Transformers (Circuits): Difference between revisions

Latest revision as of 12:32, 8 December 2015

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.

Background

Inductance

Currents can be induced (produced) by changing the current through a coil. This is due to the changing magnetic field [math]\displaystyle{ \textstyle (dB/dt) }[/math] produced by varying the current through the coil. We know from the Maxwell-Faraday Law of Maxwell's Equations:

[math]\displaystyle{ |emf| = \oint \overrightarrow{E}_{NC} \cdot d\overrightarrow{l} = \left | \frac{d\phi_{mag}}{dt} \right \vert }[/math]

Or that a changing magnetic field through an area produces a non-Coloumb electric field.

Mathematical Formulae

Before moving on to a discussion of the mathematics of transformers, here are some formulas it will be helpful to recall:

Magnetic Field Inside a Solenoid: [math]\displaystyle{ B=\frac{\mu_0 N I}{d} }[/math]

Where [math]\displaystyle{ \textstyle N }[/math] is the number of coils and [math]\displaystyle{ \textstyle d }[/math] is the length of the solenoid.

Magnitude of self-induced emf: [math]\displaystyle{ \textstyle \left|emf_{ind}\right \vert=L\left|\frac{d I}{d t} \right \vert }[/math]

Where [math]\displaystyle{ L }[/math] is the proportionality constant called the "inductance" or "self-inductance" which equals [math]\displaystyle{ \textstyle \frac{\mu_0 N^2}{d}\pi R^2 }[/math]

Expanding this, we get the self-induced emf in a solenoid is: [math]\displaystyle{ \textstyle emf= \frac{\mu_0 N^2}{d}\pi R^2 \frac{d I}{d t} }[/math]

Finally, remember your units. [math]\displaystyle{ emf }[/math] is measured in volts, self-inductance is [math]\displaystyle{ \textstyle(V•s/A) }[/math] or the "henry" (H), and [math]\displaystyle{ B }[/math] is measured in Tesla (T) or [math]\displaystyle{ \textstyle(\frac{kg}{s^2 A}) }[/math]

How They Work

Conversion from high to low, or low to high voltage can be accomplished using the principles discussed above. Consider a solenoid with [math]\displaystyle{ N_1=100 }[/math] turns around a hollow cylinder of length [math]\displaystyle{ d=.3 m }[/math]. Now wrap [math]\displaystyle{ N_2 = 200 }[/math] turns around this solenoid to form the secondary coil. If an alternating current is run through the primary coil, we get a non-zero [math]\displaystyle{ \textstyle\frac{d I}{d t} }[/math]We can now calculate the potential difference across each coil.

Primary Coil

As stated above, the induced emf in the primary coil is [math]\displaystyle{ L\left|\frac{d I}{d t} \right \vert }[/math]. Expanding this and substituting [math]\displaystyle{ A=\pi R^2 }[/math] for the area, we get a potential difference across the primary coil of [math]\displaystyle{ \textstyle A(\mu_0 N_1^2 /d)dI/dt }[/math].

Secondary Coil

A current is induced in the secondary coil by the changing magnetic field produced by the primary coil. The magnetic field is [math]\displaystyle{ \textstyle B = \mu_0 N_1 I/d }[/math] and it is changing across area [math]\displaystyle{ A }[/math] (which is only the area of the inner coil, not the outer secondary coil. So the emf in one turn of the secondary coil is [math]\displaystyle{ A dB/dt }[/math]. We have [math]\displaystyle{ N_2 }[/math] secondary coils, so the emf is [math]\displaystyle{ N_2 AdB/dt }[/math]. If we expand out our [math]\displaystyle{ dB/dt }[/math] term, we can get the emf across the second coil in a formula similar to the emf across the primary coil: [math]\displaystyle{ emf_{sec} = N_2A(\mu_0N_1/d)dI/dt }[/math].

