Swapping sources -- Source Transformation

In this topic, we will revisit the voltage divider and reuse the elements to transform the circuit into an equivalent current-source representation.   This transformation is commonly used to help simplify circuits for easier analysis. 

 Let's jump right in and compare the figures below.  Believe it or not, both circuits displayed in Figure 1 are electrically equivalent as they produce the same voltage across and current through the load resistor, R_LOAD.   

Figure 1.  Source transformation (a) Voltage source (b) Current source

This may take some getting used to, but to help drive this concept home I have provided the simple proof below.  Let's go back to our voltage divider analysis days and derive the voltage across and current through R_LOAD.

 Voltage source:

Using KCL, we can generate the expression for the current-source representation. 

Current source: 

Notice that the current, I_X, as well as the voltage, V_X, are equivalent.  Source transformations come in handy when you want to separate or simplify a network connected to another network or load.  The equivalent resistance of the simplified network, R_S, is the same value for both voltage and current transformations.  Equation 1 and 2 display the relationship between the voltage-source and current-source transformation, and can be used to easily swap from one source to another.  

Voltage source transformation:

Current source transformation:

Now that you have a good understanding of source transformation, let's proceed into Norton’s and Thevenin's Theorem.

Norton's Theorem:

Any network containing independent voltage and current sources, along with various resistances is electrically equivalent to an ideal current source, I_NORTON, in parallel with a single resistor, R_NORTON.

 Procedure:

• Find the Norton current, I_NORTON, by shorting the A and B terminals together.
• Disconnect the short and leave A and B open.  Calculate the voltage across nodes A and B  -- this voltage is labeled V_{AB_OPEN}.
• R_NORTON = V_{AB_OPEN} / I_NORTON

Thevenin's theorem:

Any network containing independent voltage and current sources, along with various resistances is electrically equivalent to a voltage source, V_THEVENIN, in series with a single resistor, R_THEVENIN.

 Procedure:

•        Leave A and B open.  Calculate the voltage across nodes A and B  -- this voltage is labeled V_THEVENIN.
•       Find the output current, I_{AB_SHORT}, by shorting the A and B terminals together.
•        R_THEVENIN = V_THEVENIN / I_{AB_SHORT}

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