Translinear Circuits

Transcription

1 Translinear Circuits Bradley A. Minch Mixed Analog-Digital VLSI Circuits and Systems Lab Franklin W. Olin College of Engineering Needham, MA April 2, 2009

2 Translinear Circuits: What s in a Name? 1 In 1975, Barrie Gilbert coined the term translinear to describe a class of circuits whose large-signal behavior hinges both on the precise exponential I/V relationship of the bipolar transistor and on the intimate thermal contact and close matching of monolithically integrated devices. The word translinear refers to the exponential I/V characteristic of the bipolar transistor its transconductance is linear in its collector current: I C = I s e V BE/U T = g m = I C = I s e V BE/U T 1 V B }{{} = I C. U T U T I C Gilbert also meant the word translinear to refer to circuit analysis and design principles that bridge the gap between the familiar territory of linear circuits and the uncharted domain of nonlinear circuits.

3 Translinear Circuits: What s in a Name? 1 In 1975, Barrie Gilbert coined the term translinear to describe a class of circuits whose large-signal behavior hinges both on the precise exponential I/V relationship of the bipolar transistor and on the intimate thermal contact and close matching of monolithically integrated devices. The word translinear refers to the exponential I/V characteristic of the bipolar transistor its transconductance is linear in its collector current: I C = I s e V BE/U T = g m = I C = I s e V BE/U T 1 V B }{{} = I C. U T U T I C Gilbert also meant the word translinear to refer to circuit analysis and design principles that bridge the gap between the familiar territory of linear circuits and the uncharted domain of nonlinear circuits.

4 Translinear Circuits: What s in a Name? 1 In 1975, Barrie Gilbert coined the term translinear to describe a class of circuits whose large-signal behavior hinges both on the precise exponential I/V relationship of the bipolar transistor and on the intimate thermal contact and close matching of monolithically integrated devices. The word translinear refers to the exponential I/V characteristic of the bipolar transistor its transconductance is linear in its collector current: I C = I s e V BE/U T = g m = I C = I s e V BE/U T 1 V B }{{} = I C. U T U T I C Gilbert also meant the word translinear to refer to circuit analysis and design principles that bridge the gap between the familiar territory of linear circuits and the uncharted domain of nonlinear circuits.

5 Gummel Plot of a Forward-Active Bipolar Transistor 2

6 Translinearity of the Forward-Active Bipolar Transistor 3 g m = I C V B = I C U T δi C g m δv B

7 The Translinear Principle 4 Consider a closed loop of base-emitter junctions of four closely matched npn bipolar transistors biased in the forward-active region and operating at the same temperature. Kirchhoff s voltage law (KVL) implies that V 1 + V 2 = V 3 + V 4 U T log I s + U T log I 2 I s = U T log I 3 I s + U T log I 4 I s log I 2 Is 2 }{{} I 2 CCW = log I 3I 4 Is 2 = }{{} I 3 I 4. CW This result is a particular case of Gilbert s translinear principle (TLP): The product of the clockwise currents is equal to the product of the counterclockwise currents.

8 Static Translinear Circuits: Geometric Mean 5 We neglect both base currents (i.e., β F = ) and the Early effect, and we assume that all transistors operate in their forward-active regions. Then, we have that TLP = I x I y = I 2 z = I z = I x I y.

9 Static Translinear Circuits: Geometric Mean 5 I z = I x I y

10 Static Translinear Circuits: Geometric Mean 5 I z = I x I y

11 Static Translinear Circuits: Squaring/Reciprocal 6 Again, we neglect both base currents (i.e., β F = ) and the Early effect, and we assume that all transistors operate in their forward-active regions. Then, we have that TLP = I 2 x = I y I z = I z = I2 x I y.

12 Static Translinear Circuits: Squaring/Reciprocal 6 I z = I2 x I y

13 Static Translinear Circuits: Squaring/Reciprocal 6 I z = I2 x I y

14 Static Translinear Circuits: Pythagorator 7 Again, we neglect both base currents and the Early effect, and we assume that all transistors operate in their forward-active regions. Then, we have that TLP 1 = I 2 x = I z1 I z = I z1 = I2 x I z TLP 2 = I 2 y = I z2 I z = I z2 = I2 y I z KCL = I z = I z1 + I z2 = I2 x I z + I2 y = I 2 z = I 2 x + I 2 y = I z = I z I 2 x + I 2 y. This circuit is called the pythagorator.

