When the positive semicycle of the input voltage exceeds the value of V_D + V₁, the diode D enters in the conduction zone. Then the voltage at the output of the operational amplifier, V_o, will be limited to this value, but with the inverted polarity, because the operational amplifier is in the inverter configuration, that is:

See Figure 66-05 for the circuit transfer characteristic graph. Note that for the negative semicycle input, the circuit output has a positive value, obeying the slope of the line whose value is given by - (R_f / R_i) . This inclination is obeyed even when the input voltage is in the positive semicycle until it reaches the value of V_D + V₁. For values higher than this the output voltage is limited to  - (V_D + V₁ ).

We can create a circuit where the output is limited to a positive value by simply reversing the direction of the diode and the battery. We can see in Figure66-06 a typical example of this circuit.

Note that in this circuit there will be no limitation on the entire positive semicycle of the input voltage. In negative semicycle, as long as the input voltage does not reach the value - (V_D + V₁ ) output voltage will obey the ratio - (R_f / R_i). Therefore the output voltage will be given by:

When the input voltage reaches the value - (V_D + V₁ ) output voltage will be limited to the value + (V_D + V₁ ). Therefore the output voltage will be given by:

See Figure 66-07 for the circuit transfer characteristic graph. Note that for the positive semicycle input, the circuit output has a negative value, obeying the slope of the line whose value is given by - (R_f / R_i) . This inclination is obeyed even when the input voltage is in the negative semicycle until it reaches the value of - (V_D + V₁ ). For values lower than this the output voltage is limited to  + (V_D + V₁ ).

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Evidently, by joining the last two circuits studied, we can elaborate a limiting circuit double , as shown in the Figure 66-08.

Based on what we have already studied, we can easily see that the upper branch is responsible for the operation in the positive semicycle of the input voltage, while the lower branch is responsible for the operation in the negative semicycle of the input voltage. In this case, the circuit works exactly as stated above. So, let's not repeat the explanation. Of course the circuit transfer characteristic graph changes slightly and we present it in the Figure 66-09.

Notably we have three possible situations for the output voltage as shown below.

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The circuits studied in this item use batteries to achieve the desired limiting point. The use of batteries is only a didactic resource. In practice, we replace it with a resistive network that meets the project objectives. Thus, in Figure 66-10 we can see a circuit that uses resistive dividers in order to achieve the limiting points. Let's look at how this circuit works.

As soon as the input voltage starts the positive semicycle, the output voltage starts its negative semicycle because the operational amplifier is in inverter configuration. And both diodes are in cut. As long as the input voltage does not exceed the threshold value for conducting D₁, the output voltage follows the circuit voltage gain, or V_o = - (R_f / R_i ) V_i. To D₁ enter in the conducting zone, the output voltage must reach a negative value such that V_a is - 0.7 V or more since the anode in D₁ is connected to ground via positive input from the operational amplifier. Therefore, to determine the value of V_a we can use the superposition theorem . So applying this theorem, we get:

To understand how we get these equations, we can rely on the circuit shown in Figure 66-11, replacing the voltage of V_D by V_a.

Now we need to determine to what value of V_o the limiting circuit will start acting. To do so, let us base on Figure 66-11 where we have the output circuit scheme for the boundary condition where D₁ will be conducting . In this case, the diode will not circulate electric current, as indicated in the figure above and, therefore, the current that circulates for R₁ is the same as that circulating for R₂. Based on this information, we can write the equations of the two meshes to determine I, or:

So the limit for diode conduction, when V_o goes through the negative semicycle, let's call it V_o^-. After an algebraic work in the equation above, we come to:

Note that when V_o reaches the required value for diode D₁ enter in the conducting zone, the voltage at point a is fixed at V_a = - 0.7 V. And since V is also constant, so the current over R₁ remains constant. Therefore, an increase in current through the diode must circulate through R₂, generating an effect that R₂ acts in parallel with R_f and the incremental gain is given by:

And when the input voltage is in the negative semicycle, the circuit transfer characteristic can be found identically to the one above. As long as the input voltage does not exceed the threshold value for conducting D₂ , the output voltage follows the circuit voltage gain, that is, V_o = - (R_f / R_i ) V_i. To calculate the value of V_b we employ the superposition method as previously done. Thus we find:

To find this equation, we rely on the circuit shown in Figure 66-12, with the removal of the battery V_D from the circuit.

To determine the value of V_o where the limiter circuit starts acting, we repeat the previous process using the same considerations made in the opportunity. Note that when the diode D₂ starts conduction, point b will have its voltage set at + 0.7V . So based on the circuit of Figure 66-12 we can write:

At the limit for diode conduction, when V_o traverses the positive semicycle, we will call V_o ⁺ . After an algebraic work in the equation above, we come to:

Here - V is also constant, so the current over R₄ remains constant. Therefore, an increase in current through the diode must circulate through R₃, producing an effect as if R₃ acts in parallel with R_f and the incremental gain is given by:

Finally we will present the graph of the circuit transfer characteristic. See the Figure 66-13 .