23 February, 2007

Electrcal Impedance, Inductive Reactance,Capacitive Reactance

Electrical Impedance :
Electrical impedance, or simply impedance, is a measure of opposition to a sinusoidal alternating electric current. The concept of electrical impedance generalizes Ohm's law to AC circuit analysis. Unlike electrical resistance, the impedance of an electric circuit can be a complex number,but the same unit, the ohm, is used for both quantities. Oliver Heaviside coined the term "impedance" in July of 1886.

Definition of Electrical impedance :
The impedance of a circuit element is defined as the ratio of the phasor voltage across the element to the phasor current through the element:

It should be noted that although Z is the ratio of two phasors, Z is not itself a phasor. That is, Z is not associated with some sinusoidal function of time.
For DC circuits, the resistance is defined by Ohm's law to be the ratio of the DC voltage across the resistor to the DC current through the resistor:

VR and IR above are DC (constant real) values.

Just as Ohm's law is generalized to AC circuits through the use of phasors, other results from DC circuit analysis such as voltag division, current division, Thevenin's theorem, and Norton's theorem generalize to AC circuits.

The electric impedance is equal to:

Impedance of different devices :

Magnitude and phase of impedance :
Complex numbers are commonly expressed in two distinct forms. The rectangular form is simply the sum of the real part with the product of j and the imaginary part:
Z = R + j X
The polar form of a complex number the real magnitude of the number multiplied by the complex phase. This can be written with exponentials, or in phasor notation :

is the magnitude of Z ( Z* denotes the complex conjugate of Z) and

is the angle.

Impedances in Series and Parallel :
Combining impedances in series, parallel, or in delta-wye configurations, is the same as for resistors. The difference is that combining impedances involves manipulation of complex numbers.
In series :
Combining impedances in series is simple:

In parallel :
Combining impedances in parallel is much more difficult than combining simple properties like resistance or capacitance, due to a multiplication term.

In rationalized form the equivalent resistance is:

Reactance :The term reactance refers to the imaginary part of the impedance.
It is important to note that the impedance of a capacitor or an inductor is a function of the frequency ω and is an imaginary quantity - however is certainly a real physical phenomenon relating the shift in phases between the voltage and current phasors due to the existence of the capacitor or inductor. Earlier it was shown that the impedance of a resistor is constant and real, in other words a resistor does not cause a phase shift between voltage and current as do capacitors and inductors.

When resistors, capacitors, and inductors are combined in an AC circuit, the impedances of the individual components can be combined in the same way that the resistances are combined in a DC circuit. The resulting equivalent impedance is in general, a complex quantity. That is, the equivalent impedance has a real part and an imaginary part. The real part is denoted with an R and the imaginary part is denoted with an X. Thus
Zeq = Req + j Xeq
Req is termed the resistive part of the impedance
Xeq is termed the reactive part of the impedance.

The reactance of an inductor and a capacitor in series is the algebraic sum of their reactances:
where XL and XC are the inductive and capacitive reactances, which are positive and negative, respectively.

Inductive reactance (symbol XL) is caused by the fact that a current is accompanied by a magnetic field; therefore a varying current is accompanied by a varying magnetic field; the latter gives an electromotive force that resists the changes in current. The more the current changes, the more an inductor resists it: the reactance is proportional to the frequency (hence zero for DC). There is also a phase difference between the current and the applied voltage.
Inductive reactance has the formula

XL is the inductive reactance, measured in ohms
ω is the angular frequency, measured in radians per second
f is the frequency, measured in hertz
L is the inductace, measured in henries.

Capacitive reactance (symbol XC) reflects the fact that electrons cannot pass through a capacitor, yet effectively alternating current (AC) can: the higher the frequency the better. There is also a phase difference between the alternating current flowing through a capacitor and the potential difference across the capacitor's electrodes.
Capacitive reactance has the formula

XC is the capacitive reactance measured in ohms
ω is the angular frequency, measured in radians per second
f is the frequency measured in hertz
C is the capacitance, measured in farads

It is therefore common to refer to a capacitor or an inductor as a reactance or equivalently, a reactive component (circuit element). Additionally, the impedance for a capacitance is negative imaginary while the impedance for an inductor is positive imaginary. Thus, a capacitive reactance refers to a negative reactance while an inductive reactance refers to a positive reactance.

A reactive component is distinguished by the fact that the sinusoidal voltage across the component is in quadrature with the sinusoidal current through the component. This implies that the component alternately absorbs energy from the circuit and then returns energy to the circuit. That is, unlike a resistance, a reactance does not dissipate power.

As the frequency approaches zero, the capacitive reactance grows without bound so that a capacitor approaches an open circuit for very low frequency sinusoidal sources. As the frequency increases, the capacitive reactance approaches zero so that a capacitor approaches a short circuit for very high frequency sinusoidal sources.

Conversely, the inductive reactance approaches zero as the frequency approaches zero so that an inductor approaches a short circuit for very low frequency sinusoidal sources. As the frequency increases, the inductive reactance increases so that an inductor approaches an open circuit for very high frequency sinusoidal sources.

For more detalis and derivations visit the following link:
http://www.aplac.hut.fi/courses/bee/exercises.pdf ---- exercise problems on inductance

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