RC Circuit - Revision history

Tommyg: Add Thevenin method and worked switched RC example

2026-10-26T13:57:04Z

Add Thevenin method and worked switched RC example

← Older revision		Revision as of 08:57, 26 October 2026
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	[[File:rccurrent.gif\|300px\|center\|]]		[[File:rccurrent.gif\|300px\|center\|]]

			== Analyzing a Switched Circuit with One Capacitor ==

			The basic series RC circuit is a useful starting point, but many circuit problems have several resistors and a switch. A reliable way to analyze any linear circuit with one capacitor is to replace the rest of the circuit, as seen from the capacitor’s terminals, with its Thevenin equivalent. The resulting circuit has one voltage source, one resistance, and the capacitor.

			This method works when the circuit contains one capacitor and resistors with constant values. It also works if several capacitors can first be combined into one equivalent capacitor. If there are multiple independent capacitors, the circuit may need more than one differential equation.

			=== The General Voltage Equation ===

			Suppose a switch changes position at time <math>t_0</math>. Define:

			* <math>V_i</math> as the voltage across the capacitor immediately before the switch changes.
			* <math>V_f</math> as the final, steady-state capacitor voltage after the switch changes.
			* <math>R_{\text{th}}</math> as the Thevenin resistance seen from the capacitor’s terminals after the switch changes.
			* <math>C</math> as the capacitance.

			For times after the switch changes, the capacitor voltage is

			[math]\displaystyle{ V_C(t)=V_f+(V_i-V_f)e^{-(t-t_0)/(R_{\text{th}}C)} }[/math]

			The time constant is

			[math]\displaystyle{ \tau=R_{\text{th}}C }[/math]

			This equation describes both charging and discharging. It also handles a capacitor that starts with a nonzero voltage and moves toward a different final voltage.

			The capacitor’s voltage cannot change instantly. Therefore,

			[math]\displaystyle{ V_C(t_0^+)=V_C(t_0^-)=V_i }[/math]

			The current can change instantly when the switch changes. If positive current is defined as entering the capacitor terminal labeled positive, then

			[math]\displaystyle{ I_C(t)=C\frac{dV_C}{dt}=\frac{V_f-V_i}{R_{\text{th}}}e^{-(t-t_0)/\tau} }[/math]

			A negative value means the actual current flows opposite to the chosen positive direction.

			=== How to Find the Thevenin Equivalent ===

			1. Draw the circuit as it is connected after the switch changes. The initial voltage <math>V_i</math> comes from the circuit before the switch changes; <math>R_{\text{th}}</math> and <math>V_f</math> come from the circuit afterward.
			2. Remove the capacitor and find the open-circuit voltage across its terminals. This is <math>V_f</math>, the voltage the capacitor approaches after a long time.
			3. Find <math>R_{\text{th}}</math> looking into the capacitor’s terminals. Turn off independent ideal voltage sources by replacing them with wires, and turn off independent ideal current sources by opening them. Leave dependent sources active.
			4. Calculate <math>\tau=R_{\text{th}}C</math>.
			5. Substitute <math>V_i</math>, <math>V_f</math>, and <math>\tau</math> into the general voltage equation.
			6. Use <math>Q=CV_C</math> or <math>I_C=C\,dV_C/dt</math> if the problem asks for charge or current.

			At one time constant, the voltage has moved about 63.2% of the way from its initial value toward its final value. At three time constants, it has moved about 95.0% of the way. At five time constants, it is more than 99% of the way there.

			=== Worked Example: A Capacitor Connected to a Voltage Divider ===

			A <math>12\ \text{V}</math> ideal battery is connected through a switch and a <math>2.0\ \text{k}\Omega</math> resistor, <math>R_1</math>, to a node. A <math>4.0\ \text{k}\Omega</math> resistor, <math>R_2</math>, and a <math>1000\ \mu\text{F}</math> capacitor are both connected from that node to ground. Before the switch closes, the capacitor is disconnected from the battery and has voltage <math>0\ \text{V}</math>. At <math>t=0</math>, the switch closes.

			Find the capacitor’s final voltage, the time constant, the voltage after one time constant, and its initial charging current.

			;Step 1: Find the initial voltage.
			The capacitor begins uncharged, so <math>V_i=0\ \text{V}</math>.

