Transformers (Circuits)

Transformer Concept Map

Electricity sent through power lines is transmitted with high voltages through long thick power lines because wires have a resistance that causes power loss at a rate proportional to the current squared. By transmitting at a high voltage, energy loss is minimized. Home appliances however operate at much lower voltages. Something is needed to convert the power to a high current, low voltage power that home appliances can use. This conversion from high voltage to low voltage, and vice versa, is accomplished by a transformer.

The Main Idea

Transformer and Faraday's Law

From Faraday's law as well as conservation of energy we see that an ideal transformer the voltage ratio is equal to the turns ratio, and power in equals power out. Transformers uses both of these to convert from either high to low or low to high voltages.

A Mathematical Model

For a "step-down" transformer (one that converts from high to low voltage and increases current):

If a solenoid is built wrapping [math]\displaystyle{ {N}_{1} }[/math] turns around a hollow cylinder for the primary coil, and wrapping [math]\displaystyle{ {N}_{2} }[/math] turns around the outside of the secondary coil, and then connecting the primary coil to a an AC power supply, the emf that will develop in the secondary coil will be as follows:

The magnetic field made by the primary coil: [math]\displaystyle{ B = \frac{\mu_0IN_1}{d} }[/math] The cross-sectional area of the solenoid is A, so the emf in one turn of the secondary coil is: [math]\displaystyle{ \frac{AdB}{dt} }[/math] The total emf in the secondary coil is [math]\displaystyle{ {N}_{2} }[/math] times the emf in one turn, so the potential difference across the secondary coil is: [math]\displaystyle{ {N}_{2}A(mu_0{N}_{1}/d)dI/dt }[/math] .

The potential difference across the primary coil is [math]\displaystyle{ \frac{LdI}{dt} }[/math], where [math]\displaystyle{ L = \frac{\mu_0AIN_1^2}{d} }[/math], so the potential difference across the primary coil is: [math]\displaystyle{ A({\mu_0}IN_1^2/d)dI/dt }[/math]

Comparing [math]\displaystyle{ emf_2={N}_{2}A(mu_0{N}_{1}/d)dI/dt }[/math] with [math]\displaystyle{ emf_1= A(mu_0{N}_{1}^2/d)dI/dt }[/math], we see that [math]\displaystyle{ emf_2= ({N}_{2}/{N}_{1})emf_1 }[/math]. The ratio of the number of turns determines the change in voltage.

Faraday's law applied to a transformer can be written as: [math]\displaystyle{ \frac{V_s}{V_p}= \frac{N_s}{N_p} }[/math], where the subscripts refer to primary and secondary coils.

Because energy is conserved and power is [math]\displaystyle{ I \Delta {E} }[/math], the smaller voltage in the secondary coil is accompanied by a larger current. This can be written as: [math]\displaystyle{ P_p= V_pI_p=V_sI_s = P_s }[/math].

In the case of a "step-up" transformer, the primary coil has few turn and the secondary many, therefore increasing the voltage and decreasing the current.

A Computational Model

How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript

Examples

Be sure to show all steps in your solution and include diagrams whenever possible

Simple

A transformer has a primary coil with 102 turns and a secondary coil of 360 turns. The AC voltage across the primary coil has a maximum of 124 V and the AC current through the primary coil has a maximum of 3 A. What are the maximum values of the voltage and current for the secondary coil?

[math]\displaystyle{ {V_s}=? {V_p}= 124V {N_s}= 360 {N_p}= 102 }[/math].

[math]\displaystyle{ \frac{V_s}{V_p}= \frac{N_s}{N_p} }[/math].

[math]\displaystyle{ \frac{V_s}{124V}= \frac{360}{102} }[/math].

[math]\displaystyle{ {V_s}= 438V }[/math].

[math]\displaystyle{ P_p= V_pI_p=V_sI_s = P_s }[/math].

[math]\displaystyle{ V_p=124V I_p=3A V_s= 438V I_s = ? }[/math].

[math]\displaystyle{ (124V)(3A)= (438V)I_s }[/math].

[math]\displaystyle{ I_s = 0.85A }[/math].

Connectedness

1. How is this topic connected to something that you are interested in?
2. How is it connected to your major?
3. Is there an interesting industrial application?

History

Put this idea in historical context. Give the reader the Who, What, When, Where, and Why. The property of induction was discovered in the 1830's but it wasn't until 1886 that William Stanley, working for Westinghouse, built the first reliable commercial transformer. It was first designed and used in both experimental and commercial systems by Ottó Bláthy, Miksa Déri, Károly Zipernowsky of the Austro-Hungarian Empire. The first AC power system that used the modern transformer was in Great Barrington, Massachusetts in 1886. In 1891 mastermind Mikhail Dobrovsky designed and demonstrated his 3 phase transformers in the Electro-Technical Exposition at Frankfurt, Germany.

DC power was mainly used in the 1880's but it was hard to transmit over distance because it requires high voltage and a thin wire or low voltage and a wide wire. High voltage on DC is very dangerous, and with low voltage the wire would be so thick that it would be impractical. With AC power, high voltage is also used to move electricity down a long wire. AC is more practical, however, because once the power reaches the destination, a transformer can be used to change the voltage down to a manageable level.

See also

Faraday's Law

This will give you a general understanding of Faraday's Law, which is the basis behind transformer technology.

Inductance

Inductance is another property of an electrical conductor derived from Faraday's law.

Gaus's Flux Theorem

Changing the flux of a magnetic field around a coil will induce voltage.

Further reading

Books, Articles or other print media on this topic

External links

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/transf.html

https://en.wikipedia.org/wiki/Transformer

http://www.edisontechcenter.org/Transformers.html

References

Chabay, R., & Sherwood, B. (2015). Electric Potential. In Matter & interactions (4th ed., Vol. Two, pp. 920). Danvers, Massachusetts: J. Wiley & sons.

http://hyperphysics.phy-astr.gsu.edu/hbase/magnetic/transf.html

http://www.edisontechcenter.org/Transformers.html