. - . - Inductive/Capacitive Reactance

Capacitance/Inductance Reactance Equations

An Alternating Current is one whose amplitude of current flow periodically rises from zero to a maximum in one direction, decreases to zero, changes its direction, rises to a maximum in the opposite direction, and decreases to zero again. This complete process, starting from zero, passing through two maximums in opposite directions, and returning to zero again, is called a cycle. The number of times per second that a current passes through the complete cycle is called the frequency of the current and is specified in Hertz. One cycle is equivalent to one hertz.

When a changing current flows through a inductor a counter-electromotive force is developed, opposing any change in the initial current. This property of an inductor causes it to offer opposition to a change in current. The measure of opposition offered by an inductor to an alternating current of a given frequency is known as its Inductive Reactance (X_L). The formula for inductive reactance, in terms of Inductance (L) and Frequency (F), is listed on the right.

Capacitors have a similar property although in this case the opposition is to any change in the voltage across the capacitor. This property is called Capacitive Reactance (X_C) and is calculated as shown in the image on the right, in terms of Capacitance (C) and Frequency (F).

The graph is a visual example of Inductive/Capacitive Reactance, for two specific component values, across two decades of frequency. The horizontal scale is Frequency in Hertz and the vertical scale is Reactance, in Ohms. A 0.002 uF Capacitor and a 1.26 mH Inductor. Note that, as frequency increases, Capacitive Reactance (X_C) decreases and Inductive Reactance (X_L) increases. While the Inductive and Capacitive Reactance equations are linear, using a logarithmic scale for frequency make things easier to see over large spans.

Note the very center of the graph (100KHz). At that point the Inductive/Capacitive Reactances are equal. This point is known as Resonance.

Capacitance, Inductance, Reactance Calculator

The calculator on the right is a interesting way of looking at reactances. Calculators like this were originally created out of cardboard and given away as promotional devices. An nice example of this would be the HP Capacitance - Inductance - Reactance Calculator. It has a built in animation that allows you to slide the image containing Frequency, Inductance, and Capacitance inside of (actually behind) the Reactance image. With this slide, you can readily see how the Frequency affects capacitive and inductive reactance.

The HP calculator inspired me to create the calculator shown below. My calculator is actually a dynamic drawing. While the HP calculator gives a wide range of frequency, inductance, and capacitance, my calculator is limited in range. This is because there are limitations with web page designs that don't exist for cardboard cutouts. And, this was just an exercise to see if I could emulate the operation of the cardboard calculator. This is also a good exercise for the reader in using logarithmic scales. Note that some scales read left to right (L, C, and X_L) while others read right to left (X_C, Frequency). This illustrates the fact that:

for any particular capacitor value, as the Frequency is increased (scale shift to the right), the Reactance, X_C, decreases and
for any particular inductance value, as the Frequency is increased (scale shift to the right) the Reactance, X_L, increases.

The black arrow in the center, marks the current Frequency. If the page is refreshed, the arrow will point to approximately 32 Hz. To change this frequency simply click on the symbols on the left and right side of the scales. The ↫ and ↬ symbols are use to shift the frequency left and right, respectively, one complete decade. The Inductance and Capacitance scales will also shift, as they are locked to the Frequency scale.

The other symbols are for less drastic scale shifting. The ⇐ and ⇒ will shift the scales left and right, respectively, but only one pixel at a time.The ⇚ and ⇛ will shift the scales left and right, respectively, four pixels at a time. At about 70 Hz, a one pixel shift translates to a 1 Hz shift. However, below 70 Hz the frequency shift is less than 1 Hz per pixel. Above 70 Hz the shift is greater than one Hz per pixel.

Capacitance, Inductance, Reactance Chart

The chart on the right is also very interesting. It has been reprinted in many books that deal with component theory. Like the Slider above, this chart is a drawing. In this chart, Frequency is listed on the bottom horizontal scale and ranges from 100 Hz to 100 MHz. The vertical scale on the left is Reactance. It can be either Inductive or Capacitive Reactance, depending on which diagonal you are using. The red diagonal lines are for Inductance and the green ones are for Capacitance. The contrasting colors helps with reading the chart. Like the Reactance Calculator above, it's easy to see how, for any particular Inductor or Capacitor, the Frequency affects the Reactance.

Inductive and capacitive reactance v.s. Frequency. Heavy lines represent multiples of 10, intermediate light lines represent multiples of five. For example, the light line between 10 uH and 100 uH represents 50 uH; the light line between 0.1 uF and 1 uF represents 0.5 uF, and so on. Intermediate values can be values within the chart range. For example, the reactance of 10 henrys at 60 Hz can be found by taking the reactance of 10 henrys at 600 Hz and dividing by 10 for the 10x times decrease in frequency.