Square-root-calculating circuits find wide use in instrumentation and measurement systems for such tasks as calculating the rms (root-mean-square) value of an arbitrary waveform, for example. Hence, designers need an effective analog square-root calculator. Because manufacturers do much of the IC fabrication in MOS technology, a MOSFET-based, analog square-root calculator seems appropriate. This Design Idea describes such a circuit, which uses only MOSFETs to provide the square-root function (Figure 1). The design is simple and versatile and can provide the output as the square root of the difference of two voltages.

The circuit uses the nested connection of MOSFETs Q₁ and Q₂. Q₂ works in the saturation region as it is diode-connected, forcing Q₁ to work in the triode region. All other MOSFETs work in the triode region. The first part of the circuit, comprising Q₃, Q₄, Q₅, and Q₆, creating the current I_O1, is basically a MOS-resistive circuit. The essential equation governing the circuit operation is:

where K₁ and K₂ represent the aspect ratios of transistors Q₁ and Q₂, respectively: K_I=(μC_OXW)/2L_I, where I=K₁=K₂. The MOSFETs creating the MOS-resistive circuit and hence responsible for the current creation are identical, having the same aspect ratio and threshold voltage. The current relates to inputs V₁ and V₂, as the following equation shows: I_O1=G(V₁−V₂), where G=2K(V_A−V_B) and represents the conductance of the MOS-resistive circuit—k=(μC_OXW)/2L—of the identical transistors forming the MOS-resistive circuit, and V_A and V_B are control voltages applied to the gate of the MOSFETs that are working in the triode. This approach provides the advantage of voltage controllability of the output; hence, the square-rooting function is voltage-controllable.

The following equation gives the output voltage:

It is evident from this equation that the output voltage, V_O, is the square root of the difference of input voltages V₁ and V₂. If you ground V₂, then the output voltage is proportional to the square root of input voltage V₁. As noted, control voltages V_A and V_B can vary the proportionality constant. Hence, you have devised a new all-MOSFET-based, voltage-controllable analog square-root calculator.

You can test the circuit using a variety of commercially available MOSFETs, such as the 2SK1228, which is available from many sources; the buffer can be a MOSFET-based op-amp buffer, such as the BUF04701 from Texas Instruments. For the operation of the circuit to be in accordance with the output-voltage equation, the four MOSFETs you use to create the MOS-resistive circuit should be identical and should work in the triode region, for which inputs V₁ and V₂ should be less than V_A−V_TH and V_B−–V_TH, respectively. The MOSFETs in the current mirror, Q₇ and Q₈, should be identical, and the diode-connected MOSFETs, Q₁ and Q₂, should be different and have different aspect ratios. You can test the circuit onboard using commercially available ICs, or you can simulate it on a computer using any standard version of Spice. The supply voltage must be in accordance with the selected components.