I worked with FindRoot too, but the answer was not validate. I have problem when work with FindRoot because it need initials values

This is the file, can you help me?

Attachments:

I reran NMinimize, giving it 0 as an objective function, along with the constraints. It returned an answer, but with a warning message that the obtained solution did not satisfy the constraints within tolerance. The answer obtained was

{0., {Subscript[i, ph] -> 1.42325, Subscript[i, o] -> 36.4052, 
  A -> -3.62571, Rs -> 0.00234389, Rp -> 0.841729}}

Those values might serve as starting values for FindRoot.

In[21]:= NMinimize[{0, 
  7.61 == 0. - (26.3 + 7.6*Rs)/Rp - 
    1. (-1 + E^((38.91741123737115* (26.3 + 7.61* Rs))/A)) Subscript[
     i, o] + Subscript[i, ph],
     8.21 == 
   0. - (8.21* Rs)/Rp - 
    1. (-1 + E^((319.51194625881715* Rs)/A)) Subscript[i, o] + 
    Subscript[i, ph], 
     0 == 0. - 32.9/Rp - 
    1. (-1 + E^(1280.3828297095108/A)) Subscript[i, o] + Subscript[i, 
    ph],
     6.62 == 
   0. - (6.62* Rs)/Rp - 
    1. (-1 + E^((239.92927434638457 *Rs)/A)) Subscript[i, o] + 
    Subscript[i, ph],
     0 == 0. - 29.9/Rp - 
    1. (-1 + E^(1083.668474766903/A)) Subscript[i, o] + Subscript[i, 
    ph]},
       {Subscript[i, ph], Subscript[i, o], A, Rs, Rp}]

During evaluation of In[21]:= NMinimize::nosat: Obtained solution does not satisfy the following constraints within Tolerance -> 0.001`: {0. +29.9/Rp+1. (-1+E^(1083.67/A)) Subscript[i, o]-Subscript[i, ph]==0,0. +32.9/Rp+1. (-1+E^(1280.38/A)) Subscript[i, o]-Subscript[i, ph]==0,6.62 +<<4>>==0,8.21 +(8.21 Rs)/Rp+1. (-1+E^(<<1>>/A)) Subscript[i, o]-Subscript[i, ph]==0,7.61 +(26.3 +7.6 Rs)/Rp+1. (-1+E^((38.9174 (<<5>> +<<1>>))/A)) Subscript[i, o]-Subscript[i, ph]==0}. >>

Out[21]= {0., {Subscript[i, ph] -> 1.42325, 
  Subscript[i, o] -> 36.4052, A -> -3.62571, Rs -> 0.00234389, 
  Rp -> 0.841729}}

Tried that and got a SystemException[MemoryAllocationFailure...]. Perhaps the problem is singular?