Applying Complex Numbers to Electrical Engineering
In math, information that expresses a single dimension, such as a linear distance, is called a scalar quantity in
mathematics. Scalar numbers are the kind that you (as students) use most often. Relating to electrical
engineering, the voltage produced by a battery, the resistance of a piece of wire (in ohms), and current through a
wire (amps) are all scalar quantities.
Key Vocabulary
• Electrical charge (essentially electricity) is the movement of electrons, usually in a closed circuit.
• Voltage (V, measured in Volts) is the difference in charge between two points.
• Current (I, measured in Amps) is the rate that the amount of charge that flows through a wire, usually
over the course of 1 second.
• Resistance (Ω omega measured in ohms, or R) is how well the material (usually wire) resists the flow of
charge (Current).
Quick Analogy
• Imagine a tank of water (like a battery) connected to a hose (the wire). Think of Water as the Charge, so
the amount of stuff that starts off in the tank that will flow through the hose (analogy for the wire). The
Voltage is like the Water Pressure at the end of the hose (wire). The more water that is in the tank, the
more Pressure (Voltage) that forces the water out of the hose. As the Water (Charge) runs out of the
tank (battery), the pressure (voltage) decreases at the end of the hose (wire).
• Same analogy, but with current. Imagine you have two tanks (batteries) that are exactly the same size.
However, the hose (wire) connected to Tank 1 is larger in diameter than the one attached to Tank 2. As
the water flows from the tank (battery) into the hose (wire), Tank 1’s hose will have a faster flow
(Current) because it can hold more water (Charge) and therefore water will flow through it at a faster
rate, creating a higher pressure (voltage) at the end of the hose.
• If you wanted to equalize those pressures (voltages), you could replace Tank 2 with a Tank that
holds more water than Tank 1. Then, this would force the water (Charge) through the second
hose (wire) at a faster rate, though a smaller flow (current) and could potentially equalize the
water pressures (voltages) from each hose (wire).
• A wall outlet has the same voltage in each socket. However, if you plug in both a massive electric
heater (Tank 1) and a cell phone charger (Tank 2) to the sockets, the heater will draw a larger amount of
electricity than the cell phone.
• Resistance, in the analogy’s case, represents the width of each hose. A smaller hose will ”resist” the
flow of water more than a larger one. In reality, a wire with a higher resistance will not allow as much
charge to flow through it.
There are two types of electricity in the world: AC & DC. DC stands for “direct current” and is where all the
electrons in a piece of wire travel in one direction, from one end of the wire to the other. The easiest example
of this is for a battery. The electrons travel from the negative end of the battery to whatever they are providing
charge to, and then return along the positive side of the wire back to the positive end of the battery. DC was
first popularized by Thomas Edison, but has long since faded out of use, as it’s not as efficient as AC. In DC,
the relationship between Voltage, Current, and Resistance is given by
𝑉 =𝐼∙𝑅
AC, or “alternating current”, holds its charge better over longer distances. The electrons are constantly in
motion, moving between a peak negative charge and a peak positive charge. Think of it like an ocean wave: at
the top of the wave the current is positive, at the middle of the curve, it’s zero, and at the bottom of the trough,
it’s negative. The current repeats this process every 1/60th of a second in the United States. In other parts of the
world, it is less. AC uses slightly different vocabulary than DC, namely that Resistance is now called
Impedence (Im-PEED-ence). However, the fascinating thing about AC circuits is that because the electrons go
forwards AND backwards, we cannot use scalars anymore because scalars only measure things that move in
one direction. In AC circuits, Voltage equals Current times Resistance. In DC circuits, we have the following
equation, where Z is the impedence:
𝑉 =𝐼∙𝑍
Although this looks similar, we say that 𝑍 = 𝑅 + 𝑖𝑋, where j represents the imaginary unit √−1, R stands for
the resistance, and X is the reactance.
1. The diagram for a circuit looks like this:
Where the Z’s are the resistors and the V is the voltage. The current (I) is what kills you
when you start playing with electricity, so it is very important to know how to calculate it.
Large voltages can hurt but cause no lasting harm, however small amounts of current are
dangerous and will cause great harm.
For the problem above, the total series resistance is the sum of the impedences. Since all the resistors are in a
line, we add them together to get the total impedence.
If Z1 = 3+2i ohms, Z2 = 2-3i ohms. The total impedence is: _________________________________
Once you know the total resistance, and since we know the voltage (from the diagram, V = 10 volts), we can
calculate the amperage.
V = I∙Z so we can re-arrange the formula to get us the formula: I = __________.
Plug in your values and simplify the problem below:
2.
In this circuit, we have 3 resistors, R1 = 4-2i ohms, R2 = 2-6i ohms, and R3
= 4+4i ohms, with a power source of 15 volts. What is the amperage?
3.
This is a different type of circuit, where we have series and parallel circuits. The
series circuits are added like before, but the two parallel resistors have to be added
differently.
1
1
+
= 𝑇𝑜𝑡𝑎𝑙 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙 𝑍
𝑍3 𝑍4
Next you can add the Parallel Z to the two series Z to find the total Z.
Z1 = 3-3i ohms, Z2 = 2+5i ohms, Z3 = 2-3i ohms, and Z4 = 2+3i ohms
What is the voltage if the amperage (I) is 4+2i amps?
A) Parallel Z sum is: (adding Z3 and Z4)
B) Series Z sum is: (Adding Z1 and Z2)
Total Z is: (adding parts A and part B)
Calculate the voltage: