Worksheets/Eulers Method Circuits - Revision history

Admin: /* Inductor */

2016-05-09T10:32:14Z

Inductor

← Older revision		Revision as of 10:32, 9 May 2016
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	[[Image:Inductor.png]]		[[Image:Inductor.png]]

	~~Google image search for "antenna~~ inductor": ~~https~~:~~//www~~.~~google~~.~~com/search?site=&tbm=isch&q=inductor+antenna~~		Here is an example of an inductor:

			[[Image:LargeInductor.jpg\|500px]]

			Inductors are also found in everyday office electronics, like wall warts:

			[[Image:WallWart.gif\|200px]]

	The inductor works by storing energy in the form of a magnetic field inside of the coil while current flows through the inductor. That magnetic field creates an electromotive force (a voltage) for a given amount of current.		The inductor works by storing energy in the form of a magnetic field inside of the coil while current flows through the inductor. That magnetic field creates an electromotive force (a voltage) for a given amount of current.

Admin: /* Inductor */

2016-05-09T10:29:52Z

Inductor

← Older revision		Revision as of 10:29, 9 May 2016
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	====Inductor====		====Inductor====

	Inductors are interesting electrical parts. Physically, they are coils - to make an inductor, you simply coil up some wire. The electrical symbol for an inductor is:		Inductors are interesting electrical parts. They resist changes in electrical current (useful in converting AC to DC power), and can electrically lengthen circuits without physically lengthening them (useful for antennas). Physically, they are simply coils - to make an inductor, you simply coil up some wire. The electrical symbol for an inductor is:

	[[Image:Inductor.png]]		[[Image:Inductor.png]]

	~~Across~~ the inductor, the voltage drop ~~depends on~~ the rate of change of ~~the~~ current:		Google image search for "antenna inductor": https://www.google.com/search?site=&tbm=isch&q=inductor+antenna

			The inductor works by storing energy in the form of a magnetic field inside of the coil while current flows through the inductor. That magnetic field creates an electromotive force (a voltage) for a given amount of current.

			Inductors can also share magnetic cores, which allows current through one inductor to induce current transfer through another. This enables circuits to transfer power from one level of voltage and current to another level of voltage and current without being physically connected to it (useful safety feature for AC to DC transformers, like your laptop or phone charger).

			Mathematically, inductors can be modeled by writing the voltage drop across the inductor as being linear to the rate of change of current:

	<math>		<math>

Admin: /* Introduction */

2016-05-09T10:10:16Z

Introduction

← Older revision		Revision as of 10:10, 9 May 2016
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	Today we will be using a spreadsheet program to implement Euler’s Method and solve a differential equation. Euler’s Method ties together the concepts of numerical integration and differential equations. We will implement Euler’s Method to solve for the current in the following circuit:		Today we will be using a spreadsheet program to implement Euler’s Method and solve a differential equation. Euler’s Method ties together the concepts of numerical integration and differential equations. We will implement Euler’s Method to solve for the current in the following circuit:

	[[Image:Circuit.png\|~~500px~~]]		[[Image:Circuit.png\|300px]]

	This circuit is described in Example 4 of Section 9.5 of the Stewart Calculus textbook.		This circuit is described in Example 4 of Section 9.5 of the Stewart Calculus textbook.

Admin: /* Inductor */

2016-05-09T10:09:50Z

Inductor

← Older revision		Revision as of 10:09, 9 May 2016
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	====Inductor====		====Inductor====

	Inductors are interesting electrical parts. Physically, they are coils - to make an inductor, you simply coil up some wire.		Inductors are interesting electrical parts. Physically, they are coils - to make an inductor, you simply coil up some wire. The electrical symbol for an inductor is:

			[[Image:Inductor.png]]

	Across the inductor, the voltage drop depends on the rate of change of the current:		Across the inductor, the voltage drop depends on the rate of change of the current:

Admin: /* The General Differential Equation */

2016-05-09T10:08:52Z

The General Differential Equation

← Older revision		Revision as of 10:08, 9 May 2016
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	==The Circuit Differential Equation==		==The Circuit Differential Equation==

	===The ~~General Differential Equation~~===		===The Circuit Components===

	On the right is a circuit containing four components: a battery (which is a source of electric current), a resistor R, an inductor L, and a switch to turn the circuit on and off.		On the right is a circuit containing four components: a battery (which is a source of electric current), a resistor R, an inductor L, and a switch to turn the circuit on and off.
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	The current - that's what we're solving for - is denoted <math>I(t)</math>.		The current - that's what we're solving for - is denoted <math>I(t)</math>.

			====Battery====

			The voltage supplied by the battery is constant with time (this is a battery providing direct current, which means it is constant with time - interesting things happen when we run the same circuit with alternating current).

