<< Chapter < Page Chapter >> Page >
V p = N p Δ Φ Δ t . size 12{V rSub { size 8{p} } = - N rSub { size 8{p} } { {ΔΦ} over {Δt} } } {}

The reason for this is a little more subtle. Lenz’s law tells us that the primary coil opposes the change in flux caused by the input voltage V p size 12{V rSub { size 8{p} } } {} , hence the minus sign. Assuming negligible coil resistance, Kirchhoff’s loop rule tells us that the induced voltage exactly equals the input voltage. Taking the ratio of these last two equations yields a useful relationship:

V s V p = N s N p . size 12{ { {V rSub { size 8{s} } } over {V rSub { size 8{p} } } } = { {N rSub { size 8{s} } } over {N rSub { size 8{p} } } } } {}

This is known as the transformer equation    , and it simply states that the ratio of the secondary to primary voltages in a transformer equals the ratio of the number of loops in their coils.

The output voltage of a transformer can be less than, greater than, or equal to the input voltage, depending on the ratio of the number of loops in their coils. Some transformers even provide a variable output by allowing connection to be made at different points on the secondary coil. A step-up transformer    is one that increases voltage, whereas a step-down transformer    decreases voltage. Assuming, as we have, that resistance is negligible, the electrical power output of a transformer equals its input. This is nearly true in practice—transformer efficiency often exceeds 99%. Equating the power input and output,

P p = I p V p = I s V s = P s . size 12{P rSub { size 8{p} } =I rSub { size 8{p} } V rSub { size 8{p} } =I rSub { size 8{s} } V rSub { size 8{s} } =P rSub { size 8{s} } } {}

Rearranging terms gives

V s V p = I p I s . size 12{ { {V rSub { size 8{s} } } over {V rSub { size 8{p} } } } = { {I rSub { size 8{p} } } over {I rSub { size 8{s} } } } } {}

Combining this with V s V p = N s N p size 12{ { {V rSub { size 8{s} } } over {V rSub { size 8{p} } } } = { {N rSub { size 8{s} } } over {N rSub { size 8{p} } } } } {} , we find that

I s I p = N p N s size 12{ { {I rSub { size 8{s} } } over {I rSub { size 8{p} } } } = { {N rSub { size 8{p} } } over {N rSub { size 8{s} } } } } {}

is the relationship between the output and input currents of a transformer. So if voltage increases, current decreases. Conversely, if voltage decreases, current increases.

Calculating characteristics of a step-up transformer

A portable x-ray unit has a step-up transformer, the 120 V input of which is transformed to the 100 kV output needed by the x-ray tube. The primary has 50 loops and draws a current of 10.00 A when in use. (a) What is the number of loops in the secondary? (b) Find the current output of the secondary.

Strategy and Solution for (a)

We solve V s V p = N s N p size 12{ { {V rSub { size 8{s} } } over {V rSub { size 8{p} } } } = { {N rSub { size 8{s} } } over {N rSub { size 8{p} } } } } {} for N s size 12{N rSub { size 8{s} } } {} , the number of loops in the secondary, and enter the known values. This gives

N s = N p V s V p = ( 50 ) 100,000 V 120 V = 4 . 17 × 10 4 . alignl { stack { size 12{N rSub { size 8{s} } =N rSub { size 8{p} } { {V rSub { size 8{s} } } over {V rSub { size 8{p} } } } } {} #" "= \( "50" \) { {"100,000 V"} over {"120"" V"} } =4 "." "17" times "10" rSup { size 8{4} } {} } } {}

Discussion for (a)

A large number of loops in the secondary (compared with the primary) is required to produce such a large voltage. This would be true for neon sign transformers and those supplying high voltage inside TVs and CRTs.

Strategy and Solution for (b)

We can similarly find the output current of the secondary by solving I s I p = N p N s size 12{ { {I rSub { size 8{s} } } over {I rSub { size 8{p} } } } = { {N rSub { size 8{p} } } over {N rSub { size 8{s} } } } } {} for I s size 12{I rSub { size 8{s} } } {} and entering known values. This gives

I s = I p N p N s = ( 10 . 00 A ) 50 4 . 17 × 10 4 = 12.0 mA . alignl { stack { size 12{I rSub { size 8{s} } =I rSub { size 8{p} } { {N rSub { size 8{p} } } over {N rSub { size 8{s} } } } } {} #" "= \( "10" "." "00 A" \) { {"50"} over {4 "." "17" times "10" rSup { size 8{4} } } } ="12" "." 0" mA" {} } } {}

Discussion for (b)

As expected, the current output is significantly less than the input. In certain spectacular demonstrations, very large voltages are used to produce long arcs, but they are relatively safe because the transformer output does not supply a large current. Note that the power input here is P p = I p V p = ( 10 . 00 A ) ( 120 V ) = 1 . 20 kW size 12{P rSub { size 8{p} } =I rSub { size 8{p} } V rSub { size 8{p} } = \( "10" "." "00"`A \) \( "120"`V \) =1 "." "20"`"kW"} {} . This equals the power output P p = I s V s = ( 12 . 0 mA ) ( 100 kV ) = 1 . 20 kW size 12{P rSub { size 8{p} } =I rSub { size 8{s} } V rSub { size 8{s} } = \( "12" "." 0`"mA" \) \( "100"`"kV" \) =1 "." "20"`"kW"} {} , as we assumed in the derivation of the equations used.

