Analog multipliers take cube root
To understand the circuit’s operation, first consider the basic transfer function of a multiplier IC:
To perform a first-order analysis, set SF to 10V and A to infinity (similar to the assumption of an ideal op amp). Thus, for any output voltage, the term in brackets must be zero. You can evaluate the closed-loop transfer function of an individual multiplier IC by substituting the X, Y, and Z inputs into the bracketed term of the equation and setting the equation to zero.
The circuit in Figure 1a configures IC1 as a squarer. This IC multiplies its X1 and Y1 inputs to form an output of VOUT2/10, according to the mathematical manipulation of this equation. IC2 is a modified square rooter. IC2 transforms its inputs (X1=VOUT, Y2=VOUT2/10V, and Z2) into (100×Z2)1/3. R1 and R2 attenuate VIN to IC2 (Z2) by 100 so the circuit’s complete transfer function is VOUT=VIN1/3.
Figure 1 You can connect two analog multipliers to perform the cube-root function (a). A 1-kHz, 19.5V peak-to-peak, triangle-wave input produces 4.28V peak-to-peak output (b).
One application for this circuit is an analog computer to solve the cubic equation y3+py2+qy+r=0. In this case, the answer involves the cube root of several coefficients as found in a common mathematics handbook. Another application is analog control of an astrodynamic positioner in celestial mechanics, in which the radius varies as 1/r3. Analytic-geometry applications include semicubical parabola, hypocycloid of four cusps, Folium of Descartes, and Cissoid of Diocles.
Figure 1b shows the circuit’s performance with a triangle-wave input and cube-root output. Gain and offset errors cause the slightly unsymmetrical output waveform. The 10V peak input scales to a 2.15V peak output. The output and input waveforms are in phase, and the output’s frequency content includes frequencies higher than the input as a result of the squaring of the waveform.
The circuit’s less-than-100V input range includes positive and negative signals, unlike square-root circuits that place unipolar restrictions on the input swing. This cube rooter’s bandwidth is from dc to 200 kHz, and you can direct couple the analog inputs. If necessary, you can trim gain and offset errors for highest accuracy using the manufacturer’s recommended potentiometer connections. (DI #2006)