Electronic – How to answer this kind of diode problem

circuit analysisdiodes

I have this problem here where I need to find V2 in 3 different cases and I have trouble of knowing how to take on the problem. Could someone please lead me into the right direction. I can't really find anything on how to go on this kind of problems.

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Best Answer

Because @Hilmar gave a very good answer, I will present a more mathematical approach using the Shockley diode equation

Well, we are trying to analyze the following circuit:

schematic

simulate this circuit – Schematic created using CircuitLab

When we use and apply KCL, we can write the following set of equations:

$$\text{I}_1=\text{I}_\text{D}+\text{I}_2\tag1$$

When we use and apply Ohm's law and the Shockley diode equation, we can write the following set of equations:

$$ \begin{cases} \text{I}_1=\frac{\text{V}_\text{in}-\text{V}_1}{\text{R}_1}\\ \\ \text{I}_\text{D}=\text{I}_\text{s}\left(\exp\left(\frac{\text{q}\text{V}_1}{\eta\text{kT}}\right)-1\right)\\ \\ \text{I}_2=\frac{\text{V}_1}{\text{R}_2} \end{cases}\tag2 $$

Assuming room-temperature, \$\text{R}_1=\text{R}_2=10\space\text{k}\Omega\$ and \$\eta=\frac{3}{2}\$, I used Mathematica to find:

In[1]:=Is = 10^(-12);
\[Eta] = 3/2;
T = 20 + (5463/20);
q = (1602176620898/1000000000000)*10^(-19);
k = (1380649/1000000)*10^(-23);
Vin = 1;
R1 = x;
R2 = x;
x = 10000;
FullSimplify[
 Solve[{I1 == ID + I2, I1 == (Vin - V1)/R1, 
   ID == Is*(Exp[(q*V1)/(\[Eta]*k*T)] - 1), I2 == V1/R2, 
   ID \[Element] Reals && I1 \[Element] Reals && I2 \[Element] Reals &&
     V1 \[Element] Reals}, {ID, I1, I2, V1}]]

Out[1]={{ID -> -(1/1000000000000) + (
    24284235261 ProductLog[(
      801088310449 E^(80108831845988310449/6071058815250000000))/
      6071058815250000000])/3204353241796000, 
  I1 -> 99999999/2000000000000 + (
    24284235261 ProductLog[(
      801088310449 E^(80108831845988310449/6071058815250000000))/
      6071058815250000000])/6408706483592000, 
  I2 -> 100000001/2000000000000 - (
    24284235261 ProductLog[(
      801088310449 E^(80108831845988310449/6071058815250000000))/
      6071058815250000000])/6408706483592000, 
  V1 -> 100000001/200000000 - (
    121421176305 ProductLog[(
      801088310449 E^(80108831845988310449/6071058815250000000))/
      6071058815250000000])/3204353241796}}

In[2]:=N[%1]

Out[2]={{ID -> 5.03226*10^-7, I1 -> 0.0000502516, I2 -> 0.0000497484, 
  V1 -> 0.497484}}
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