WEEK OF 8/3/2014

 

An ideal resistor has zero reactance, while ideal inductors and capacitors consist entirely of reactance. The magnitude of the reactance of an inductor is proportional to frequency,

while the magnitude of the reactance of a capacitor is inversely proportional to frequency.

 

Analysis[edit]

In phasor analysis, reactance is used to compute amplitude and phase changes of sinusoidal alternating current going through the circuit element. It is denoted by the symbol \scriptstyle{X}.

Both reactance \scriptstyle{X} and resistance \scriptstyle{R} are components of impedance \scriptstyle{Z}.

Z = R + jX\,
where

Both capacitive reactance \scriptstyle{X_C} and inductive reactance \scriptstyle{X_L} contribute to the total reactance \scriptstyle{X}.

{X = X_L - X_C = \omega L -\frac {1} {\omega C}}
where

Although \scriptstyle{X_L} and \scriptstyle{X_C} are both positive by convention, the capacitive reactance \scriptstyle{X_C} makes a negative contribution to total reactance.

Hence,

Capacitive reactance[edit]

Main article: Capacitance

Capacitive reactance is an opposition to the change of voltage across an element. Capacitive reactance \scriptstyle{X_C} is inversely proportional to the signal frequency \scriptstyle{f} (or angular frequency ω) and the capacitance \scriptstyle{C}.[1]

X_C = \frac {1} {\omega C} = \frac {1} {2\pi f C} [2]

A capacitor consists of two conductors separated by an insulator, also known as a dielectric.

At low frequencies a capacitor is open circuit, as no current flows in the dielectric. A DC voltage applied across a capacitor causes positive charge to accumulate on one side and negative charge to accumulate on the other side; the electric field due to the accumulated charge is the source of the opposition to the current. When the potential associated with the charge exactly balances the applied voltage, the current goes to zero.

Driven by an AC supply, a capacitor will only accumulate a limited amount of charge before the potential difference changes polarity and the charge dissipates. The higher the frequency, the less charge will accumulate and the smaller the opposition to the current.

 

 

Inductive Reactance against Frequency

reactance against frequency

The slope shows that the “Inductive Reactance” of an inductor increases as the supply frequency across it increases.

Therefore Inductive Reactance is proportional to frequency. ( XL α ƒ )
 

Then we can see that at DC an inductor has zero reactance (short-circuit), at high frequencies an inductor has infinite reactance (open-circuit).

Inductive Reactance Example No1

A coil of inductance 150mH and zero resistance is connected across a 100V, 50Hz supply. Calculate the inductive reactance of the coil and the current flowing through it.

inductive reactance example 1

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Capacitors are one of the standard components in electronic circuits. Moreover, complicated combinations of capacitors often occur in practical circuits. It is, therefore, useful to have a set of rules for finding the equivalent capacitance of some general arrangement of capacitors. It turns out that we can always find the equivalent capacitance by repeated application of two simple rules. These rules related to capacitors connected in series and in parallel.

 

Figure 15: Two capacitors connected in parallel.
\begin{figure} \epsfysize =2.5in \centerline{\epsffile{cap1.eps}} \end{figure}

The equivalent capacitance of two capacitors connected in parallel is the sum of the individual capacitances.

 

\begin{figure} \epsfysize =1in \centerline{\epsffile{cap2.eps}} \end{figure}
Consider two capacitors connected in series: i.e., in a line such that the positive plate of one is attached to the negative plate of the other--see Fig. 16.
In fact, let us suppose that the positive plate of capacitor 1 is connected to the ``input'' wire, the negative plate of
capacitor 1 is connected to the positive plate of capacitor 2, and the negative plate of capacitor 2 is connected to the ``output'' wire.
What is the equivalent capacitance between the input and output wires?
\begin{displaymath} \frac{1}{C_{\rm eq}} = \frac{1}{C_1} + \frac{1}{C_2}. \end{displaymath}

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