Problem 64-8    Source:   Problem elaborated by the author's of the site.

Look at the circuit shown in Figure 64-08.1. Assume that V_Z1 = 12 V, V_Z2 = 5 V, R_L = 200 Ω, P_Z1 = 1 W, P_Z2 = 0.5 W, and as input voltage we have two sources in series, where V_i = 22 V and V_S = 5 sin ω t V. On the other hand, R_L and the zener Z₂ are part of a lithium battery charger, whose nominal voltage is 3.7 V. When the lithium battery reaches a voltage of 4.0 V, R_C must ensure that no current flows greater than 10 mA (charge maintenance current) by the battery. Determine the values of R_S1, R_S2 and R_C for the circuit to function properly.

Solution of the Problem 64-8   

Let's start solving this problem by analyzing the circuit formed by the zener Z₂ and R_C. According to the problem statement, R_C must ensure that a current greater than 10 mA does not flow through the battery when it reaches a voltage of 4 V. Then the value of R_C will be:

Therefore, 10 mA is the minimum current in the load ( I_Lmin ) and the maximum current can be calculated assuming the worst case that occurs when the battery has 0 V at its terminals. Therefore, the maximum current is I_Lmax = 5 - 0 / 100 = 50 mA. After these calculations, we can adopt a current of 70 mA circulating through R_S2. Thus, when the battery has a voltage of 0 V, a current of 50 mA will flow through it and a current of 70 - 50 = 20 mA will flow through the zener. When the battery reaches 4 V, a current of 10 mA will flow through it and a current of 70 - 10 = 60 mA will flow through the zener. All these values are in accordance with the characteristics of Z₂, as it supports a maximum current of 100 mA. So, the value of R_S2 is

To calculate the value of R_S1 we must take into account that the input voltage, due to V_S, varies between a minimum value and a maximum value. Let's determine these values.

Note that for the calculation above we used the fact that the function sine varies between - 5 and + 5, in the case of this problem.

The current passing through R_L is

With this value we can calculate the load current for Z₁, or

So, we conclude that for Z₁ we have a constant load current ( I_L ) and an input voltage variable. This falls under   CASE 3. Therefore, to find the value of R_S2 we will use the equations eq. 64-12 and eq. 64-13. First, we must find the minimum and maximum value of the current that can flow through the zener Z₁. Using eq. 64-1, we have:

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 = 32.14   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 maximum (worst case) potential difference over R_S1 is equal to V_R = 27 - 12 = 15 V. Therefore, the maximum current flowing through this resistor is:

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

In other words, as the zener supports a maximum current of 83.33 mA, we conclude that this zener is not suitable for the project. To overcome this problem, you must choose a zener that dissipates a power of 5 W, as this will present a maximum current of

Thus, we conclude that a 5 W zener will support the current of 324.5   mA.