Voltage Ratio

We now can see that the ratio of the secondary to primary emf is [math]\displaystyle{ emf_{pri}/emf_{sec} }[/math]. This yields:

[math]\displaystyle{ \frac {N_2A(\mu_0N_1/d)dI/dt}{A(\mu_0 N_1^2 /d)dI/dt} }[/math] which cancels and leaves [math]\displaystyle{ \frac {N_2}{N_1} }[/math] or in this case [math]\displaystyle{ \frac{200}{100} }[/math].

This transformer would create a emf 2 times the emf in the primary coil. Because we can't create energy from nothing, power ([math]\displaystyle{ I\Delta V }[/math]) must be conserved; the double voltage in the secondary coil is accompanied by a current of half the strength of the primary coil.

The transformer described above is called a "step-up" transformer because it "ups" the voltage. There are also "step-down" transformers which reduce the voltage and have fewer turns on the secondary coil than primary coil.

Circuit Diagrams

Below are animated circuit diagrams for a step-up transformer, step-down transformer, and even a somewhat pointless transformer that has a 1:1 voltage ratio. The right side panel lists the properties of the transformer. The primary inductance and coupling coefficient are beyond the scope of my knowledge. However I did enough research to know that the Coupling Coefficient is some property of a transformer derived from the self inductance of each coil. I believe for most discussions on transformers, the "ideal" coupling coefficient is 100%. The waveform is graphing the voltage of the left loop (blue) and voltage of the right loop (green).

A step-up transformer that doubles voltage and halves current

A step-down transformer that halves voltage and double current

A 1:1 transformer. Here the green line isn't visible on the waveform because the voltages are identical.

Connectedness

1. How is this topic connected to something that you are interested in?

Although Dr. Greco said he found the circuits chapter and material dull, as a Computer Science major with an interest in Electrical Engineering and tinkering with hardware, I thought it was a great practical part of the course. Our textbook Electric Potential. In Matter & interactions (4th ed.) introduces transformers in the context of induction and Faraday in a more conceptual sense. I believed they should also be shown alongside circuits to demonstrate their practicality. It really helped with my understanding the view the setup and waveform of a circuit involving transformers.

I even tried to add a bridge rectifier and regulator to the circuit to show the AC being transformed back into DC. My circuits knowledge is somewhat limited and I was unable to properly connect this circuit.

2. How is it connected to your major?

In today's world of open source hardware and software, it is very important for a Computer Science major to understand circuits beyond a basic level and be able to incorporate circuits and moving parts into their work. With Raspberry Pis, Arduinos, and everything else the internet has to offer, a CS major is severely limiting themselves by not studying topics like theses.

3. Is there an interesting industrial application?

I think the interesting thing about industrial applications is how transformers appear in such a simple context in many power cords for appliances today. I remember back to my first interaction with one: Years ago at a friend's house we were playing songs on his electrical keyboard. At a point, we decided to move the keyboard and in the process broke the plastic casing off the transformer of the power cord. I was young, so didn't recognize it as a danger or issue at the time. I can vividly remember the copper coils (I think). At any rate, I plugged it in and was touching the coils and got an arm numbing shock. But my shock boils down to just a simple transformer that any Physics 2 student could make and understand.

iCircuit - Electronic Circuit Simulator and Designer (available for Windows, Mac, and iOS)

