Methodology for Designing Voltage Generator Circuits with Shunt Regulators

Basic Circuit

The basic circuit considered is as follows:

The circuit is powered by a voltage \(v_g\) to generate the voltage \(v_s\), which is the regulated voltage. The regulator \(R\) is a shunt regulator.

The characteristics of the circuit elements are as follows:

  • Shunt voltage regulator, \(R\)
    • has a reference voltage \(v_r\) at the regulation pin
    • has a current \(I_\text{KA}\) with an operating range between \(I_\text{KA,min}\) and \(I_\text{KA,max}\).
    • has a current \(i_r\) at the regulation terminal, defined as positive towards the output and much lower than \(I_\text{KA}\)
  • Supply voltage, \(v_g\)
    • has a range between \(v_\text{g,min}\) and \(v_\text{g,max}\).
  • Load
    • draws a current \(I_\text{L}\) that varies between \(I_\text{L,min}\) and \(I_\text{L,max}\).

Operation

The key point of the circuit is the resistor \(R_s\), which establishes the potential difference between the input voltage \(v_g\) (unregulated) and the regulated voltage \(v_s\). The potential difference is equal to the product of the resistor value and the current flowing through it. The function of the shunt regulator in this circuit is to act as a dependent current source, so as to make the necessary current flow through \(R_s\) to ensure the value of \(v_s\).

From this, a first necessary condition results:

The range of values of \(I_\text{L}\) cannot be greater than the range of values of \(I_\text{KA}\). \(I_\text{L,max} - I_\text{L,min} \leq I_\text{KA,max} - I_\text{KA,min}\)

Proof by contradiction: if the variation of \(I_\text{L}\) is greater than the variation of \(I_\text{KA}\), then, when the load current varies from one extreme to the other (from \(I_\text{L,min}\) to \(I_\text{L,max}\), for example), this variation must be compensated by a variation of opposite sign in \(I_\text{KA}\). If the range of \(I_\text{KA}\) is not sufficient to cover this variation (and if it is smaller, it is not), then the circuit loses the ability to regulate the voltage \(v_s\), as it cannot keep the value of the current \(i_s\) through \(R_s\) constant.

Circuit Analysis

The current \(I_s\) flowing through \(R_s\) is given by:

\[I_s = I_\text{KA} + I_F + I_L\]

Knowing that

\[v_g - v_s = R_s I_s\]

we then have:

\[v_g - v_s = R_s \left( I_\text{KA} + I_F + I_L \right)\]

or

\[v_g - v_s = R_s \left( I_\text{KA} + \frac{v_s}{R_1 + R_2} + I_L \right)\]

Neglecting the effect of \(i_r\), we have that, in the previous equation, there are a set of values that are fixed in the correct operation of the circuit (\(v_s\) and \(I_F\)) and other quantities that may vary (\(v_g\) and \(I_L\)). The effect of variations in the latter will be compensated by variations in \(I_\text{KA}\):

  • variations in \(I_L\) will be compensated by symmetrical variations in \(I_\text{KA}\) to keep the current \(I_S\) constant, such that \(\Delta I_L = - \Delta I_\textrm{KA}\);
  • variations in \(v_g\) will be compensated by variations in \(I_\text{KA}\), such that \(\Delta v_g = R_S \Delta I_\text{KA} \).

Rearranging the equation, we have:

\[\frac{v_g - v_s}{R_s} = I_\text{KA} + I_F + I_L\]

Extreme Situation 1

Now considering one of the extreme situations, when \(v_g = v_{g\text{min}}\). The situation is described by

\[\frac{v_{g\text{,min}} - v_s}{R_s} = I_\text{KA} + I_F + I_L\]

Under these conditions, the term \( I_\text{KA} + I_L\) assumes the minimum value. However, it is necessary that:

\[\text{min} \left\{ I_\text{KA} + I_L \right\} \geq I_\text{L,max} + I_\text{KA, min}\]

That is, in the condition where the current through \(R_S\) is minimum, it must be sufficient to supply current to the load (in any condition) and guarantee at least the minimum value of current \(I_\text{KA}\).

From this, it follows:

\[\frac{v_{g\text{,min}} - v_s}{R_s} - I_F \geq I_\text{L,max} + I_\text{KA, min}\]

Solving for \(R_S\):

\[R_s \leq \frac{v_{g\text{,min}} - v_s}{I_\text{L,max} + I_\text{KA, min} + I_F}\]

Extreme Situation 2

Now considering the situation where \(v_g = v_{g\text{max}}\). The situation is described by

\[\frac{v_{g\text{,max}} - v_s}{R_s} = I_\text{KA} + I_F + I_L\]

Under these conditions, the term \( I_\text{KA} + I_L\) assumes the maximum value. In these circumstances, it is necessary to ensure that, when the load draws the minimum and this current is absorbed by the regulator, the current in the regulator does not exceed \(I_\text{KA, max}\), that is:

\[\text{max} \left\{ I_\text{KA} + I_L \right\} \leq I_\text{L,min} + I_\text{KA, max}\]

In this circumstance, we have:

\[\frac{v_{g\text{,max}} - v_s}{R_s} - I_F \leq I_\text{L,min} + I_\text{KA, max}\]

Solving for \(R_S\), we obtain:

\[R_s \geq \frac{v_{g\text{,max}} - v_s}{I_\text{L,min} + I_\text{KA, max} + I_F} \tag{1}\]

Combining the Two Situations

Now combining the two previous results, we have the condition that the resistor \(R_S\) must meet for regulation in the circuit:

\[\frac{v_{g\text{,max}} - v_s}{I_\text{L,min} + I_\text{KA, max} + I_F} \leq R_s \leq \frac{v_{g\text{,min}} - v_s}{I_\text{L,max} + I_\text{KA, min} + I_F}\]