User:DJIndica/Euler-Cromer algorithm

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Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating current. The concept of electrical impedance generalizes Ohm's law to AC circuit analysis, describing not only the relative magnitudes of the voltage and current, but also the relative phases. In general impedance is a complex quantity $\tilde{Z}$ , of which the real part is the resistance $R$ and the imaginary part is the reactance $χ$ . The term "impedance" was coined by Oliver Heaviside July of 1886.

A graphical representation of the complex impedance plane. Note that while reactance

G

can be either positive or negative, resistance

R

is always positive. The concept of negative resistance is sometimes used to describe the operation of amplifiers as they supply power to a circuit, particularly in the design of oscillators.Actual size

$\tilde{Z} = R + jG \quad$

It is far more instructive to use the polar form.

$\tilde{Z} = |Z|e^{j\theta} \quad$

1 Ohm's law
2 Complex voltage and current
3 Examples
4 See also
5 References

[edit] Ohm's law

We can understand this by substituting it into Ohm's law

$\tilde{V} = \tilde{I}\tilde{Z} = \tilde{I}|Z|e^{j\theta} \quad$

An AC supply connected across a device with impedance

Z

, with an AC voltmeter and ammeter in parallel, and series respectively. An AC supply applying a voltage

V

across a load

Z

, driving a current

I

Actual size

The magnitude of the impedance $|\tilde{Z}|$ acts just like resistance, giving the drop in voltage amplitude across an impedance $\tilde{Z}$ for a given current $I$ . The phase factor tells us that the voltage lags the current by a phase of $θ$ .

[edit] Complex voltage and current

In order to simplify calculations, the voltage and current are commonly represented as complex quantities denoted as $\tilde{V}$ and $\tilde{I}$ ; we must bear in mind that these are real quantities and take only the real part at the end of a calculation.

$\tilde{V} = V_0e^{j(\omega t + \phi_V)}$

$\tilde{I} = I_0e^{j(\omega t + \phi_I)}$

Impedance is defined as the ratio of these quantities.

$Z = {\tilde{V} \over \tilde{I}}$

Substituting these into Ohm's law we have

$\Rightarrow V_0e^{j(\omega t + \phi_V)} = I_0e^{j(\omega t + \phi_I)}|Z|e^{j\theta}$

Taking the real parts of both sides we have

$\Rightarrow V_0\cos(\omega t + \phi_V) = I_0|Z|\cos(\omega t + \phi_I + \theta)$

Equating the magnitudes and phases we have

$V_0 = I_0|Z| \quad$

$\phi_V = \phi_I + \theta \quad$

The magnitude equation is the familiar Ohm's law applied to the voltage and current amplitudes, while the second equation defines the phase relationship.

[edit] Examples

It is instructive to examine two specific examples; capacitors and inductors (by far the most important examples).

$Z_L = j\omega L \quad$

$Z_C = {1 \over j\omega C}$

We observe that

$j = \cos{\left({\pi \over 2}\right)} + j\sin{\left({\pi \over 2}\right)} = e^{j{\pi \over 2}}$

${1 \over j} = -j = \cos{\left(-{\pi \over 2}\right)} + j\sin{\left(-{\pi \over 2}\right)} = e^{j(-{\pi \over 2})}$

Thus we can rewrite the inductor and capacitor impedance equations in polar form

$Z_L = \omega Le^{j{\pi \over 2}}$

$Z_C = {1 \over \omega C}e^{j(-{\pi \over 2})}$

The magnitude tells us the change in voltage amplitude for a given current amplitude through our impedance, while the exponential factors give the phase relationship.

The phase angles in the equations for the impedance of inductors and capacitors indicate that the voltage across a capacitor leads the current through it by a phase of

π / 2

, while the voltage across an inductor lags the current through it by

π / 2

. The identical voltage and current amplitudes tell us that the magnitude of the impedance is equal to one.Actual size

[edit] See also

[edit] References

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User:DJIndica/Euler-Cromer algorithm

From Wikipedia, the free encyclopedia

Contents

[edit] Ohm's law

[edit] Complex voltage and current

[edit] Examples

[edit] See also

[edit] References

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