RC Circuit: Difference between revisions

Latest revision as of 23:15, 27 November 2016

Claimed by Sean Fischer Fall 2016

The Main Idea

An RC circuit is a circuit that contains a battery with a known emf, a resistor (R), and a capacitor (C). An RC circuit can be in either series or parallel. The figure in the top right of the page shows an RC circuit. The capacitor stores electric charge (Q)

RC Circuits use a DC (direct current) voltage source and the capacitor is uncharged at its initial state. In the figure below, you see an RC circuit with a switch. When the switch is closed, the capacitor will begin to charge as the current can now flow throughout the circuit. To discharge the capacitor, you simply disconnect the switch.

Remember that potential difference across the capacitor is delta [math]\displaystyle{ {V} = {Q}/{C} }[/math], where Q is charge on the plate and C is the capacitance. When the switch is closed, voltage on the capacitor rises rapidly at first, due to the high current at [math]\displaystyle{ {time} = {0} }[/math]. The voltage opposes the battery, increasing from zero to the max emf when the capacitor is fully charged. The current decreases from its initial value of [math]\displaystyle{ {I}_{0} = {emf}/{R} }[/math] to zero as the voltage on the capacitor reaches the same value as the emf. The current is initially at its max at time [math]\displaystyle{ {t} = {0} }[/math]. Once the potential difference across the plates of the capacitor equals the battery's voltage supply, current will stop flowing through the circuit. This is known as the steady state of an RC circuit; it is reached when time goes to infinity.

Using derived calculus, the equation for voltage versus time when the capacitor is charged through resistor R is [math]\displaystyle{ {V} = {emf(1-e^{\frac{-t}{RC}})} }[/math]. [math]\displaystyle{ {V} }[/math] is defined as the voltage across the capacitor. [math]\displaystyle{ {emf} }[/math] is equal to the emf of the DC voltage source. The units of [math]\displaystyle{ {RC} }[/math] are in seconds. [math]\displaystyle{ {τ} = {RC} }[/math]. [math]\displaystyle{ {τ} }[/math] is the constant of time in the RC circuit.

The smaller the resistance, the faster a capacitor will be charged. It takes longer to charge than to discharge. This is because a larger current flows through a smaller resistance ([math]\displaystyle{ {I} = {V/R} }[/math]). Also the smaller the capacitor (C), the less time it will need to charge. [math]\displaystyle{ {τ} = {RC} }[/math] explains both of these.

Kirchoff's Node Rule is important to RC Circuits because finding the current flow in the different states of an RC Circuit often relies on nodes when the capacitor is in a parallel circuit.

Kirchhoff’s loop rule explains that the sum of changes in potential around any closed loop must be zero.

A Mathematical Model

These three equations are helpful in solving and understanding RC circuit problems

To find the equation for voltage versus time when charging a capacitor through a resistor, you start with rearranging the energy equation and solving for I:

The graph of Voltage vs. Time in a RC Circuit is given by:

A Computational Model

Using a simulation where R and C were equal, the capacitors were charged and then discharged and the two images show the voltage and current of this versus time.

For the gif below, the capacitor is charged and then discharged. This shows the voltage over time. The red line is the voltage and the gray line is the emf. (May have to click image to see gif)

For the gif below, the capacitor is charged and then discharged. This shows the current over time. The blue line is the current. (May have to click image to see gif)

Examples

Simple

In a circuit where time t = RC, the factor [math]\displaystyle{ {e^{\frac{-t}{RC}}} }[/math] has fallen from value [math]\displaystyle{ {e^0} = {1} }[/math] to the value:

e^(-t/RC) = e^1 = 1/e = 1/2.718 = 0.37

Calculate the time constant for the RC circuit with R = 12 ohms and C = 1 farad.

Solution:

[math]\displaystyle{ {RC}= {12*1} = {12 seconds} }[/math]

Middling

Using the information from the problem above:

Show that the power dissipated in the bulb at t=RC is only 14% of the original power?