			;Step 2: Find the final voltage.
			After a long time, the capacitor current is zero. The resistors form a voltage divider, so

			[math]\displaystyle{ V_f=12\ \text{V}\left(\frac{4.0\ \text{k}\Omega}{2.0\ \text{k}\Omega+4.0\ \text{k}\Omega}\right)=8.0\ \text{V} }[/math]

			;Step 3: Find the Thevenin resistance.
			Turn off the ideal battery by replacing it with a wire. From the capacitor’s terminals, <math>R_1</math> and <math>R_2</math> are in parallel:

			[math]\displaystyle{ R_{\text{th}}=R_1\parallel R_2=\frac{R_1R_2}{R_1+R_2}=\frac{(2.0)(4.0)}{2.0+4.0}\ \text{k}\Omega=1.33\ \text{k}\Omega }[/math]

			;Step 4: Find the time constant.
			Since <math>1000\ \mu\text{F}=1.00\times10^{-3}\ \text{F}</math>,

			[math]\displaystyle{ \tau=R_{\text{th}}C=(1.33\times10^3\ \Omega)(1.00\times10^{-3}\ \text{F})=1.33\ \text{s} }[/math]

			;Step 5: Write the voltage as a function of time.
			For <math>t\geq0</math>,

			[math]\displaystyle{ V_C(t)=8.0\left(1-e^{-t/(1.33\ \text{s})}\right)\ \text{V} }[/math]

			At one time constant,

			[math]\displaystyle{ V_C(\tau)=8.0(1-e^{-1})\ \text{V}\approx5.06\ \text{V} }[/math]

			;Step 6: Find the initial current into the capacitor.
			Using the current direction into the capacitor’s positive terminal,

			[math]\displaystyle{ I_C(0^+)=\frac{V_f-V_i}{R_{\text{th}}}=\frac{8.0\ \text{V}}{1.33\times10^3\ \Omega}\approx6.0\ \text{mA} }[/math]

			The current decreases as the capacitor charges. After a long time, the capacitor current approaches zero, even though current still flows through the two resistors in the voltage divider.

			=== What to Check in a Circuit Sketch ===

			* Mark the capacitor’s positive terminal and use the same voltage direction throughout the calculation.
			* Use the circuit before switching to determine <math>V_i</math>. Capacitor voltage is continuous at the instant the switch changes.
			* Use the circuit after switching to find <math>V_f</math> and <math>R_{\text{th}}</math>.
			* At steady state in a DC circuit, an ideal capacitor carries no current. It acts like an open circuit after it has finished charging.
			* The resistance in <math>\tau=R_{\text{th}}C</math> is the resistance seen from the capacitor’s terminals. It may combine several resistors and is not always the resistor drawn next to the capacitor.
			* Use seconds for time, ohms for resistance, and farads for capacitance. Since <math>\Omega\cdot\text{F}=\text{s}</math>, the time constant has units of seconds.

			=== Practice Check ===

			A capacitor initially has voltage <math>6.0\ \text{V}</math>. After a switch changes position, the Thevenin equivalent seen by the capacitor is <math>10\ \text{V}</math> in series with <math>3.0\ \text{k}\Omega</math>. The capacitance is <math>200\ \mu\text{F}</math>.

			# Find the time constant.
			# Find the capacitor voltage after one time constant.
			# Find the initial current into the positive terminal.

			;Answers:

			[math]\displaystyle{ \tau=(3.0\times10^3\ \Omega)(200\times10^{-6}\ \text{F})=0.60\ \text{s} }[/math]

			[math]\displaystyle{ V_C(\tau)=10+(6.0-10)e^{-1}\approx8.53\ \text{V} }[/math]

			[math]\displaystyle{ I_C(0^+)=\frac{10-6.0}{3.0\times10^3}\approx1.33\ \text{mA} }[/math]

			=== Further Reading ===

			OpenStax, ''University Physics Volume 2'', Section 10.5, “RC Circuits”: https://openstax.org/books/university-physics-volume-2/pages/10-5-rc-circuits

	==Examples==		==Examples==

Adam at 09:48, 2 December 2025

2025-12-02T09:48:05Z

← Older revision		Revision as of 04:48, 2 December 2025
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	Claimed by Adam Teske Fall 2025		Claimed by Adam Teske Fall 2025

	An RC circuit (resistor-capacitor circuit) is a type of [[Non Steady State\|non steady state]] circuit		An RC circuit (resistor-capacitor circuit) is a type of [[Non Steady State\|non steady state]] circuit.
	[[File:Rc_circuit.JPG\|400px\|right\|]]		[[File:Rc_circuit.JPG\|400px\|right\|]]
	==The Main Idea==		==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)~~		The number of independent energy storage elements in the RC circuit determine the circuit's order, for this course first order RC circuits are the most prevalent. First order RC circuits can have series and parallel elements. The figure in the top right of the page depicts an RC circuit.