	The voltage supplied by the battery is denoted <math>E(t)</math> and is a constant function:		The voltage supplied by the battery is denoted <math>E(t)</math> and is a constant function:
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	</math>		</math>

	Across the resistor, the voltage ~~dorps by7~~ an amount proportional to the resistance and the current:		====Resistor====

			Using Ohm's Law E = I R, we can find the voltage drop across a resistor - the voltage drop E is a linear function of current I and resistance R.

			Across the resistor, the voltage drops by an amount proportional to the resistance and the current:

	<math>		<math>
	\Delta V_{resistor} = R I		\Delta V_{resistor} = R I
	</math>		</math>

			====Inductor====

			Inductors are interesting electrical parts. Physically, they are coils - to make an inductor, you simply coil up some wire.

	Across the inductor, the voltage drop depends on the rate of change of the current:		Across the inductor, the voltage drop depends on the rate of change of the current:
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	\Delta V_{inductor} = L \dfrac{dI}{dt}		\Delta V_{inductor} = L \dfrac{dI}{dt}
	</math>		</math>

			===The General Differential Equation===

			We are writing a differential equation for current I(t). But because current is not conserved, we have to write a balance equation for voltage, E, then write voltage in terms of current to get current as a function of time.

	Balancing the supplied voltage with the voltage drops gives us our differential equation:		Balancing the supplied voltage with the voltage drops gives us our differential equation:

Admin: /* Content */

2016-05-09T10:03:15Z

Content

← Older revision		Revision as of 10:03, 9 May 2016
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	Today we will be using a spreadsheet program to implement Euler’s Method and solve a differential equation. Euler’s Method ties together the concepts of numerical integration and differential equations. We will implement Euler’s Method to solve for the current in the following circuit:		Today we will be using a spreadsheet program to implement Euler’s Method and solve a differential equation. Euler’s Method ties together the concepts of numerical integration and differential equations. We will implement Euler’s Method to solve for the current in the following circuit:

			[[Image:Circuit.png\|500px]]

	This circuit is described in Example 4 of Section 9.5 of the Stewart Calculus textbook.		This circuit is described in Example 4 of Section 9.5 of the Stewart Calculus textbook.

Admin: /* Introduction */

2016-05-08T18:00:22Z

Introduction

← Older revision		Revision as of 18:00, 8 May 2016
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	This circuit is described in Example 4 of Section 9.5 of the Stewart Calculus textbook.		This circuit is described in Example 4 of Section 9.5 of the Stewart Calculus textbook.

			Would be nice to have just a little bit more motivation on what inductors can be used to do. What role do they play in power transformers and rectifier circuits?

	==Euler's Method==		==Euler's Method==

Admin at 17:59, 8 May 2016

2016-05-08T17:59:38Z

← Older revision		Revision as of 17:59, 8 May 2016
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	Part 4c: Resistor of 200 Ohms		Part 4c: Resistor of 200 Ohms



			=References=

			Stewart Example 4 Section 9.5

			=Flags=

			{{WorksheetFlag}}

Admin: /* Worksheet Questions */

2016-05-08T17:58:57Z

Worksheet Questions

← Older revision		Revision as of 17:58, 8 May 2016
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	Question 3: Create a chart with values of current as a function of time. Add proper labels (with units) to your axes, and a title to your chart explaining what is shown.		Question 3: Create a chart with values of current as a function of time. Add proper labels (with units) to your axes, and a title to your chart explaining what is shown.

	Question 4: What affect does resistance have on the value of <~~code~~>I_{\infty}</math>? Implement Euler's Method for the three resistances shown below. Replicate your chart of current versus time for each resistor.		Question 4: What affect does resistance have on the value of <math>I_{\infty}</math>? Implement Euler's Method for the three resistances shown below. Replicate your chart of current versus time for each resistor.

	Part 4a: Resistor of 5 Ohms		Part 4a: Resistor of 5 Ohms

Admin: /* Worksheet Questions */

2016-05-08T17:58:42Z

Worksheet Questions

← Older revision		Revision as of 17:58, 8 May 2016
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	* Voltage of 120 V, E = 120		* Voltage of 120 V, E = 120

	Question 2: Implement the solution to this differential equation in an Euler's Method spreadsheet program. When you are finished, you should have a column of values that give you the current as a function of time. What is the limiting value of current, <math>I_{\infty} = \lim_{t \rightarrow \infty} I(t)</math>?		Question 2: Implement the solution to this differential equation in an Euler's Method spreadsheet program. When you are finished, you should have a column of values that give you the current as a function of time. What is the limiting value of current, <math>I_{\infty} = \lim_{t \rightarrow \infty} I(t)</math>?

			Question 3: Create a chart with values of current as a function of time. Add proper labels (with units) to your axes, and a title to your chart explaining what is shown.

			Question 4: What affect does resistance have on the value of <code>I_{\infty}</math>? Implement Euler's Method for the three resistances shown below. Replicate your chart of current versus time for each resistor.

			Part 4a: Resistor of 5 Ohms

			Part 4b: Resistor of 50 Ohms

			Part 4c: Resistor of 200 Ohms