15 Dynamic Translinear Circuits: First-Order Low-Pass Filter 8 Next, consider the dynamic translinear circuit shown below, comprising a translinear loop and a capacitor. We shall again neglect base currents and the Early effect. We also assume that all transistors operate in the forward-active region. Then, we have that TLP = I τ I x = I p I y I c = C dv dt = C d dt ( U T log I ) y I s KCL = I c +I τ = I p = CU T I y = I p = I τi x I y = CU T I y di y dt di y dt +I τ = I τi x I y = CU T di y } I {{ τ } dt + I y = I x = τ di y dt + I y = I x. τ This circuit is a first-order log-domain filter.

16 Dynamic Translinear Circuits: RMS-to-DC Converter 9 TLP = I 2 wi τ = I p I 2 z = I p = I τi 2 w I 2 z I c = C d dt ( 2U T log I ) z I s =2 CU T I z di z dt KCL = I c + I τ = I p = 2CU T I z di z dt + I τ = I τi 2 w I 2 z = CU T } I {{ τ } τ 2 I z di z dt + I2 z = I 2 w

17 Dynamic Translinear Circuits: RMS-to-DC Converter 9 = τ ( ) di z 2 I z dt + I 2 z = I 2 w = τ d ( ) I 2 dt z + I 2 z = Iw 2 = τ d ( ) I 2 z + I2 z = I2 w dt }{{} }{{} }{{} I y I y I x I z = I y }{{} root τ di y dt + I y = I x }{{} mean I x = I2 w }{{ } square

18 Why Translinear Circuits? 10

19 Why Translinear Circuits? 10 Translinear circuits are universal. Conjecture: In principle, we can realize any system whose description we can write down as a nonlinear ODE with time as its independent variable as a dynamic translinear circuit. Translinear circuits are synthesizable via highly structured methods. They should be very amenable to the development both of CAD tools and of reconfigurable FPAA architectures for rapid prototyping and deployment of translinear analog signal processing systems. Translinear circuits are fundamentally large-signal circuits. Linear dynamic translinear circuits are linear because of device nonlinearities rather than in spite of them. Translinear circuits are tunable electronically over a wide dynamic range of parameters (e.g., gains, corner frequencies, quality factors). Translinear circuits are robust. Carefully designed translinear circuits are temperature insensitive and do not depend on device or technology parameters.

20 Why Translinear Circuits? 10 Translinear circuits are universal. Conjecture: In principle, we can realize any system whose description we can write down as a nonlinear ODE with time as its independent variable as a dynamic translinear circuit. Translinear circuits are synthesizable via highly structured methods. They should be very amenable to the development both of CAD tools and of reconfigurable FPAA architectures for rapid prototyping and deployment of translinear analog signal processing systems. Translinear circuits are fundamentally large-signal circuits. Linear dynamic translinear circuits are linear because of device nonlinearities rather than in spite of them. Translinear circuits are tunable electronically over a wide dynamic range of parameters (e.g., gains, corner frequencies, quality factors). Translinear circuits are robust. Carefully designed translinear circuits are temperature insensitive and do not depend on device or technology parameters.

21 Why Translinear Circuits? 10 Translinear circuits are universal. Conjecture: In principle, we can realize any system whose description we can write down as a nonlinear ODE with time as its independent variable as a dynamic translinear circuit. Translinear circuits are synthesizable via highly structured methods. They should be very amenable to the development both of CAD tools and of reconfigurable FPAA architectures for rapid prototyping and deployment of translinear analog signal processing systems. Translinear circuits are fundamentally large-signal circuits. Linear dynamic translinear circuits are linear because of device nonlinearities rather than in spite of them. Translinear circuits are tunable electronically over a wide dynamic range of parameters (e.g., gains, corner frequencies, quality factors). Translinear circuits are robust. Carefully designed translinear circuits are temperature insensitive and do not depend on device or technology parameters.

