Difference between revisions of "Resistors (Ohm's Law), Capacitors, and Inductors"
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<math>R_{total}=\frac{R_1 R_2}{R_1+R_2}</math> |
<math>R_{total}=\frac{R_1 R_2}{R_1+R_2}</math> |
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===Proof for Resistors in Parallel equation=== |
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[[Image:Parallel_resistors.jpg]] |
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We can prove the equation for parallel resistors by using Kirchhoff's voltage and current laws: |
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<math>V_s=V_{R1}=V_{R2}\,</math> (KVL) |
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<math>I_s=I_1+I_2\,</math> (KCL) |
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Plugging in the constitutive law for resistors in the second equation yields: |
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<math>I_s=\frac{V_s}{R_1}+\frac{V_s}{R_2}=V_s\left(\frac{1}{R_1}+\frac{1}{R_2}\right)=V_s\left(\frac{R_1+R_2}{R_1R_2}\right) |
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</math> |
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<math> |
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R_{eq}=\frac{V_s}{I_s}=\frac{R_1R_2}{R_1+R_2} |
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</math> |
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==Capacitors== |
==Capacitors== |
Revision as of 11:24, 14 June 2006
Resistors
The symbol for a resistor:
The relationship between the current through a conductor with resistance and the voltage across the same conductor is described by Ohm's law:
where V is the voltage across the conductor, I is the current through the conductor, and R is the resistance of the conductor.
The power dissipated by the resistor is equal to the voltage multiplied by the current:
If is measured in amps and in volts, then the power is in watts.
By plugging in different forms of , we can rewrite as:
or
Resistors in series or parallel
To find the combined resistance of resistors connected in series, simply add the resistances:
If the resistors are connected in parallel, the equation is:
For two resistors in parallel, this simplifies to:
Proof for Resistors in Parallel equation
We can prove the equation for parallel resistors by using Kirchhoff's voltage and current laws:
(KVL)
(KCL)
Plugging in the constitutive law for resistors in the second equation yields:
Capacitors
The symbol for a capacitor:
A capacitor is a device that stores electric charges. (Do not confuse this with a battery; a battery creates an electric potential difference, but doesn't store charge. A battery is analogous to a water pump, while a capacitor is analogous to a water tank.) At steady state, a capacitor has a constant voltage across it and zero current through it. In this state, the capacitor acts like an open circuit.
The unit of measurement for the capacitance of a capacitor is the farad, which is equal to 1 coulomb per volt.
The charge(q), voltage (v), and capacitance(C) of a capacitor are related as follows:
where q(t) and v(t) are the values for charge and voltage, expressed as a function of time.
Differentiating both sides with respect to time gives:
Rearranging and then integrating with respect to time give:
If we assume that the charge, voltage, and current of the capacitor are zero at , our equation reduces to:
or
,
which is equivalent to the first equation.
Assuming the same zero initial states as before, the energy stored in a capacitor (in joules) is given by the equation:
Inductors
The symbol for an inductor:
An inductor stores energy in the form of a magnetic field, usually by means of a coil of wire. An inductor resists change in the current flowing through it; an inductor at steady state has a constant current flowing through it and no voltage across it. In this state, the inductor acts like a short circuit.
The relationship between the voltage across the inductor is linearly related by a factor L, the inductance, to the time rate of change of the current through the inductor. The unit for inductance is the henry, and is equal to a volt-second per ampere.
The relationship between the voltage and the current is as follows:
If we multiply both sides by dt, we get:
Integrating both sides from to gives:
which is equal to:
assuming that the voltage, current and energy of the inductor are all zero at reduces the equation to
The energy stored in the inductor is given by:
References
Hayt, William H. Jr., Jack E. Kemmerly, and Steven M. Durbin. Engineering Circuit Analysis. 6th ed. New York:McGraw-Hill, 2002.