Circuit Theory/Source Transformations

Source Transformations

Independent current sources can be turned into independent voltage sources, and vice-versa, by methods called "Source Transformations." These transformations are useful for solving circuits. We will explain the two most important source transformations, Thevenin's Source, and Norton's Source, and we will explain how to use these conceptual tools for solving circuits.

Black Boxes

A circuit (or any system, for that matter) may be considered a black box if we don't know what is inside the system. For instance, most people treat their computers like a black box because they don't know what is inside the computer (most don't even care), all they know is what goes in to the system (keyboard and mouse input), and what comes out of the system (monitor and printer output).

Black boxes, by definition, are systems whose internals aren't known to an outside observer. The only methods that an outside observer has to examine a black box is to send input into the systems, and gauge the output.

Thevenin's Theorem

Let's start by drawing a general circuit consisting of a source and a load, as a block diagram:

Let's say that the source is a collection of voltage sources, current sources and resistances, while the load is a collection of resistances only. Both the source and the load can be arbitrarily complex, but we can conceptually say that the source is directly equivalent to a single voltage source and resistance (figure (a) below).

(a)	(b)

We can determine the value of the resistance R_s and the voltage source, v_s by attaching an independent source to the output of the circuit, as in figure (b) above. In this case we are using a current source, but a voltage source could also be used. By varying i and measuring v, both v_s and R_s can be found using the following equation:

$v=v_{s}+iR_{s}\,$

There are two variables, so two values of i will be needed. See Example 1 for more details. We can easily see from this that if the current source is set to zero (equivalent to an open circuit), then v is equal to the voltage source, v_s. This is also called the open-circuit voltage, v_oc.

This is an important concept, because it allows us to model what is inside a unknown (linear) circuit, just by knowing what is coming out of the circuit. This concept is known as Thévenin's Theorem after French telegraph engineer Léon Charles Thévenin, and the circuit consisting of the voltage source and resistance is called the Thévenin Equivalent Circuit.

Norton's Theorem

Recall from above that the output voltage, v, of a Thévenin equivalent circuit can be expressed as

$v=v_{s}+iR_{s}\,$

Now, let's rearrange it for the output current, i:

$i=-{\frac {v_{s}}{R_{s}}}+{\frac {v}{R_{s}}}$

This is equivalent to a KCL description of the following circuit. We can call the constant term v_s/R_s the source current, i_s.

The equivalent current source and the equivalent resistance can be found with an independent source as before (see Example 2).

When the above circuit (the Norton Equivalent Circuit, after Bell Labs engineer E.L. Norton) is disconnected from the external load, the current from the source all flows through the resistor, producing the requisite voltage across the terminals, v_oc. Also, if we were to short the two terminals of our circuit, the current would all flow through the wire, and none of it would flow through the resistor (current divider rule). In this way, the circuit would produce the short-circuit current i_sc (which is exactly the same as the source current i_s).

Circuit Transforms

We have just shown turns out that the Thévenin and Norton circuits are just different representations of the same black box circuit, with the same Ohm's Law/KCL equations. This means that we cannot distinguish between Thévenin source and a Norton source from outside the black box, and that we can directly equate the two as below:

$\equiv$

We can draw up some rules to convert between the two:

• The values of the resistors in each circuit are conceptually identical, and can be called the equivalent resistance, R_eq:

$R_{s_{n}}=R_{s_{t}}=R_{s}=R_{eq}$

• The value of a Thévenin voltage source is the value of the Norton current source times the equivalent resistance (Ohm's law):

$v_{s}=i_{s}r\,$

If these rules are followed, the circuits will behave identically. Using these few rules, we can transform a Norton circuit into a Thévenin circuit, and vice versa. This method is called source transformation. See Example 3.

Open Circuit Voltage and Short Circuit Current

The open-circuit voltage, v_oc of a circuit is the voltage across the terminals when the current is zero, and the short-circuit current i_sc is the current when the voltage across the terminals is zero:

The open circuit voltage	The short circuit current

We can also observe the following:

• The value of the Thévenin voltage source is the open-circuit voltage:

$v_{s}=v_{oc}\,$

• The value of the Norton current source is the short-circuit current:

$i_{s}=i_{sc}\,$

We can say that, generally,

$R_{eq}={\frac {v_{oc}}{i_{sc}}}$

Why Transform Circuits?

How are Thevenin and Norton transforms useful?

Describe a black box characteristics in a way that can predict its reaction to any load.

Find the current through and voltage across any device by removing the device from the circuit! This can instantly make a complex circuit much simpler to analyze.

Stepwise simplification of a circuit is possible if voltage sources have a series impedance and current sources have a parallel impedance.