April 10, 1997

Analog multipliers take cube roots

Clayton B Grantham, Burr-Brown Corp, Tucson, AZ

You can configure analog multipliers for more real-time arithmetic operations than just multiplication. Analog multipliers can also perform division, squaring, square rooting, linearizing, percentage computation, modulation, demodulation, and many more analog-signal calculations. The circuit in Figure 1a, using a common multiplier IC, connects two analog multipliers to provide an analog output that is proportional to the cube root of the input voltage.

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 V_OUT²/10, according to the mathematical manipulation of this equation. IC₂ is a modified square rooter. IC₂ transforms its inputs (X₁=V_OUT, Y₂=V_OUT²/10V, and Z₂) into (100´Z₂)^1/3. R₁ and R₂ attenuate V_IN to IC₂ (Z₂) by 100 so the circuit’s complete transfer function is V_OUT=V_IN^1/3.

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)

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