iron atoms are very responsive to magnetic fields, even the atoms that are outside the primary solenoid will align with this magnetic field (largely because they are feeling the effects of their neighboring iron atoms). Because the magnetic field in the primary solenoid is oscillating (due to the alternating current), this means that the magnetic field in all of the iron ring is also constantly changing. For snapshot in time, the field may look like this:+As with the step-up transformer, the two solenoids are wrapped around the same iron ring. When current exists in the primary solenoid, it creates a magnetic field inside the solenoid, aligned along the iron ring. The magnetic field from the solenoid will cause all of the atoms within the iron to align with the magnetic field from the primary solenoid. Because iron atoms are very responsive to magnetic fields, even the atoms that are outside the primary solenoid will align with this magnetic field (largely because they are feeling the effects of their neighboring iron atoms). Because the magnetic field in the primary solenoid is oscillating (due to the alternating current), this means that the magnetic field in all of the iron ring is also constantly changing. For snapshot in time, the field may look like this:
  
-{{ 184_notes:14_step_down_b_field.png?600 |Step-down Transformer with B-field}}+[{{ 184_notes:14_step_down_b_field.png?600 |Step-down Transformer with B-field}}]
  
 The iron is able to align its atoms with the magnetic field much faster than the current alternates between directions, which is why we draw the magnetic field the same everywhere. The iron also greatly amplifies the magnetic field that the primary solenoid would produce in air, so even though BP contains magnetic field contributions from the primary solenoid //and// from the iron, the contribution from the iron is far greater. For this reason, we approximate the magnetic field as the same at all locations in the iron. By this approximation, BP=BS. The iron is able to align its atoms with the magnetic field much faster than the current alternates between directions, which is why we draw the magnetic field the same everywhere. The iron also greatly amplifies the magnetic field that the primary solenoid would produce in air, so even though BP contains magnetic field contributions from the primary solenoid //and// from the iron, the contribution from the iron is far greater. For this reason, we approximate the magnetic field as the same at all locations in the iron. By this approximation, BP=BS.
  
-Since the magnetic field is changing with the alternating current, (in the primary solenoid), this will induce a voltage (VS) in the secondary solenoid. We can use Faraday's Law to write this as:+Since the magnetic field in the iron ring is changing with the alternating current, (in the primary solenoid), this will induce a voltage (VS) in the secondary solenoid. We can use Faraday's Law to write this as:
 −VS=dΦBSdt −VS=dΦBSdt
  
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 −VS=ddt(BSASNS) −VS=ddt(BSASNS)
  
-Since NS and AS are constant with respect to time (not add/taking away loops or increasing/decreasing the area), these terms can come out of the derivative:+Since NS and AS are constant with respect to time (not adding/taking away loops or increasing/decreasing the area), these terms can come out of the derivative:
 −VS=NSASddt(BS)  −VS=NSASddt(BS) 
-//__If we assume that the torus has a constant cross-sectional area__//, then AP=AS and we already said that BP=BS. This means we can rewrite the flux through the secondary solenoid in terms of the magnetic field and area of the primary solenoid.+//__If we assume that the iron ring has a constant cross-sectional area__//, then AP=AS and we already said that BP=BS. This means we can rewrite the flux through the secondary solenoid in terms of the magnetic field and area of the primary solenoid.
 −VS=NSAPddt(BP)  −VS=NSAPddt(BP) 
  
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 VS=VPNSNP VS=VPNSNP
  
-Remember, we need to step down from the 240 kV power line to a 120 V line. This is a factor of 2000. One way to achieve this would be to set $N_P = 10,andN_S = 20000.Duetothehugestepdown,itmaybeeveneasiertodesignaseriesofstep−downtransformers,sothatwedon′thavetohavesuchalargenumberofturnsforthesecondarysolenoid.MaybeapplyafactorofN_S/N_P=40foronetransformer,andthenN_S/N_P=50forasecondtransformer.Youshouldbeabletoconvinceyourselfthatthiswouldbephysicallyequivalenttojustonestep−downtransformerwithN_S/N_P=2000$.+Remember, we need to step down from the 240 kV power line to a 120 V line. This is a factor of 2000. One way to achieve this would be to set $N_P = 20000,andN_S = 10.Duetothehugestepdown,itmaybeeveneasiertodesignaseriesofstep−downtransformers,sothatwedon′thavetohavesuchalargenumberofturnsforthesecondarysolenoid.MaybeapplyafactorofN_S/N_P=1/40foronetransformer,andthenN_S/N_P=1/50forasecondtransformer.Youshouldbeabletoconvinceyourselfthatthiswouldbephysicallyequivalenttojustonestep−downtransformerwithN_S/N_P=1/2000$.
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  • Last modified: 2017/11/29 16:56
  • by tallpaul