Problem 61-1    Source:   Problem n° 4 - 1ª test - Fisica II-C - Ufrgs - 2016-2 Prof. Dr. Ricardo Francke.

In the Table 61-01.1 we see the characteristics of a diode relative to the curve I x V.

a) Show that the current grows exponentially.

b) What would be the current for a voltage of 0.2 volts?

c) What if the voltage was 1.0 volt?

Solution of the Problem 61-1   

Item a   

With the use of eq 61-04 developed in item Theory, this problem can be solved. The equation for clarity is shown below.

For a given device operating at ambient temperature, it is known that the value given by 2,3 η V_T is a constant and here it will be represented by the letter K. In this way, we must calculate the value of K for each interval and find out if it holds a constant value and this will characterize the angular coefficient of the equation of a line. Thus, the value of K_n will be determined by the relation:

Taking the first two rows of the table and making   V₂ = 0.691 V ,   V₁ = 0.604 V , I₂ = 0.209 mA   and   I₁ = 0.0136 mA. Now, calculating the value of K₁ using the above equation, we find:

The index 1 is due to n = 1. To calculate K₂, you use the third and second line of the table assigning  V₂ = 0.749 V ,  V₁ = 0.691 V ,  I₂ = 1.590 mA   and  I₁ = 0.641 mA. Performing the calculation:

Repeating this systematic for the other values, we find:

Note that the calculated values are very close. If the intention is to be rigorous, we can determine the equation of the line that adapts to these values, using specific methods, such as linear regression or Lagrange, etc ... The intention here is to obtain an approximation that leads to a practical result. Thus, we simply chose to calculate the mean of the values and establish a K_ave which will serve the purposes. Like this:

But what does this value of K_ave mean? This value says that with each increment of voltage of the order of 0.06V or 60 mV on the diode, on average, the current on the diode will increase by 10 times. This justifies an exponential growth in the current on the diode. In the item b and c a eq. 61-04 has been manipulated algebraically and shows this feature clearly.

Item b   

To respond to the item b and calculate the value I₁, is taken as a reference to first line of the table, assuming that V₂ = 0.604 V , V₁ = 0.2 V and I₂ = 0.0136 mA. Manipulating algebraically the eq 61-04:

Performing the numerical substitution of the values and calculating:

It can be stated, categorically, that this diode is in the cut zone, given the tiny current that flows through it.

Item c   

Using the same methodology, but using the last row of the table, since V₂ = 1 V and V₁ = 0.83 V. In this case, you must find the value of I₂, which is the value corresponding to V₂. Then:

After the substitution by the numerical values and the calculation is carried out:

For this current value, it can be stated that this diode will certainly short-circuit.

Complementation   

Since we have already calculated the value of K_ave, we can calculate the value of η and see if it is within the standards established by the theory. Not forgetting the value of V_T = 25 mV = 0.025 V. Then:

Since the value of η must be between 1 and 2, the value found in our calculations above, η = 1.0435, is in accordance with the theory.