22 Why Translinear Circuits? 10 Translinear circuits are universal. Conjecture: In principle, we can realize any system whose description we can write down as a nonlinear ODE with time as its independent variable as a dynamic translinear circuit. Translinear circuits are synthesizable via highly structured methods. They should be very amenable to the development both of CAD tools and of reconfigurable FPAA architectures for rapid prototyping and deployment of translinear analog signal processing systems. Translinear circuits are fundamentally large-signal circuits. Linear dynamic translinear circuits are linear because of device nonlinearities rather than in spite of them. Translinear circuits are tunable electronically over a wide dynamic range of parameters (e.g., gains, corner frequencies, quality factors). Translinear circuits are robust. Carefully designed translinear circuits are temperature insensitive and do not depend on device or technology parameters.

23 Why Translinear Circuits? 10 Translinear circuits are universal. Conjecture: In principle, we can realize any system whose description we can write down as a nonlinear ODE with time as its independent variable as a dynamic translinear circuit. Translinear circuits are synthesizable via highly structured methods. They should be very amenable to the development both of CAD tools and of reconfigurable FPAA architectures for rapid prototyping and deployment of translinear analog signal processing systems. Translinear circuits are fundamentally large-signal circuits. Linear dynamic translinear circuits are linear because of device nonlinearities rather than in spite of them. Translinear circuits are tunable electronically over a wide dynamic range of parameters (e.g., gains, corner frequencies, quality factors). Translinear circuits are robust. Carefully designed translinear circuits are temperature insensitive and do not depend on device or technology parameters.

24 Simple EKV Model of the Saturated nmos Transistor 11 We model the saturation current of an nmos transistor by I sat = SI s log 2( 1+e (κ(v G V T0 ) V S )/2U T ) SI s e (κ(v G V T0 ) V S )/U T, κ(v G V T0 ) V S < 0 SI s 4U 2 T (κ (V G V T0 ) V S ) 2, κ(v G V T0 ) V S > 0, where U T = kt q, S = W L, I s = 2μC oxut 2 C ox, and κ =. κ C ox + C dep Weak inversion operation corresponds to I sat SI s, moderate inversion operation corresponds to I sat SI s, and strong inversion operation to I sat SI s. Note that SI s is approximately twice the saturation current at threshold.

25 Saturation Current of an nmos Transistor 12

26 Translinearity of the Saturated nmos Transistor 13 g m = I sat V G = κi sat U T,I sat SI s δi sat g m δv G

27 Weak Inversion is Suited to Audio Signal Processing 14 For integrated continous-time filters, typically f c = g m 2πC. On-chip capacitors are typically on the order of 1 pf or of 10 pf. If we choose a reasonable value for C, sayc = 5/π pf 1.59 pf, then we find that weak inversion maps onto the audio band.

28 Weak Inversion is Suited to Audio Signal Processing 14 For integrated continous-time filters, typically f c = g m 2πC. On-chip capacitors are typically on the order of 1 pf or of 10 pf. If we choose a reasonable value for C, sayc = 5/π pf 1.59 pf, then we find that weak inversion maps onto the audio band.

29 Weak-Inversion MOS Translinear Principle When designing weak-inversion MOS translinear circuits, if we restrict ourselves to using translinear loops that alternate between clockwise and counterclockwise elements, we obtain Gilbert s original TLP, with no dependence on the body effect (i.e., κ). 15 stacked loop alternating loop TLP: I κ 2 = I κ 3 I 4 TLP: I 3 = I 2 I 4 This restriction does not limit the class of systems that we can implement. However, designs based on alternating loops generally consume more current but operate on a lower power supply voltage than do designs based on stacked loops.

30 Static Translinear Circuits: Squaring/Reciprocal 16 TLP = I 2 x = I y I z = I z = I2 x I y

31 Static Translinear Circuits: Squaring/Reciprocal 16 I z = I2 x I y

32 Static Translinear Circuits: Squaring/Reciprocal 16 I z = I2 x I y

33 Static Translinear Circuit Synthesis: Pythagorator 17 Synthesize a two-dimensional vector-magnitude circuit implementing r = x 2 + y 2, where x>0 and y>0. We represent each signal as a ratio of a signal current to the unit current: x I x, y I y, and r I r. We substitute these into the original equation and rearrange to obtain (Ix I r = ) 2 + ( Iy ) 2 = I 2 r = I 2 x + I 2 y = I r = I2 x I r }{{} I r1 + I2 y I r }{{} I r2

