Why does the Darlington pair have a higher input resistance?

On July 7, in one of our discussions, LvW asked me an interesting question about the famous Darlington pair: "By the way: Some days ago I stumbled over the question: What are the advantages/disadvantages of the Darlington-pair? Do you have a short answer?"

It was an exact hit because this circuit was one of my favorite circuit solutions. I immediately replied: "I have also thought a lot about the Darlington pair... I have even found an interesting phenomenon in its operation - the first transistor introduces negative feedback between the collector and the base of the second transistor. That is why, its collector voltage cannot be below 0.7 V (disadvantage)." A big advantage of this 2-transistor circuit is that it behaves like a three-pin transistor (collector, base, emitter). This allows ordinary transistors to be replaced by Darlington "transistors" (TIP) and v.v. I have often seen my son do it during car repairs...

But the warm summer came, we went on vacation and the discussion ended...

...

Esterday I was pleasantly surprised to see that LvW resumed the discussion:

"I think, it is nothing else than an emitter follower (seen from T1) with an active emitter resistance (in put resistance of T2). As a consequence, the input resistance at the base node of T1 has increased correspondingly (doubled). This is the only advantage of the Darlington pair if compared with a single BJT. Some people think that the drastic increase of the current gain beta=beta1*beta2 would be an enormous improvement - however, this is not so important. The only positive consequence is - as mentioned - the increase of the input resistance. We should not overlook that the transconductance gm - if compared with a single transistor having the same collector current Ic as T2 - is reduced by 50%. Hence, the voltage gain of a single transistor can be made larger (same Ic) - but for the price of a smaller input resistance."

Here is my answer:

My explanation

I completely agree with you. I will only explain it more "philosophically" as an example of applying a "circuit philosophy":) I will use the simplest possible words, without any terms and clichés... as though we do not know anything about emitter follower, series negative feedback, Miller theorem etc... we only know the simplest electrical concepts... In other words, I will try to answer a question like, "Why is the grass green:)?

So, what is the trick here?

If there was only T1 with a grounded emitter (simple common-emitter stage), the input voltage source would pass a relatively high base current through T1's base-emitter junction.

The clever trick here is that "something" like a (non-linear) resistor is inserted between T1's emitter and ground... and the amplified base current (i.e., the collector current) passes through it. We can think of this current-supplied element as a voltage source connected contrary in series to the input voltage. As a result, T1's base current decreases many times while the input voltage is the same... and the input resistance artificially increases.

So, the conclusion is: Whatever we put in the emitter of a transistor, as long as it has some resistance (linear or nonlinear), the input resistance increases. Interesting are the cases when the "thing" is nonlinear and it behaves like a "dynamic resistor". Then it either helps or hinders the input voltage in its quest to change the base current.

In our case (an emitter follower with voltage-stabilizing emitter "resistor"), when the input voltage increases, the base, collector and emitter currents increase as well... but the dynamic resistance (of T2's base-emitter junction) decreases. So, it helps the input voltage source to increase T1's base current... and, as a result, decreases the input resistance.

In other cases (an emitter follower with current-stabilizing emitter "resistor"), when the input voltage increases, the base, collector and emitter currents increase as well... but now the dynamic resistance (emitter "current source") increases, too. It hinders the input voltage source to increase T1's base current... and, as a result, the input resistance increases.