Trans Linear Principle

In this blog, I want to introduce an interesting but very handy topic, known as "Trans Linear Principle (TLP)". After giving an overview of this, slowly we shall take up some problems, and in the process of solving that problem, we shall try to understand how TLP actually works.

Trans Linear Principle (TLP):

“In a closed loop containing an even number of ideal junctions, arranged so that there are an equal number of clockwise-facing and counter-clockwise-facing polarities, with no further voltage generators inside the loop, the product of the current densities in the clockwise direction is equal to the product of the current densities in the counter-clockwise direction.”

                                                                                                    - Barrie Gilbert

 

 

Now, before applying any fancy principle, we should be aware of the conditions and assumptions behind that principle, otherwise there is a possibility of blunder.

So there are some conditions to apply this principle:

• There must be an even number of base-emitter junctions in the loop.
• Half of the junctions must be oriented in one direction and half in the other.
• There should not be further voltage generators inside the loop.

Also there are some assumptions before using the results:

• All emitter areas are equal.
• All transistors are operating in forward active with large beta.
• All transistors have infinite output resistance i.e. ignoring early effect

Proof: 

By KVL,

\(V_{BE1}+V_{BE2}=V_{BE3}+V_{BE4}\)

Suppose the transistors follow the ideal equation

\(V_{BE}=V_{T}ln\left ( \frac{I_C}{I_S} \right )\)

Substituting,

$V_{T}ln\left ( \frac{I_1}{I_S} \right )+V_{T}ln\left ( \frac{I_2}{I_S} \right )=V_{T}ln\left ( \frac{I_C}{I_3} \right )+V_{T}ln\left ( \frac{I_4}{I_S} \right )$

Therefore, 

$I_1 I_2=I_3 I_4$

Mathematically, if a loop of base-emitter voltages satisfies,

$\sum_{CW}^{}V_{BE}=\sum_{CCW}^{}V_{BE}$

Then according to TLP, it is also true that,

$\prod_{CW}^{}J_{C}=\prod_{CCW}^{}J_{C}$

Bonus: 

If you want to dig further down into this, if you wonder what will happen if there is finite beta or if the emitter areas are not equal or there is finite output resistance, then you can watch this famous video where the master himself explaining Trans Linear circuits! The video was recorded in February, 1991.

If you have understood this, try to solve Problem-1, Problem-2 by observation using TLP. You will understand, how easily you can solve big circuits without writing and solving so many equations.