@@ Line 1: / Line 1: @@
-==Thermodynamics==
+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.
-This topics focuses on energy work of a system but it can only deal with a large scale response to heat in a system.  '''Thermodynamics''' is the study of the work, heat and energy of a system.  The smaller scale gas interactions can explained using the kinetic theory of gases.  There are three fundamental laws that go along with the topic of thermodynamics.  They are the zeroth law, the first law, and the second law.  These laws help us understand predict the the operation of the physical system.  In order to understand the laws, you must first understand thermal equilibrium.  [[Thermal equilibrium]] is reached when a object that is at a higher temperature is in contact with an object that is at a lower temperature and the first object transfers heat to the latter object until they approach the same temperature and maintain that temperature constantly.  It is also important to note that any thermodynamic system in thermal equilibrium possesses internal energy.
+==Background==
-===Zeroth Law===
+===Inductance===
+Currents can be induced (produced) by changing the current through a coil. This is due to the changing magnetic field <math>\textstyle (dB/dt)</math> produced by varying the current through the coil. We know from the Maxwell-Faraday Law of Maxwell's Equations:
-The zeroth law states that if two systems are at thermal equilibrium at the same time as a third system, then all of the systems are at equilibrium with each other.  If systems A and C are in thermal equilibrium with B, then system A and C are also in thermal equilibrium with each other.  There are underlying ideas of heat that are also important.  The most prominent one is that all heat is of the same kind.  As long as the systems are at thermal equilibrium, every unit of internal energy that passes from one system to the other is balanced by the same amount of energy passing back.  This also applies when the two systems or objects have different atomic masses or material.
+<math align="center">|emf| = \oint \overrightarrow{E}_{NC} \cdot d\overrightarrow{l} = \left | \frac{d\phi_{mag}}{dt} \right \vert </math>
-====A Mathematical Model====
+Or that a changing magnetic field through an area produces a non-Coloumb electric field.
-======Prereading======
-Here are some formulas it will be good to remember before reading about transformers.
-Magnetic field inside a solenoid
-<math>
-\overrightarrow{B} = \frac{&mu_0 N I}{d}
-</math>
-====A Computational Model====
+===Mathematical Formulae===
+Before moving on to a discussion of the mathematics of transformers, here are some formulas it will be helpful to recall:
-How do we visualize or predict using this topic. Consider embedding some vpython code here [https://trinket.io/glowscript/31d0f9ad9e Teach hands-on with GlowScript]
+* Magnetic Field Inside a Solenoid: <math>B=\frac{\mu_0 N I}{d}</math>
+::Where <math>\textstyle N</math> is the number of coils and <math>\textstyle d</math> is the length of the solenoid.
-===First Law===
+* Magnitude of self-induced emf: <math>\textstyle \left|emf_{ind}\right \vert=L\left|\frac{d I}{d t} \right \vert</math>
+:: Where <math>L</math> is the proportionality constant called the "inductance" or "self-inductance" which equals <math>\textstyle \frac{\mu_0 N^2}{d}\pi R^2</math>
-The first law of thermodynamics defines the internal energy (E) as equal to the difference between heat transfer (Q) ''into'' a system and work (W) ''done by'' the system.  Heat removed from a system would be given a negative sign and heat applied to the system would be given a positive sign.  Internal energy can be converted into other types of energy because it acts like potential energy.  Heat and work, however, cannot be stored or conserved independently because they depend on the process.  This allows for many different possible states of a system to exist.  There can be a process known as the adiabatic process in which there is no heat transfer.  This occurs when a system is full insulated from the outside environment.  The implementation of this law also brings about another useful state variable, '''enthalpy'''.
+*Expanding this, we get the self-induced emf in a solenoid is: <math>\textstyle emf= \frac{\mu_0 N^2}{d}\pi R^2 \frac{d I}{d t}</math>
-====A Mathematical Model====
+* Finally, remember your units. <math>emf</math> is measured in volts, self-inductance is <math>\textstyle(V•s/A)</math> or the "henry" (H), and <math>B</math> is measured in Tesla (T) or <math>\textstyle(\frac{kg}{s^2 A})</math>
-E2 - E1 = Q - W
+==How They Work==
+Conversion from high to low, or low to high voltage can be accomplished using the principles discussed above. Consider a solenoid with <math>N_1=100</math> turns around a hollow cylinder of length <math>d=.3 m</math>. Now wrap <math>N_2 = 200</math> turns around this solenoid to form the secondary coil. If an ''alternating current'' is run through the primary coil, we get a non-zero <math>\textstyle\frac{d I}{d t}</math>We can now calculate the potential difference across each coil.
