# 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:

![](General_Source-Load_Circuit.svg "General_Source-Load_Circuit.svg")

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).

  -------------------------------------------------------- ------------------------------------------------------------------------------
  ![](Thevenin_Equivalent.svg "Thevenin_Equivalent.svg")   ![](Thevenin_Equivalent_Under_Test.svg "Thevenin_Equivalent_Under_Test.svg")
  *(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~*.

![](Norton_Equivalent_Under_Test.svg "Norton_Equivalent_Under_Test.svg")

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:

  ---------------------------------------------------- ---------- --------------------------------------------------
  center   $\equiv$   center
  ---------------------------------------------------- ---------- --------------------------------------------------

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):

$$v_s=i_sr\,$$

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:

  ---------------------------------------------------------- ------------------------------------------------------------
  ![](Open_Circuit_Voltage.svg "Open_Circuit_Voltage.svg")   ![](Short_Circuit_Current.svg "Short_Circuit_Current.svg")
  *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.
