Each device in a series circuit has the same current, and each device in a parallel circuit has the same voltage. Therefore, finding the current in each device in a circuit is easier when the devices are all connected in parallel, and finding the voltage is easier when they're all connected in series.

Through a circuit *transformation*, or makeover, you can treat a complex circuit as
though all its devices were arranged the same way - in parallel or in series - by appropriately
changing the independent source to either a current or voltage source.

Changing the practical voltage source to an equivalent current source (or vice versa) requires the following conditions:

**•** The resistors must be equal in both circuits.

**•** The source transformation must be constrained by V_{s} = i_{s}R.

The constraining equation, V_{s} = i_{s} R, looks like Ohm’s law, which
should help you remember what to do when transforming between independent voltage and current sources.

**Converting to a Parallel Circuit with a Current Source**

Transformation techniques let you convert a practical voltage source with a resistor connected in series to a current source with a resistor connected in parallel. Therefore, you can convert a relatively complex circuit to an equivalent circuit if all the devices in the external circuit are connected in parallel. You can then find the current of individual devices by applying the current divider techniques that I discuss later in "Cutting to the Chase Using the Current Divider Technique."

When switching from a voltage source to a current source, the resistors have to be equal in
both circuits, and the source transformation must be constrained by V_{s} = i_{s} R.
Solving the constraint equation for i_{s} allows you to algebraically convert the practical
voltage source into a current source:

i_{s} = V_{s}/R

**Figure 4-2:** Transforming a voltage source into a current source

Figure 4-3 shows the conversion with some numbers plugged in. Both circuits contain the same 3-kΩ
resistor, and the source voltage in Circuit A is 15 volts. With this information, you can find the source
current, i_{s}, for the transformed Circuit B.

**Figure 4-3:** A numerical example of transforming a voltage source into a current source.

Use the constraint equation to find the source current in Circuit B. Here's what you get when you plug in the numbers:

i_{s} = V_{s}/R = 15 V/3 KΩ = 5 mA

**Changing to a Series Circuit With a Voltage Source**

You can convert a current source connected in parallel with a resistor to a voltage source connected in series with a resistor. You use this technique to form an equivalent circuit when the external circuit has devices connected in series.

Converting a practical current source connected with a resistor in parallel to a voltage source connected with a resistor in series follows the conditions for equivalent circuits:

**•** The resistors must be equal in both circuits.

**•** The source transformation must be constrained by V_{s} = i_{s}R.

**Figure 4-4:** Transforming a current source into a voltage source.

Figure 4-4 illustrates how to convert a current source into a voltage source.

**Figure 4-5:** Numerical example of transforming a current source into a voltage source.

You can use the constraint equation to find the source voltage for Circuit B. Plugging in the numbers produces the following:

v_{s} = i_{s}R = (5 mA)(3 kΩ) = 15 V

Suppose you have a complex circuit that has a current source, a resistor connected in parallel, and an external circuit with multiple resistors connected in series. You can transform the circuit so that it has a voltage source connected with all the resistors in series.

**Figure 4-6:** Transforming a complex circuit into a series circuit.

Consider Circuit A in Figure 4-6, where the right side of Terminals A and B consists of two resistors connected in series. On the left side of Terminals A and B is a practical current source modeled as an ideal current source in parallel with a resistor.

You want all the devices to be connected in series, so you need to move R when you transform the
circuit. To transform the circuit, change the current source to a voltage source and move R so that
it's connected in series rather than in parallel. When you use the constraint equation v_{s}
= i_{s}R to find the source voltage, remember that R is the resistor you moved.

Circuit B is a series circuit where all the devices share the same current. You can find the voltage through R, R1, and R2 using voltage divider techniques.

**About the Author**

John Santiago retired from the military in 2003 with 26 years of service in the United States Air Force (USAF) traveling over 23 countries. John has served in a variety of leadership positions in technical program management, acquisition, and operation research support. While assigned in Europe for three years with the USAF, he spearheaded more than 40 international scientific and engineering conferences/workshops as a steering committee member.

John has experience in many engineering disciplines and missions, including control and modeling of large, flexible space structures; communications systems; electro-optics; high energy lasters; missile seekers/sensors for precision-guided munitions; image processing/recognition; information technologies; space, air, and missile warning; missile defense; and homeland defense.

One of John's favorite assignments was serving as associate professor at the USAF Academy during his tour from 1984 though 1989. John is currently a professor of Electrical and Systems Engineering at Colorado Technical University, where he has taught over 26 different undergraduate and graduate courses in electrical and systems engineering.

Some of his awards include Faculty of the Year at Colorado Technical University in 2008; USAF Academy Outstanding Military Educator in 1989; and USAF Academy Outstanding Electrical Engineering Educator in 1998.

During his USAF career, John received his PhD in Electrical Engineering from the University of New Mexico; his Master of Science in National Resource Strategy at the Industrial College of the Armed Forces; his Master of Science in Electrical Engineering from the Air Force Institute of Technology, specializing in electro-optics and his Bachelor from the University of California, Los Angeles (UCLA). More information about John's background and experience is available at FreedomUniversity.TV

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