Calculating characteristics of a step-down transformer

A battery charger meant for a series connection of ten nickel-cadmium batteries needs to have a 15.0 V output to charge the batteries. It uses a step-down transformer with a 200-loop primary and a 120 V input. (a) How many loops should there be in the secondary coil? (b) If the charging current is 16.0 A, what is the input current?

Strategy and Solution for (a)

You would expect the secondary to have a small number of loops. Solving V s V p = N s N p size 12{ { {V rSub { size 8{s} } } over {V rSub { size 8{p} } } } = { {N rSub { size 8{s} } } over {N rSub { size 8{p} } } } } {} for N s size 12{N rSub { size 8{s} } } {} and entering known values gives

N s = N p V s V p = ( 200 ) 15 . 0 V 120 V = 25 . alignl { stack { size 12{N rSub { size 8{s} } =N rSub { size 8{p} } { {V rSub { size 8{s} } } over {V rSub { size 8{p} } } } } {} #" "= \( "200" \) { {"15" "." "0 V"} over {"120 V"} } ="25" {} } } {}

Strategy and Solution for (b)

The current input can be obtained by solving I s I p = N p N s size 12{ { {I rSub { size 8{s} } } over {I rSub { size 8{p} } } } = { {N rSub { size 8{p} } } over {N rSub { size 8{s} } } } } {} for I p size 12{I rSub { size 8{p} } } {} and entering known values. This gives

I p = I s N s N p = ( 16 . 0 A ) 25 200 = 2 . 00 A . alignl { stack { size 12{I rSub { size 8{p} } =I rSub { size 8{s} } { {N rSub { size 8{s} } } over {N rSub { size 8{p} } } } } {} #" "= \( "16" "." "0 A" \) { {"25"} over {"200"} } =2 "." "00"" A" {} } } {}

Discussion

The number of loops in the secondary is small, as expected for a step-down transformer. We also see that a small input current produces a larger output current in a step-down transformer. When transformers are used to operate large magnets, they sometimes have a small number of very heavy loops in the secondary. This allows the secondary to have low internal resistance and produce large currents. Note again that this solution is based on the assumption of 100% efficiency—or power out equals power in ( P p = P s size 12{P rSub { size 8{p} } =P rSub { size 8{s} } } {} )—reasonable for good transformers. In this case the primary and secondary power is 240 W. (Verify this for yourself as a consistency check.) Note that the Ni-Cd batteries need to be charged from a DC power source (as would a 12 V battery). So the AC output of the secondary coil needs to be converted into DC. This is done using something called a rectifier, which uses devices called diodes that allow only a one-way flow of current.

Section summary

  • Transformers use induction to transform voltages from one value to another.
  • For a transformer, the voltages across the primary and secondary coils are related by
    V s V p = N s N p , size 12{ { {V rSub { size 8{s} } } over {V rSub { size 8{p} } } } = { {N rSub { size 8{s} } } over {N rSub { size 8{p} } } } } {}
    where V p size 12{V rSub { size 8{p} } } {} and V s size 12{V rSub { size 8{s} } } {} are the voltages across primary and secondary coils having N p size 12{N rSub { size 8{p} } } {} and N s size 12{N rSub { size 8{s} } } {} turns.
  • The currents I p size 12{I rSub { size 8{p} } } {} and I s size 12{I rSub { size 8{s} } } {} in the primary and secondary coils are related by I s I p = N p N s size 12{ { {I rSub { size 8{s} } } over {I rSub { size 8{p} } } } = { {N rSub { size 8{p} } } over {N rSub { size 8{s} } } } } {} .
  • A step-up transformer increases voltage and decreases current, whereas a step-down transformer decreases voltage and increases current.

Problems&Exercises

A plug-in transformer, like that in [link] , supplies 9.00 V to a video game system. (a) How many turns are in its secondary coil, if its input voltage is 120 V and the primary coil has 400 turns? (b) What is its input current when its output is 1.30 A?

(a) 30.0

(b) 9 . 75 × 10 2 A size 12{9 "." "75" times "10" rSup { size 8{ - 2} } `A} {}

An American traveler in New Zealand carries a transformer to convert New Zealand’s standard 240 V to 120 V so that she can use some small appliances on her trip. (a) What is the ratio of turns in the primary and secondary coils of her transformer? (b) What is the ratio of input to output current? (c) How could a New Zealander traveling in the United States use this same transformer to power her 240 V appliances from 120 V?

A digital recorder uses a plug-in transformer to convert 120 V to 12.0 V, with a maximum current output of 200 mA. (a) What is the current input? (b) What is the power input? (c) Is this amount of power reasonable for a small appliance?

(a) 20.0 mA

(b) 2.40 W

(c) Yes, this amount of power is quite reasonable for a small appliance.

(a) What is the voltage output of a transformer used for rechargeable flashlight batteries, if its primary has 500 turns, its secondary 4 turns, and the input voltage is 120 V? (b) What input current is required to produce a 4.00 A output? (c) What is the power input?

(a) The plug-in transformer for a laptop computer puts out 7.50 V and can supply a maximum current of 2.00 A. What is the maximum input current if the input voltage is 240 V? Assume 100% efficiency. (b) If the actual efficiency is less than 100%, would the input current need to be greater or smaller? Explain.

(a) 0.063 A

(b) Greater input current needed.

A multipurpose transformer has a secondary coil with several points at which a voltage can be extracted, giving outputs of 5.60, 12.0, and 480 V. (a) The input voltage is 240 V to a primary coil of 280 turns. What are the numbers of turns in the parts of the secondary used to produce the output voltages? (b) If the maximum input current is 5.00 A, what are the maximum output currents (each used alone)?

A large power plant generates electricity at 12.0 kV. Its old transformer once converted the voltage to 335 kV. The secondary of this transformer is being replaced so that its output can be 750 kV for more efficient cross-country transmission on upgraded transmission lines. (a) What is the ratio of turns in the new secondary compared with the old secondary? (b) What is the ratio of new current output to old output (at 335 kV) for the same power? (c) If the upgraded transmission lines have the same resistance, what is the ratio of new line power loss to old?

(a) 2.2

(b) 0.45

(c) 0.20, or 20.0%

If the power output in the previous problem is 1000 MW and line resistance is 2.00 Ω , what were the old and new line losses?

Questions & Answers

current economic plans (MDGS) needs
Ajijola Reply
I don't know what is happening
surajkumar
What is economic
Joeali Reply
What is the importance of study economics
Wilma
Economic is the study of how humans make decisions in face of sacristy
Wilma
economics is the study of how humans makes decision in the face of scarcity
Kpienta
economics is the study of human behaviour when faced with difficult situation example when goods and services are scarcity.
Sydney
what is Economic
Dauda Reply
what is 4ps of economic?
thomas Reply
production place Price product
Benedict
Criticism of elasticity
Siddikur Reply
what is unemployment
Gyamfi Reply
ohk thanks
Gyamfi
why is unemployment rapid in the country
Gyamfi
I need more explanation
Odo
what is unemployment
Munanag Reply
not working
Bethel
some one who is willing qualified to work but can't find job
jackie
Bethel...explain? please
Abubakar
some one who is willing to work but can't find job
Hawa
Yes true
Brian
which one please
Hawa
unemployment refers to the ability for someone who is capable and willing to work but could not find a job..
Mnoko
some one who not able to find a job
Dennis
please what is the secret of learning?
thomas
What is stock market?
JOHN Reply
explain the various types of cost curve
Ruth Reply
Short-run average fixed cost (SRAFC) Short-run average total cost (SRAC or SRATC) Short-run average variable cost (AVC or SRAVC) Short-run fixed cost (FC or SRFC) Short-run marginal cost (SRMC) Short-run total cost (SRTC)
Romy
what's economic development and growth
Popoola Reply
what do you understand by Ceteris Paribus?
Gabriel Reply
the external factor will remained constant, except the price
Hasib
everything being equal
Chenwi
explain the uses of microeconomics
Nikita Reply
uses of microeconomics
Nikita
Adam Smith's definition of economics
Sylvia Reply
what is economic deficit
Amjad
this is a situation whereby a nation's outcome or available resources are not enough to the people thereby causing scarcity
Ariel
prices of Quality demanded is equal to Quality supplied
NABUBOLO Reply
it's quantity demand and quantity supplied that's called equilibrium
Romy
no
NABUBOLO
they deal With prices
NABUBOLO
define the elasticity
NABUBOLO
explain different types of elasticity
NABUBOLO
oops 😬 you are right you talk about quality I tell about quantity
Romy
elasticity is the measurement of the percentage change of one economic variable in response to a change in another
Romy
Cross Elasticity of Demand (XED) Income Elasticity of Demand (YED) Price Elasticity of Supply (PES)
Romy
anything else?
Romy
I need to know everything about theory of consumer behavior
Grace
Romy, what is microeconomic?
thomas
What is Economic please
Dauda
Thomas, microeconomics is the study of how consumers, workers, and firms interact to generate outcomes in specific markets
Kieran
Dauda, economics is the study of people and choices. it is on one side the study of wealth and on the more important side, a part of the study if man
Kieran
How does one analyze a market where both demand and supply shift?
Gabriel Reply
That's equilibrium market
Ramon
but an equlibrum can appear twice on the same market... both in Movement along the Demand/supply curve of shift in the Curve
Gabriel
I Mean on the same curve..
Gabriel
how can consumer surplus be calculated
Franklyn
While the American heart association suggests that meditation might be used in conjunction with more traditional treatments as a way to manage hypertension
Beverly Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play




Source:  OpenStax, Concepts of physics. OpenStax CNX. Aug 25, 2015 Download for free at https://legacy.cnx.org/content/col11738/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Concepts of physics' conversation and receive update notifications?

Ask