-==Second Law==
+===Primary Coil===
+As stated [[#Mathematical Formulae|above]], the induced emf in the primary coil is <math>L\left|\frac{d I}{d t} \right \vert</math>. Expanding this and substituting <math>A=\pi R^2</math> for the area, we get a potential difference across the primary coil of <math>\textstyle A(\mu_0 N_1^2 /d)dI/dt</math>.
-The second law states that there is another useful variable of heat, entropy (S).  Entropy can be described as the disorder or chaos of a system, but in physics, we will just refer to it as another variable like enthalpy or temperature.  For any given physical process, the combined entropy of a system and the environment remains a constant if the process can be reversed.  The second law also states that if the physical process is irreversible, the combined entropy of the system and the environment must increase.  Therefore, the final entropy must be greater than the initial entropy.
+===Secondary Coil===
+A current is induced in the secondary coil by the changing magnetic field produced by the primary coil. The magnetic field is <math>\textstyle B = \mu_0 N_1 I/d</math> and it is changing across area <math>A</math> (which is only the area of the inner coil, not the outer secondary coil. So the emf in one turn of the secondary coil is <math>A dB/dt</math>. We have <math>N_2</math> secondary coils, so the emf is <math>N_2  AdB/dt</math>. If we expand out our <math>dB/dt</math> term, we can get the emf across the second coil in a formula similar to the emf across the primary coil: <math>emf_{sec} = N_2A(\mu_0N_1/d)dI/dt</math>.
-===Mathematical Models===
+===Voltage Ratio===
+We now can see that the ratio of the secondary to primary emf is <math>emf_{pri}/emf_{sec}</math>. This yields:
-delta S = delta Q/T
+<math>\frac {N_2A(\mu_0N_1/d)dI/dt}{A(\mu_0 N_1^2 /d)dI/dt}</math> which cancels and leaves <math>\frac {N_2}{N_1}</math> or in this case <math>\frac{200}{100}</math>.
-Sf = Si (reversible process)
-Sf > Si (irreversible process)
-===Examples===
+This transformer would create a emf 2 times the emf in the primary coil. Because we can't create energy from nothing, power (<math>I\Delta V</math>) must be conserved; the double voltage in the secondary coil is accompanied by a current of half the strength of the primary coil.
-'''Reversible process''': Ideally forcing a flow through a constricted pipe, where there are no boundary layers. As the flow moves through the constriction, the pressure, volume and temperature change, but they return to their normal values once they hit the downstream.  This return to the variables' original values allows there to be no change in entropy.  It is often known as an isentropic process.
+The transformer described above is called a "step-up" transformer because it "ups" the voltage. There are also "step-down" transformers which reduce the voltage and have fewer turns on the secondary coil than primary coil.
-'''Irreversible process''': When a hot object and cold object are put in contact with each other, eventually the heat from the hot object will transfer to the cold object and the two will reach the same temperature and stay constant at that temperature, reaching equilibrium.  However, once those objects are separated, they will remain at that equilibrium temperature until something else acts upon it.  The objects do not go back to their original temperatures so there is a change in entropy.
+==Circuit Diagrams==
+Below are animated circuit diagrams for a step-up transformer, step-down transformer, and even a somewhat pointless transformer that has a 1:1 voltage ratio. The right side panel lists the properties of the transformer. The primary inductance and coupling coefficient are beyond the scope of my knowledge. However I did enough research to know that the ''Coupling Coefficient'' is some property of a transformer derived from the self inductance of each coil. I believe for most discussions on transformers, the "ideal" coupling coefficient is 100%. The waveform is graphing the voltage of the left loop (blue) and voltage of the right loop (green).
+[[File:step_up.gif|500px|border|frame|center|A step-up transformer that doubles voltage and halves current]]
+[[File:step_down.gif|500px|border|frame|center|A step-down transformer that halves voltage and double current]]
+[[File:constant.gif|500px|border|frame|center|A 1:1 transformer. Here the green line isn't visible on the waveform because the voltages are identical.]]
 ==Connectedness==
-#How is this topic connected to something that you are interested in?
+:'''1. How is this topic connected to something that you are interested in?'''
-#How is it connected to your major?