34 Static Translinear Circuit Synthesis: Pythagorator 17 Synthesize a two-dimensional vector-magnitude circuit implementing r = x 2 + y 2, where x>0 and y>0. We represent each signal as a ratio of a signal current to the unit current: x I x, y I y, and r I r. We substitute these into the original equation and rearrange to obtain (Ix I r = ) 2 + ( Iy ) 2 = I 2 r = I 2 x + I 2 y = I r = I2 x I r }{{} I r1 + I2 y I r }{{} I r2

35 Static Translinear Circuit Synthesis: Pythagorator 17 Synthesize a two-dimensional vector-magnitude circuit implementing r = x 2 + y 2, where x>0 and y>0. We represent each signal as a ratio of a signal current to the unit current: x I x, y I y, and r I r. We substitute these into the original equation and rearrange to obtain (Ix I r = ) 2 + ( Iy ) 2 = I 2 r = I 2 x + I 2 y = I r = I2 x I r }{{} I r1 + I2 y I r }{{} I r2

36 Static Translinear Circuit Synthesis: Pythagorator 17 Synthesize a two-dimensional vector-magnitude circuit implementing r = x 2 + y 2, where x>0 and y>0. We represent each signal as a ratio of a signal current to the unit current: x I x, y I y, and r I r. We substitute these into the original equation and rearrange to obtain (Ix I r = ) 2 + ( Iy ) 2 = I 2 r = I 2 x + I 2 y = I r = I2 x I r }{{} I r1 + I2 y I r }{{} I r2

37 Static Translinear Circuit Synthesis: Pythagorator 17 Synthesize a two-dimensional vector-magnitude circuit implementing r = x 2 + y 2, where x>0 and y>0. We represent each signal as a ratio of a signal current to the unit current: x I x, y I y, and r I r. We substitute these into the original equation and rearrange to obtain (Ix I r = ) 2 + ( Iy ) 2 = I 2 r = I 2 x + I 2 y = I r = I2 x I r }{{} I r1 + I2 y I r }{{} I r2

38 Static Translinear Circuit Synthesis: Pythagorator 17 Synthesize a two-dimensional vector-magnitude circuit implementing r = x 2 + y 2, where x>0 and y>0. We represent each signal as a ratio of a signal current to the unit current: x I x, y I y, and r I r. We substitute these into the original equation and rearrange to obtain (Ix I r = ) 2 + ( Iy ) 2 = I 2 r = I 2 x + I 2 y = I r = I2 x I r }{{} I r1 + I2 y I r }{{} I r2

39 Static Translinear Circuit Synthesis: Pythagorator 18 TLP: I r1 I r = I 2 x I r2 I r = I 2 y KCL: I r = I r1 +I r2

40 Static Translinear Circuit Synthesis: Pythagorator 18 TLP: I r1 I r = I 2 x I r2 I r = I 2 y KCL: I r = I r1 +I r2

41 Static Translinear Circuit Synthesis: Pythagorator 18 TLP: I r1 I r = I 2 x I r2 I r = I 2 y KCL: I r = I r1 +I r2

42 Static Translinear Circuit Synthesis: Pythagorator 18 TLP: I r1 I r = I 2 x I r2 I r = I 2 y KCL: I r = I r1 +I r2

43 Static Translinear Circuit Synthesis: Pythagorator 18 TLP: I r1 I r = I 2 x I r2 I r = I 2 y KCL: I r = I r1 +I r2

44 Static Translinear Circuit Synthesis: Pythagorator 18 TLP: I r1 I r = I 2 x I r2 I r = I 2 y KCL: I r = I r1 +I r2

45 Static Translinear Circuit Synthesis: Vector Normalizer 19 Synthesize a two-dimensional vector-normalization circuit implementing u = x x2 + y 2 and v = y where x>0 and y>0. x2 + y2, Each equation shares r x 2 + y 2, which we can use to decompose the system as u = x r, v = y r, and r = x 2 + y 2, where x>0 and y>0. We represent each signal as a ratio of a signal current to the unit current: x I x, y I y, u I u, v I v, and r I r.

