Translinear circuits were originally developed for nonlinear analog signal processing, as circuits with inputs and outputs in the form of currents.^{1} Another simplified interpretation of a translinear circuit stems from the fact that the transconductance of a bipolar transistor is linearly proportional to its collector current. Essentially, a translinear circuit works in the current domain rather than the voltage domain, requiring a voltage-to-current converter prior to translinear processing and a current-to-voltage converter following translinear processing. As will be shown, translinear circuits can offer some significant benefits in terms of signal processing.

By working in the current domain, many complex mathematical functions can be implemented that are not possible with the standard linear analog building blocks. Most translinear building blocks have been designed to work in the low MHz range. With advances in highspeed analogto- digital-converter (ADC) and digital-signal-processing (DSP) technologies, however, most complex high-frequency signal processing can be done in the digital domain.

The first step in developing a translinear circuit is to identify the translinear loop. This is a closed loop with an even number of forward biased junctions, arranged so that there are an equal number of clockwise facing and counterclockwise facing polarities. The product of the current densities in the clockwise direction is equal to the product of the current densities in the counterclockwise direction. The most basic form of this is illustrated in * Fig. 1*. The closed loop starts from the emitter of device Q

_{1}, goes through the base to emitter junctions of transistors Q

_{2}, Q

_{3}, and Q

_{4}, and returns to transistor Q

_{1}. The clockwise junctions come from Q

_{2}and Q

_{4}. The counterclockwise junctions come from Q

_{1}and Q

_{3}. According to translinear theory, the current flow into the four transistors are related by Eq. 1.

This can be easily derived by adding the voltage around the closed loop. The sum of the voltage around the loop is zero, so that the Vbe term with its collector current can be expressed by Eq. 2.

This can be readily rewritten into the form of Eqs. 3 (above) and 4

Equation 1 is a key result for the translinear circuit. For example, if I_{c1} is the output and I_{c2} and I_{c4} are the input, by fixing I_{c3}, an analog multiplier is implemented, without any approximation. If I_{c2} and I_{c4} are identical, then a square function is implemented. Going one step further, by swapping the input and the output, a square-root function is implemented. Translinear techniques offer an almost unlimited number of circuit design possibilities.

Translinear circuit techniques can be used to develop an automaticgain- control (AGC) circuit at RF.^{4} One key design parameter in an AGC is to have its gain vary linearly with its control voltage. A voltage-mode AGC typically requires a curvefitting calibration to achieve linear gain control over its dynamic range. The reason for this is because a voltage- mode AGC depends on process parameters such as the transconductance, which is not linear. The gain of the circuit in * Fig. 2* depends on the ratio of I

_{4}/I

_{3}which is highly linear. By doubling the ratio of I

_{4}/I

_{3}, the voltage gain will also be doubled. The expression will be derived below. The translinear loop is formed with the transistors from Q

_{1}to Q

_{4}. The base current is ignored for now. The current flows from each transistor as represented by Eqs. 5-8:

Translinear theory can be applied to these equations to derive Eq. 9:

By substituting Eqs. 5, 6, 7, and 8 into Eq. 9, and multiplying out the terms, the relations of Eq. 10 result (below), which can be simplified to the forms of Eqs. 11 and 12:

The relation of the input voltage to current and output current to voltage can be shown as Eqs. 13 and 14.

By substituting Eqs. 13 and 14 into Eq. 12, the final expression for the output voltage, V_{out}, to the input voltage, V_{in}, can be derived as shown in Eq. 15. If I_{3} is fixed, the gain can be changed by varying I_{4} in a linear manner..

A simplified schematic diagram of the GHz-frequency RMS power detector is shown in Fig. 3.^{5} By applying translinear principles,^{2,3} it is assumed the input voltage has been converted to an input current. The input to the RMS detector is differential. There are two identical translinear loops. Each half of the input feeds one translinear loop. The first translinear loop uses devices Q_{1a} to Q_{4a}. The input current to this loop is (I_{0} + i_{in}). The output current is the current flows through Q_{4a}. The second translinear loop uses transistors Q_{1b} to Q_{4b}. The input current to the second loop is (I_{0} i_{in}). The output current flows through Q_{4b}. The two output currents are added to feed the input of the operational amplifier with a 2I_{0} current offset. This will be explained following the derivation.

With Eq. 1, the two output currents can be shown as Eqs. 16 and 17:

The quantity within the angle brackets represents a time-averaged function. The sum of the two output currents can be derived to be Eq. 18:

Equation 18 is the key result for the RMS power detector. It shows that the output current is proportional to the input current squared, which is the desired relationship for an RMS detector. The scaling factor is 2/I_{0}. There is an offset of 2I_{0} in Eq. 18. At the current output node, the signal is no longer within the GHz frequency range. That is why a 2I_{0} current source is added at the opamp's negative terminal to cancel out the current offset generated by the two translinear loops. The output current can be converted to an output voltage using the opamp in an inverse configuration. Reference 6 shows an input voltage to current converter.

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A micromixer can also be treated as a translinear circuit (see www.mwrf.com for schematic, Fig. 4). First, I_{3} I_{1} = I_{RF} must be proved, using translinear theory.^{7} Because of the current mirror, current I_{3} is the same as I_{2}. The derivation finds expressions for I_{1} and I_{2}. Then it remains to prove that I_{2} I_{1} = I_{RF}.

The translinear loop is formed with Q_{2}, Q_{1}, Q_{4}, and Q_{5}. According to Eq, 1,1the current flowing in those four transistors must meet the conditions shown in Eq. 19:

Equation 19 is then rearranged into standard quadratic equation form (a polynomial equation of the second order) as shown in Eq. 20. The variable to be solved is I_{1} as shown in Eq. 21.

To simplify the expression, use a modulation factor λ = I_{ R}F/2I_{5}. Only the positive solution from Eq. 21 is needed and can be reduced to Eq. 22:

Similarly, current I_{2} can be derived as shown in Eq. 23:

With Eqs. 22 and 23 in place, the value of IRF can be derived easily:

The expression in Eq. 24 demonstrates that I_{RF} can indeed create the differential feeding current to the core switching stage (Q_{6} to Q_{9}).

**REFERENCES**

1. Barrie Gilbert, "Translinear Circuits: A Proposed classifications," Electronics Letters, Vol. 11, No.1, 1975.

2. K. Kumwachara et al., "A dual translinear-based true RMS-to-DC converter," IEEE Transactions on Instrumentation and Measurement, Vol. 47, April 1998.

3. J. Mulder, "Dynamic translinear circuits an overview," Proceedings of the ISIC, Singapore, September 1997.

4. John B. Groe et al., "Translinear variable gain amplifier," United States Patent No. 7,132,889, Nov. 7, 2006.

5. QiZhang Yin et al., "A Translinear-based RF RMS Detector For Embedded Test," ISCAS, May, 2004.

6. R. Caprio, "Precision differential voltage-current convertor," Electronic Letters, Vol. 9, March, 1973.

7. Barrie Gilbert, "The MICROMIXER: A Highly Linear Variant of the Gilbert Mixer Using a Bisymmetric Class-AB Input Stage," IEEE Journal of Solid-State Circuits, September 1997.