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I have read that a long tailed pair of transistors has a tanh characteristic I,V curve.

How do we get from the exponential function of a single transistor Ie = Ies(e^(Vbe/Vt)-1) to a tanh function of a LTP?

It would be nice to have method of deriving this equation that shows intuitively what is happening in the circuit.

Here are some relationships that should be relevant here.
sinh(x) = 1/2 e^x - e^-x
cosh(x) = 1/2 e^x + e^-x
e^x = sinh(x) + cosh(x)
tanh = sinh(x)/cosh(x)

I was reading this page about OTAs from which the images below are taken, it has a description of how to get a function for a LTP, but no mention of the tanh function.

long tailed pair I V curve of long tailed pair

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  • \$\begingroup\$ Does not Grey and Meyer provide this math? \$\endgroup\$
    – analogsystemsrf
    Commented Sep 17, 2017 at 15:02
  • \$\begingroup\$ Thank you, I was not aware of the relevant books, I am teaching myself online. If I was at university they would have told me what to read. I did a search for Grey and Meyer, it looks relevant. \$\endgroup\$
    – John Spence
    Commented Sep 17, 2017 at 15:12

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