Solution:

Using [math]\displaystyle{ {IΔV} = {RI^2} }[/math], reduction in current by factor of 0.37 gives a reduction in power by a factor of (0.37)^2 = 0.1369

Difficult

The best example of difficult RC circuit problems come from past exams. Take a look at this RC circuit.

The first step in these problems is to identify all the parts on the circuit and draw in nodes at necessary locations. You want to choose spots for the nodes where the wire splits so you can apply Kirchoff's law and relate the current at these locations.

Once you correctly label the diagram you can begin the problem.

1. Write down the three energy conservation equations for the circuit. Also write down the charge conservation equation for the circuit you would need to determine the current through the bulbs and capacitor. You do not need to solve these equations.

2.Determine the initial current passing through each bulb.

3. Determine the current passing through each bulb for a long time after the switch has been closed.

Connectedness

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

I enjoy listening to music and RC circuits are used in speakers and headphones. They allow the current to be regulated and for the music to to played at the right pitch and volume.

2. How is it connected to your major?

I'm a Mechanical Engineering major and RC currents are directly related to every job, but in warehouses where heavy machinery and appliances are involved, RC currents are used in parts either at the warehouse or ones that are being assembled. I would be designing the machines not wiring the curcuits, but it is important to understand how the RC circuits are used in order to properly design and implement the machines.

3. Is there an interesting industrial application?

RC circuits are used as filters for almost all types of electronics. The initial input is taken across the RC circuit and the output is taken across the capacitor. This allows the capacitor in the circuit to be charged to different values and used when the desired one is obtained. It is an easy way to allow multiple energy outputs across a single circuit rather than using a bunch of different circuits.

History

The RC circuits have been in use for a long time. Georg Simon Ohm, was someone who spent alot of time researching RC Circuits and Ohm's Law was founded by him. He did his work from 1830s-1840s and he received recognition, after a long time of being mocked by other scientists. Once his work was recognized and accepted by the scientific community people began to realize the many applications of RC circuits as filters and began to further research and implement them in their own work.