-#Is there an interesting industrial application?
+Although Dr. Greco said he found the circuits chapter and material dull, as a Computer Science major with an interest in Electrical Engineering and tinkering with hardware, I thought it was a great practical part of the course. Our textbook ''Electric Potential. In Matter & interactions (4th ed.)'' introduces transformers in the context of induction and Faraday in a more conceptual sense. I believed they should also be shown alongside circuits to demonstrate their practicality. It really helped with my understanding the view the setup and waveform of a circuit involving transformers.
+I even tried to add a bridge rectifier and regulator to the circuit to show the AC being transformed back into DC. My circuits knowledge is somewhat limited and I was unable to properly connect this circuit.
+:'''2. How is it connected to your major?'''
-==History==
+In today's world of open source hardware and software, it is very important for a Computer Science major to understand circuits beyond a basic level and be able to incorporate circuits and moving parts into their work. With Raspberry Pis, Arduinos, and everything else the internet has to offer, a CS major is severely limiting themselves by not studying topics like theses.
-Thermodynamics was brought up as a science in the 18th and 19th centuries.  However, it was first brought up by Galilei, who introduced the concept of temperature and invented the first thermometer.  G. Black first introduced the word 'thermodynamics'.  Later, G. Wilke introduced another unit of measurement known as the calorie that measures heat.   The idea of thermodynamics was brought up by Nicolas Leonard Sadi Carnot.  He is often known as "the father of thermodynamics".  It all began with the development of the steam engine during the Industrial Revolution.  He devised an ideal cycle of operation.  During his observations and experimentations, he had the incorrect notion that heat is conserved, however he was able to lay down theorems that led to the development of thermodynamics.  In the 20th century, the science of thermodynamics became a conventional term and a basic division of physics.  Thermodynamics dealt with the study of general properties of physical systems under equilibrium and the conditions necessary to obtain equilibrium.
+:'''3. Is there an interesting industrial application?'''
-== See also ==
+I think the interesting thing about industrial applications is how transformers appear in such a simple context in many power cords for appliances today. I remember back to my first interaction with one: Years ago at a friend's house we were playing songs on his electrical keyboard. At a point, we decided to move the keyboard and in the process broke the plastic casing off the transformer of the power cord. I was young, so didn't recognize it as a danger or issue at the time. I can vividly remember the copper coils (I think). At any rate, I plugged it in and was touching the coils and got an arm numbing shock. But my shock boils down to just a simple transformer that any Physics 2 student could make and understand.
-Are there related topics or categories in this wiki resource for the curious reader to explore?  How does this topic fit into that context?
+==See also==
+#[[Faraday's Law]] This will give you a general understanding of Faraday's Law, which is the basis behind transformer technology.
+#[[Inductance]] A more in depth look at Inductance, a direct consequence of Faraday's Law.
+#[[Gauss's Flux Theorem]] Changing the flux of a magnetic field around a coil will induce voltage.
+#[[Transformers from a physics standpoint]] Detail on the material properties and physics of transformers, outside the scope of circuits.
 ===Further reading===
@@ Line 60: / Line 76: @@
 Books, Articles or other print media on this topic
-===External links===
+==References==
-Internet resources on this topic
+Chabay, R., & Sherwood, B. (2015). Electric Potential. In Matter & interactions (4th ed., Vol. Two, pp. 917-921). Danvers, Massachusetts: J. Wiley & sons.
-==References==
-https://www.grc.nasa.gov/www/k-12/airplane/thermo0.html
+[http://icircuitapp.com iCircuit - Electronic Circuit Simulator and Designer (available for Windows, Mac, and iOS)]
-http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/thereq.html
-https://www.grc.nasa.gov/www/k-12/airplane/thermo2.html
-http://www.phys.nthu.edu.tw/~thschang/notes/GP21.pdf
-http://www.eoearth.org/view/article/153532/
-[[Category:Which Category did you place this in?]]
+[[Category: Simple Circuits]]

Transformers (Circuits): Difference between revisions

Latest revision as of 12:32, 8 December 2015

Contents

Background

Inductance

Mathematical Formulae

How They Work

Primary Coil

Secondary Coil

Voltage Ratio

Circuit Diagrams

Connectedness

See also

Further reading

References

Navigation menu