	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~~.		RC Circuits contain charging and discharging phases. For charging phases, a voltage source is connected to the capacitor, with capacitance C and max charge of Q. When the switch is disengaged, the capacitor discharges its stored energy to the resistors with resistance R.

			The time it takes for the capacitor to charge and discharge can be represented by the circuit's time constant, τ.

	[[File:Rc_switch.JPG\|400px\|center\|]]		[[File:Rc_switch.JPG\|400px\|center\|]]

Adam at 04:54, 1 December 2025

2025-12-01T04:54:30Z

← Older revision		Revision as of 23:54, 30 November 2025
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	Claimed by Adam Teske Fall 2025		Claimed by Adam Teske Fall 2025

	An RC circuit (resistor-capacitor circuit) is a type of ~~Non Steady state circuit~~ [[Non Steady State\|non steady state]]		An RC circuit (resistor-capacitor circuit) is a type of [[Non Steady State\|non steady state]] circuit
	~~[[Rest Mass Energy\|rest energy]]~~
	[[File:Rc_circuit.JPG\|400px\|right\|]]		[[File:Rc_circuit.JPG\|400px\|right\|]]
	==The Main Idea==		==The Main Idea==

Adam at 04:48, 1 December 2025

2025-12-01T04:48:39Z

← Older revision		Revision as of 23:48, 30 November 2025
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	Claimed by Adam Teske Fall 2025		Claimed by Adam Teske Fall 2025

			An RC circuit (resistor-capacitor circuit) is a type of Non Steady state circuit [[Non Steady State\|non steady state]]
	[[Rest Mass Energy\|rest energy]]		[[Rest Mass Energy\|rest energy]]
	[[File:Rc_circuit.JPG\|400px\|right\|]]		[[File:Rc_circuit.JPG\|400px\|right\|]]

Adam at 04:27, 1 December 2025

2025-12-01T04:27:33Z

← Older revision		Revision as of 23:27, 30 November 2025
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	Claimed by Adam Teske Fall 2025		Claimed by Adam Teske Fall 2025

			[[Rest Mass Energy\|rest energy]]
	[[File:Rc_circuit.JPG\|400px\|right\|]]		[[File:Rc_circuit.JPG\|400px\|right\|]]
	==The Main Idea==		==The Main Idea==

Adam at 03:53, 1 December 2025

2025-12-01T03:53:41Z

← Older revision		Revision as of 22:53, 30 November 2025
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	Claimed by ~~Sean Fischer~~ Fall ~~2016~~		Claimed by Adam Teske Fall 2025
	[[File:Rc_circuit.JPG\|400px\|right\|]]		[[File:Rc_circuit.JPG\|400px\|right\|]]
	==The Main Idea==		==The Main Idea==

Sfischer8: /* References */

2016-11-28T04:15:54Z

References

← Older revision		Revision as of 23:15, 27 November 2016
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	https://openstaxcollege.org/textbooks/college-physics		https://openstaxcollege.org/textbooks/college-physics

			http://www.animations.physics.unsw.edu.au/jw/RCfilters.html

	[[Category:Simple Circuits]]		[[Category:Simple Circuits]]

Sfischer8: /* History */

2016-11-28T04:15:01Z

History

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	==History==		==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.		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 ==		== See also ==

Sfischer8: /* Connectedness */

2016-11-28T04:13:24Z

Connectedness

← Older revision		Revision as of 23:13, 27 November 2016
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	2. How is it connected to your major?		2. How is it connected to your major?
	:: I'm ~~an ISyE~~ 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'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?		3. Is there an interesting industrial application?
	:: RC circuits are used to ~~hear certain sounds so that in~~ an ~~assembly line something can be sorted or passed through~~ to ~~another area~~.		:: 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==		==History==

Sfischer8: /* Connectedness */

2016-11-28T04:03:48Z

Connectedness

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	1. 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?
	:: 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.		:: 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.
			[[File:Speakercirc.jpg]]

	2. How is it connected to your major?		2. How is it connected to your major?