@@ Line 1: / Line 1: @@
-Work in progress by Joe Welch
+Claimed by Sean Fischer Fall 2016
+[[File:Rc_circuit.JPG|400px|right|]]
+==The Main Idea==
+An RC circuit is a circuit that contains a battery with a known emf, a resistor (R), and a capacitor (C). An RC circuit can be in either series or parallel. The figure in the top right of the page shows an RC circuit. The capacitor stores electric charge (Q)
+RC Circuits use a DC (direct current) voltage source and the capacitor is uncharged at its initial state. In the figure below, you see an RC circuit with a switch. When the switch is closed, the capacitor will begin to charge as the current can now flow throughout the circuit. To discharge the capacitor, you simply disconnect the switch.
+[[File:Rc_switch.JPG|400px|center|]]
+Remember that potential difference across the capacitor is delta <math>{V} = {Q}/{C}</math>, where Q is charge on the plate and C is the capacitance. When the switch is closed, voltage on the capacitor rises rapidly at first, due to the high current at <math> {time} = {0}</math>. The voltage opposes the battery, increasing from zero to the max emf when the capacitor is fully charged. The current decreases from its initial value of <math>{I}_{0} = {emf}/{R}</math> to zero as the voltage on the capacitor reaches the same value as the emf. The current is initially at its max at time <math>{t} = {0}</math>. Once the potential difference across the plates of the capacitor equals the battery's voltage supply, current will stop flowing through the circuit. This is known as the steady state of an RC circuit; it is reached when time goes to infinity.
+Using derived calculus, the equation for voltage versus time when the capacitor is charged through resistor R is <math>{V} = {emf(1-e^{\frac{-t}{RC}})}</math>. <math>{V}</math> is defined as the voltage across the capacitor. <math>{emf}</math> is equal to the emf of the DC voltage source. The units of <math>{RC}</math> are in seconds. <math>{τ} = {RC}</math>. <math>{τ}</math> is the constant of time in the RC circuit.
+The smaller the resistance, the faster a capacitor will be charged. It takes longer to charge than to discharge. This is because a larger current flows through a smaller resistance (<math>{I} = {V/R}</math>). Also the smaller the capacitor (C), the less time it will need to charge. <math>{τ} = {RC}</math> explains both of these.
-==The Main Idea==
+[[File:images.png|100px|left|]]
+Kirchoff's Node Rule is important to RC Circuits because finding the current flow in the different states of an RC Circuit often relies on nodes when the capacitor is in a parallel circuit.
-The figure below shows a capacitor, ( C ) in series with a resistor, ( R ) forming a RC Charging Circuit connected across a DC battery supply ( Vs ) via a mechanical switch. When the switch is closed, the capacitor will gradually charge up through the resistor until the voltage across it reaches the supply voltage of the battery. The manner in which the capacitor charges up is also shown below.
-[[File:rc1.gif|200px|thumb|left|]]
-Let us assume left, that the capacitor, C is fully “discharged” and the switch (S) is fully open. These are the initial conditions of the circuit, then t = 0, i = 0 and q = 0. When the switch is closed the time begins at t = 0 and current begins to flow into the capacitor via the resistor.
+[[File:images2.jpeg|100px|right|]]
+Kirchhoff’s loop rule explains that the sum of changes in potential around any closed loop must be zero.
-Since the initial voltage across the capacitor is zero, ( Vc = 0 ) the capacitor appears to be a short circuit to the external circuit and the maximum current flows through the circuit restricted only by the resistor R.
-As the capacitor charges up, the potential difference across its plates slowly increases with the actual time taken for the charge on the capacitor to reach its maximum possible voltage. The maximum voltage the capacitor reaches is equal to the voltage supply of the battery. Once the potential difference across the plates of the capacitor equals the battery's voltage supply, current will stop flowing through the circuit. This is known as the steady state of an RC circuit; it is reached when time goes to infinity.
 ===A Mathematical Model===
+These three equations are helpful in solving and understanding RC circuit problems
+[[File:Rc123.jpg]]
+To find the equation for voltage versus time when charging a capacitor through a resistor, you start with rearranging the energy equation and solving for I:
+[[File:Rc234.jpg]]
-ΔV = I * R
-[[File:images.png|100px|thumb|left|]]
+The graph of Voltage vs. Time in a RC Circuit is given by:
-Node Rule-Current is assigned (positive or negative) quantity reflecting direction towards or away from a node.
+[[File:Outputrc.jpg]]
-This principle can be solved by adding all the currents up to equal 0. The Node Rule is important to RC Circuits
-because finding the current flow in the different states of an RC Circuit often relies on nodes when the
-capacitor is in parallel.
+===A Computational Model===
+Using a simulation where R and C were equal, the capacitors were charged and then discharged and the two images show the voltage and current of this versus time.
+For the gif below, the capacitor is charged and then discharged. This shows the voltage over time. The red line is the voltage and the gray line is the emf. (May have to click image to see gif)
+[[File:rcvoltage.gif|300px|center|]]
