How to Use Imaginary Operator "j"
February 1971 Radio-Electronics

February 1971 Radio-Electronics [Table of Contents] Wax nostalgic about and learn from the history of early electronics. See articles from Radio-Electronics, published 1930-1988. All copyrights hereby acknowledged.

Complex numbers have served the function of weeding out prospective electronics technician and electrical engineer degree-seeking people for a long time. I do not recall ever seeing such a beast until taking college courses. In high school and USAF tech school, we calculated reactive circuit parameters using well-established formulas that already accounted for the "imaginary" part of complex impedance. You can only go so far with circuit analysis without complex number math, though. All of the electronics magazines at some time (often every couple of years) ran articles introducing readers to the manipulation of the real and imaginary parts of reactive impedance. I have posted many of them here on RF Cafe (do a site search to find them). This 1971 Radio-Electronics magazine piece is yet another in a long line of them.

Both books referenced in the article can often be found on eBay for a reasonable price.

How to Use Imaginary Operator "j"

By John Collins

In almost any articles about impedance, reactance, resonant circuits or phase angles - in fact, any feature of ac networks - you run into the letter "j. Expressions like "Z = R + jX_L," or "E = 12 - j16" keep turning up. You may learn that j is an operator (meaning that it indicates an operation), that the operation is a multiplication by 1 and that the result is an imaginary number.

This information may have been interesting, but did little to explain how or why j is used to describe electrical quantities in ac circuits, or why we should bother about imaginary numbers at all. An example may help to clear up the mystery. Suppose we have a 120-volt ac generator connected in series with a resistor and capacitor, as in Fig. 1. A voltmeter shows a drop of 96 volts across the resistor.

What is the voltage drop across the capacitor?

A beginner might guess 24 volts (120-96). But the voltmeter shows that the actual drop is 72 volts. The sum of the drops across the two components is much greater than the input voltage!

How is this possible? Look at the vector diagram, Fig. 2. Voltage E across the resistor is in phase with the current in the circuit (rises and falls in step with it) and is plotted on the horizontal axis. Voltage E, across the capacitor does not keep in step, however. Because the capacitor has to be fully charged before the voltage comes up to full, voltage lags behind current in a capacitive circuit.

In an ordinary ac circuit, this lag is a quarter of a cycle. The current is at maximum just as the capacitor starts to charge, and voltage is at its lowest. As the capacitor charges, the voltage across it rises and current drops, till at full charge voltage is maximum and current zero. Then, as is the habit of alternating current, the voltage starts to go down, and the capacitor to discharge, till it is discharging at maximum when the source voltage is zero.

This quarter-cycle lag (or current lead) is put down mathematically as a 90° lag, and is plotted on the Y axis at 90° from the X axis. Mathematicians have a convention by which capacitive reactance is plotted below the origin (the point of intersection of the two axes) and called negative, while inductive reactance is called positive and plotted above the origin.

The resultant voltage E_T can be represented by the diagonal of the rectangle in which E_R and E_C are adjacent sides.

If the figure is drawn to scale, the diagonal will be approximately equal to 120 when the sides of the rectangle are 96 and 72. The relationship between the voltages would be indicated this way: E_T = 96 - j72. The j simply indicates that the line (or vector) representing the reactive part of the voltage must be rotated 90° to the vector for the simple resistive part, to show its true relationship.

If we replace the capacitor with an inductor of equal reactance, the voltage drops will be the same, but the voltage across the inductor will lead the circuit current by 90 °. because the magnetic field building up around the inductor opposes the rising current. It is represented by a line on the Y axis above the origin. The equation then becomes E_T = 96 + j72.

In ac circuits it is almost always necessary to deal with vectors, which are quantities having both magnitude and direction. A typical example of a vector is the diagonal in Fig. 2, which has a magnitude of 120 and a direction at an angle below the X axis. Every vector can be described by a complex number consisting of a real term, plotted on the X axis, and an imaginary term, plotted on the Y axis: 96 - j72, for example, is a complex number.

It may sometimes be necessary to add, subtract, multiply or divide numbers containing j operators. Addition and subtraction are most common, and the easiest. Simply add the real and imaginary parts separately. Thus 6 - j8 added to 5 - j3 comes to 11 - j5. Multiplication and division are almost as easy, and the method may be learned without trouble from math books for electronic technicians, such as Nelson Cooke's Basic Mathematics for Electronics (McGraw-Hill, $9.95) or Crowhurst's Basic Mathematics, 3 volumes (Rider, $12.75).