A translinear circuit is a circuit that carries out its function using the translinear principle. These are current-mode circuits that can be made using transistors that obey an exponential current-voltage characteristic—this includes bipolar junction transistors (BJTs) and CMOS transistors in weak inversion. Translinearity, in a broad sense, is linear dependence of transconductance on current, which occurs in components with exponential current-voltage relationship.
History and etymology The word translinear (TL) was invented by Barrie Gilbert in 1975 to describe circuits that used the exponential current-voltage relation of BJTs. By using this exponential relationship, this class of circuits can implement multiplication, amplification and power-law relationships. When Barrie Gilbert described this class of circuits he also described the translinear principle (TLP) which made the analysis of these circuits possible in a way that the simplified view of BJTs as linear current amplifiers did not allow. TLP was later extended to include other elements that obey an exponential current-voltage relationship (such as CMOS transistors in weak inversion).
The Translinear Principle The translinear principle (TLP) is that in a closed loop containing an even number of translinear elements (TEs) with an equal number of them arranged clockwise and counter-clockwise, the product of the currents through the clockwise TEs equals the product of the currents through the counter-clockwise TEs or ∏ n ϵ C W I n = ∏ n ϵ C C W I n {\displaystyle \,\!\prod _{n\epsilon CW}I_{n}=\prod _{n\epsilon CCW}I_{n}}
The TLP is dependent on the exponential current-voltage relationship of a circuit element. Thus, an ideal TE follows the relationship
I = λ I s e η V / U T {\displaystyle I=\lambda I_{s}e^{\eta V/U_{T}}}
where I s {\displaystyle I_{s}} is a pre-exponential scaling current, λ {\displaystyle \lambda } is a dimensionless multiplier to I s {\displaystyle I_{s}} , η {\displaystyle \eta } is a dimensionless multiplier to the gate-emitter voltage and U T {\displaystyle U_{T}} is the thermal voltage k T / q {\displaystyle kT/q} . In a circuit, TEs are described as either clockwise (CW) or counterclockwise (CCW). If the arrow on the emitter points clockwise, it is considered a CW TE, if it points counterclockwise, it is considered a CCW TE. Consider an example:
By Kirchhoff's voltage law, the voltage around the loop that goes from V r e f {\displaystyle V_{ref}} to V r e f {\displaystyle V_{ref}} must be 0. In other words, the voltage drops must equal the voltage increases. When a loop that only goes through the emitter-gate connections of TEs exists, we call it a translinear loop. Mathematically, this becomes
∑ n ϵ C W V n = ∑ n ϵ C C W V n {\displaystyle \,\!\sum _{n\epsilon CW}V_{n}=\sum _{n\epsilon CCW}V_{n}}
Because of the exponential current-voltage relationship, this implies TLP:
∏ n ϵ C W I n = ∏ n ϵ C C W I n {\displaystyle \,\!\prod _{n\epsilon CW}I_{n}=\prod _{n\epsilon CCW}I_{n}}
this is effectively because current is used as the signal. Because of this, voltage is the log of the signal and addition in the log domain is like multiplication of the original signal (i.e. l o g ( a ) + l o g ( b ) = l o g ( a b ) {\displaystyle log(a)+log(b)=log(ab)} ). The translinear principle is the rule that, in a translinear loop, the product of the currents through the CW TEs is equal to the product of the currents through the CCW TEs. For a detailed derivation of the TLP, and physical interpretations of the parameters in the ideal TE law, please refer to or.
Example Translinear Circuits
Squaring Circuit
According to TLP,
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