Problem 64-7    Source:   Problem elaborated by the author's of the site.

Look at the circuit shown in Figure 64-07.1. Assume that V_Z1 = 12 V, V_Z2 = 5 V, R_L1 = 200 Ω, P_Z1 = 1 W, P_Z2 = 0.5 W and the input voltage is constant in value V_i = 18 V. On the other hand, the charge R_L2 consumes a variable current between 20 mA and 50 mA. Determine the values of R_S1 and R_S2 for the circuit to function properly.

Solution of the Problem 64-7   

Let's start solving this problem by analyzing the zener V_Z2, where we know that the current in the load is variable, so the current through the zener is also is variable. Consulting the theoretical part we see that we are facing   CASE 2. Then we will use the equations eq. 64-10 and eq. 64-11. Initially, we must calculate the maximum current that the zener supports, as we know its power and working voltage. Let's use eq. 64-01, shown below.

As the minimum zener current was not provided, we will adopt the rule of thumb

To calculate the value of R_S2min we will use eq. 64-10. Note that, in this case, the input voltage V_i assumes the value of the voltage V_i = V_Z1 = 12 V.

And to calculate the value of R_S2max we will use eq. 64-11.

A very interesting value for choosing R_S2 is the commercial value

With this, we will guarantee a constant current circulating by R_S2, as the voltage at its extremes is fixed and equal to 12 - 5 = 7 V. Then we can calculate the value of I₂.

Note that when load 2 consumes 20 mA, then the current through the zener is 70 - 20 = 50 mA. And when load 2 consumes 50 mA, then the current flowing through the zener is equal to 70 - 50 = 20 mA. In other words, values within the zener parameters. And it is also possible to calculate the value of I₁.

Note that I₁ and I₂ are constant currents. In this way, the zener Z₁ "sees" a constant charge represented in the circuit by the orange dashed rectangle. Its value is equal to

As the load is constant and the input voltage is constant, we are faced with   CASE 1. So, let's use eq. 64-08and eq. 64-09. However, first it is necessary to calculate the maximum and minimum current of Z₁. As the zener power is equal to 1 W and its voltage 12 V, using the eq. 64-01, we obtain:

And for the minimum current we will use the rule of thumb

Therefore, the value of R_S1min is

And the value of R_S1max is

Thus, taking the arithmetic mean between the two values we have R_S1 = 35.75   Ω. We can choose the commercial value

Addendum   

Let's check if the value of R_S1 makes the zener work within its characteristics. We know that the potential difference over R_S1 is equal to V_R = 18 - 12 = 6 V. Therefore, the current flowing through this resistor is:

Then, the constant current that will flow through Z1 will be:

In other words, a value perfectly within the technical characteristics of the zener that supports a maximum current of 83.33 mA.