Ohm's law states that the
current through a conductor between two points is directly
proportional to the potential difference across the two points. Introducing the constant of proportionality, the
resistance,
[1] one arrives at the usual mathematical equation that describes this relationship:
[2]
where
I is the current through the conductor in units of
amperes,
V is the potential difference measured
across the conductor in units of
volts, and
R is the
resistance of the conductor in units of
ohms. More specifically, Ohm's law states that the
R in this relation is constant, independent of the current.
[3]
The law was named after the German physicist
Georg Ohm,
who, in a treatise published in 1827, described measurements of applied
voltage and current through simple electrical circuits containing
various lengths of wire. He presented a slightly more complex equation
than the one above (see History section below) to explain his experimental results. The above equation is the modern form of Ohm's law.
In physics, the term
Ohm's law is also used to refer to various generalizations of the law originally formulated by Ohm. The simplest example of this is:
where
J is the
current density at a given location in a resistive material,
E is the electric field at that location, and
σ is a material dependent parameter called the
conductivity. This reformulation of Ohm's law is due to
Gustav Kirchhoff
In
circuit analysis, three equivalent expressions of Ohm's law are used interchangeably:
Each equation is quoted by some sources as the defining relationship of Ohm's law,
or all three are quoted,
or derived from a proportional form,
or even just the two that do not correspond to Ohm's original statement may sometimes be given.
The interchangeability of the equation may be represented by a triangle, where V (
voltage) is placed on the top section, the I (
current) is placed to the left section, and the R (
resistance)
is placed to the right. The line that divides the left and right
sections indicate multiplication, and the divider between the top and
bottom sections indicates division (hence the division bar).
Resistive circuits
Resistors
are circuit elements that impede the passage of electric charge in
agreement with Ohm's law, and are designed to have a specific resistance
value
R. In a schematic diagram the resistor is shown as a
zig-zag symbol. An element (resistor or conductor) that behaves
according to Ohm's law over some operating range is referred to as an
ohmic device (or an
ohmic resistor) because Ohm's law and a single value for the resistance suffice to describe the behavior of the device over that range.
Ohm's law holds for circuits containing only resistive elements (no
capacitances or inductances) for all forms of driving voltage or
current, regardless of whether the driving voltage or current is
constant (
DC) or time-varying such as
AC. At any instant of time Ohm's law is valid for such circuits.
Resistors which are in
series or in
parallel
may be grouped together into a single "equivalent resistance" in order
to apply Ohm's law in analyzing the circuit. This application of Ohm's
law is illustrated with examples in "
How To Analyze Resistive Circuits Using Ohm's Law" on
wikiHow.
Reactive circuits with time-varying signals
When reactive elements such as capacitors, inductors, or transmission
lines are involved in a circuit to which AC or time-varying voltage or
current is applied, the relationship between voltage and current becomes
the solution to a
differential equation,
so Ohm's law (as defined above) does not directly apply since that form
contains only resistances having value R, not complex impedances which
may contain capacitance ("C") or inductance ("L").
Equations for
time-invariant AC circuits take the same form as Ohm's law, however, the variables are generalized to
complex numbers and the current and voltage waveforms are
complex exponentials.
[26]
In this approach, a voltage or current waveform takes the form
, where
t is time,
s is a complex parameter, and
A is a complex scalar. In any
linear time-invariant system, all of the currents and voltages can be expressed with the same
s
parameter as the input to the system, allowing the time-varying complex
exponential term to be canceled out and the system described
algebraically in terms of the complex scalars in the current and voltage
waveforms.
The complex generalization of resistance is
impedance, usually denoted
Z; it can be shown that for an inductor,
and for a capacitor,
We can now write,
where
V and
I are the complex scalars in the voltage and current respectively and
Z is the complex impedance.
This form of Ohm's law, with
Z taking the place of
R, generalizes the simpler form. When
Z is complex, only the real part is responsible for dissipating heat.
In the general AC circuit,
Z varies strongly with the frequency parameter
s, and so also will the relationship between voltage and current.
For the common case of a steady
sinusoid, the
s parameter is taken to be
, corresponding to a complex sinusoid
.
The real parts of such complex current and voltage waveforms describe
the actual sinusoidal currents and voltages in a circuit, which can be
in different phases due to the different complex scalars.
Linear approximations
Ohm's law is one of the basic equations used in the
analysis of electrical circuits. It applies to both metal conductors and circuit components (
resistors)
specifically made for this behaviour. Both are ubiquitous in electrical
engineering. Materials and components that obey Ohm's law are described
as "ohmic"
which means they produce the same value for resistance (R = V/I)
regardless of the value of V or I which is applied and whether the
applied voltage or current is DC (
direct current) of either positive or negative polarity or AC (
alternating current).
In a true ohmic device, the same value of resistance will be
calculated from R = V/I regardless of the value of the applied voltage
V. That is, the ratio of V/I is constant, and when current is plotted as
a function of voltage the curve is
linear (a straight line). If
voltage is forced to some value V, then that voltage V divided by
measured current I will equal R. Or if the current is forced to some
value I, then the measured voltage V divided by that current I is also
R. Since the plot of I versus V is a straight line, then it is also true
that for any set of two different voltages V
1 and V
2 applied across a given device of resistance R, producing currents I
1 = V
1/R and I
2 = V
2/R, that the ratio (V
1-V
2)/(I
1-I
2)
is also a constant equal to R. The operator "delta" (Δ) is used to
represent a difference in a quantity, so we can write ΔV = V
1-V
2 and ΔI = I
1-I
2.
Summarizing, for any truly ohmic device having resistance R, V/I =
ΔV/ΔI = R for any applied voltage or current or for the difference
between any set of applied voltages or currents.
The
I–V curves of four devices: Two
resistors, a diode, and a battery. The two resistors follow Ohm's law: The plot is a straight line through the origin. The other two devices do
not follow Ohm's law.
There are, however, components of electrical circuits which do not obey Ohm's law; that is, their relationship between current and voltage (their I–V curve) is nonlinear (or non-ohmic). An example is the p-n junction diode (curve at right). As seen in the figure, the current does not increase linearly with applied voltage for a diode. One can determine a value of current (I) for a given value of applied voltage (V) from the curve, but not from Ohm's law, since the value of "resistance" is not constant as a function of applied voltage. Further, the current only increases significantly if the applied voltage is positive, not negative. The ratio V/I for some point along the nonlinear curve is sometimes called the static, or chordal, or DC, resistance,but as seen in the figure the value of total V over total I varies depending on the particular point along the nonlinear curve which is chosen. This means the "DC resistance" V/I at some point on the curve is not the same as what would be determined by applying an AC signal having peak amplitude ΔV volts or ΔI amps centered at that same point along the curve and measuring ΔV/ΔI. However, in some diode applications, the AC signal applied to the device is small and it is possible to analyze the circuit in terms of the dynamic, small-signal, or incremental resistance, defined as the one over the slope of the V–I curve at the average value (DC operating point) of the voltage (that is, one over the derivative of current with respect to voltage). For sufficiently small signals, the dynamic resistance allows the Ohm's law small signal resistance to be calculated as approximately one over the slope of a line drawn tangentially to the V-I curve at the DC operating point.
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