+For the gif below, the capacitor is charged and then discharged. This shows the current over time. The blue line is the current. (May have to click image to see gif)
+[[File:rccurrent.gif|300px|center|]]
-[[File:images2.jpeg|100px|thumb|right|]]
-Loop Rule-Similar to the node rule; however, the voltage jumps and drops are added up around the circuit to equal to 0./
 ==Examples==
-Be sure to show all steps in your solution and include diagrams whenever possible
+===Simple===
+In a circuit where time t = RC, the factor <math>{e^{\frac{-t}{RC}}}</math> has fallen from value <math>{e^0} = {1}</math> to the value:
+e^(-t/RC) = e^1 = 1/e = 1/2.718 = 0.37
+Calculate the time constant for the RC circuit with R = 12 ohms and C = 1 farad.
+Solution:
+<math> {RC}= {12*1} = {12 seconds}</math>
-===Simple===
-In the steady state what is the potential difference across the plates of the capacitor?
 ===Middling===
+Using the information from the problem above:
+Show that the power dissipated in the bulb at t=RC is only 14% of  the original power?
+Solution:
+Using <math>{IΔV} = {RI^2}</math>, reduction in current by factor of 0.37 gives a reduction in power by a factor of (0.37)^2 = 0.1369
 ===Difficult===
+The best example of difficult RC circuit problems come from past exams. Take a look at this RC circuit.
+[[File:Primary1.jpg]]
+The first step in these problems is to identify all the parts on the circuit and draw in nodes at necessary locations. You want to choose spots for the nodes where the wire splits so you can apply Kirchoff's law and relate the current at these locations.
+[[File:Primary2.jpg]]
+Once you correctly label the diagram you can begin the problem.
+. Write down the three energy conservation equations for the circuit. Also write down the charge conservation equation for the circuit you would need to determine the current through the bulbs and capacitor. You do not need to solve these equations.
+.Determine the initial current passing through each bulb.
+. Determine the current passing through each bulb for a long time after the switch has been closed.
+[[File:Workwork.jpg]]
 ==Connectedness==
-#How is this topic connected to something that you are interested in?
+. How is this topic connected to something that you are interested in?
-#How is it connected to your major?
+:: I enjoy listening to music and RC circuits are used in speakers and headphones. They allow the current to be regulated and for the music to to played at the right pitch and volume.
-#Is there an interesting industrial application?
+[[File:Speakercirc.jpg]]
+. How is it connected to your major?
+:: I'm a Mechanical Engineering major and RC currents are directly related to every job, but in warehouses where heavy machinery and appliances are involved, RC currents are used in parts either at the warehouse or ones that are being assembled. I would be designing the machines not wiring the curcuits, but it is important to understand how the RC circuits are used in order to properly design and implement the machines.
+. Is there an interesting industrial application?
+:: RC circuits are used as filters for almost all types of electronics. The initial input is taken across the RC circuit and the output is taken across the capacitor. This allows the capacitor in the circuit to be charged to different values and used when the desired one is obtained. It is an easy way to allow multiple energy outputs across a single circuit rather than using a bunch of different circuits.
+[[File:Fil42.gif]]
 ==History==
-Put this idea in historical context. Give the reader the Who, What, When, Where, and Why.
+The RC circuits have been in use for a long time. Georg Simon Ohm, was someone who spent alot of time researching RC Circuits and Ohm's Law was founded by him. He did his work from 1830s-1840s and he received recognition, after a long time of being mocked by other scientists. Once his work was recognized and accepted by the scientific community people began to realize the many applications of RC circuits as filters and began to further research and implement them in their own work.
 == See also ==
-Are there related topics or categories in this wiki resource for the curious reader to explore?  How does this topic fit into that context?
+===Further reading===
+[[Charge in a RC Circuit]]
+[[Current in a RC circuit]]
+[[Loop Rule]]
-===Further reading===
+[[Node Rule]]
-Books, Articles or other print media on this topic
+[[Current]]
 ===External links===
-Internet resources on this topic
+http://hyperphysics.phy-astr.gsu.edu/hbase/electric/rcimp.html
+http://www.allaboutcircuits.com/textbook/alternating-current/chpt-4/series-resistor-capacitor-circuits/
+http://buphy.bu.edu/~duffy/semester2/c11_RC.html
+http://ocw.mit.edu/courses/physics/8-022-physics-ii-electricity-and-magnetism-fall-2006/lecture-notes/lecture32.pdf
 ==References==
-This section contains the the references you used while writing this page
+http://www.compadre.org/portal/items/detail.cfm?ID=9986
+https://www.pa.msu.edu/courses/1997spring/PHY232/lectures/kirchoff/examples.html
+https://openstaxcollege.org/textbooks/college-physics
+http://www.animations.physics.unsw.edu.au/jw/RCfilters.html
-[[Category:Which Category did you place this in?]]
+[[Category:Simple Circuits]]

RC Circuit: Difference between revisions

Latest revision as of 23:15, 27 November 2016

Contents

The Main Idea

A Mathematical Model

A Computational Model

Examples

Simple

Middling

Difficult

Connectedness

History

See also

Further reading

External links

References

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