46 Static Translinear Circuit Synthesis: Vector Normalizer 19 Synthesize a two-dimensional vector-normalization circuit implementing u = x x2 + y 2 and v = y where x>0 and y>0. x2 + y2, Each equation shares r x 2 + y 2, which we can use to decompose the system as u = x r, v = y r, and r = x 2 + y 2, where x>0 and y>0. We represent each signal as a ratio of a signal current to the unit current: x I x, y I y, u I u, v I v, and r I r.

47 Static Translinear Circuit Synthesis: Vector Normalizer 19 Synthesize a two-dimensional vector-normalization circuit implementing u = x x2 + y 2 and v = y where x>0 and y>0. x2 + y2, Each equation shares r x 2 + y 2, which we can use to decompose the system as u = x r, v = y r, and r = x 2 + y 2, where x>0 and y>0. We represent each signal as a ratio of a signal current to the unit current: x I x, y I y, u I u, v I v, and r I r.

48 Static Translinear Circuit Synthesis: Vector Normalizer 19 We substitute these into the original equations and rearrange to obtain I u = I x/ I r / and I v = I y/ I r / = I u I r = I x and I v I r = I y

49 Static Translinear Circuit Synthesis: Vector Normalizer 19 We substitute these into the original equations and rearrange to obtain I u = I x/ I r / and I v = I y/ I r / = I u I r = I x and I v I r = I y

50 Static Translinear Circuit Synthesis: Vector Normalizer 19 We substitute these into the original equations and rearrange to obtain and I u = I x/ I r / and I v = I y/ I r / = I u I r = I x and I v I r = I y (Ix I r = ) 2 + ( Iy ) 2 = I 2 r = I 2 x + I 2 y = I r = I2 x I r }{{} I r1 + I2 y I r }{{} I r2

51 Static Translinear Circuit Synthesis: Vector Normalizer 19 We substitute these into the original equations and rearrange to obtain and I u = I x/ I r / and I v = I y/ I r / = I u I r = I x and I v I r = I y (Ix I r = ) 2 + ( Iy ) 2 = I 2 r = I 2 x + I 2 y = I r = I2 x I r }{{} I r1 + I2 y I r }{{} I r2

52 Static Translinear Circuit Synthesis: Vector Normalizer 19 We substitute these into the original equations and rearrange to obtain and I u = I x/ I r / and I v = I y/ I r / = I u I r = I x and I v I r = I y (Ix I r = ) 2 + ( Iy ) 2 = I 2 r = I 2 x + I 2 y = I r = I2 x I r }{{} I r1 + I2 y I r }{{} I r2

53 Static Translinear Circuit Synthesis: Vector Normalizer 19 We substitute these into the original equations and rearrange to obtain and I u = I x/ I r / and I v = I y/ I r / = I u I r = I x and I v I r = I y (Ix I r = ) 2 + ( Iy ) 2 = I 2 r = I 2 x + I 2 y = I r = I2 x I r }{{} I r1 + I2 y I r }{{} I r2

54 Static Translinear Circuit Synthesis: Vector Normalizer 19 TLP: I r1 I r = I 2 x I r2 I r = I 2 y I u I r = I x I v I r = I y KCL: I r = I r1 +I r2

55 Static Translinear Circuit Synthesis: Vector Normalizer 19 TLP: I r1 I r = I 2 x I r2 I r = I 2 y I u I r = I x I v I r = I y KCL: I r = I r1 +I r2

56 Static Translinear Circuit Synthesis: Vector Normalizer 19 TLP: I r1 I r = I 2 x I r2 I r = I 2 y I u I r = I x I v I r = I y KCL: I r = I r1 +I r2

57 Static Translinear Circuit Synthesis: Vector Normalizer 19 TLP: I r1 I r = I 2 x I r2 I r = I 2 y I u I r = I x I v I r = I y KCL: I r = I r1 +I r2

58 Static Translinear Circuit Synthesis: Vector Normalizer 19 TLP: I r1 I r = I 2 x I r2 I r = I 2 y I u I r = I x I v I r = I y KCL: I r = I r1 +I r2

59 Static Translinear Circuit Synthesis: Vector Normalizer 19 TLP: I r1 I r = I 2 x I r2 I r = I 2 y I u I r = I x I v I r = I y KCL: I r = I r1 +I r2

60 Static Translinear Circuit Synthesis: Vector Normalizer 19 TLP: I r1 I r = I 2 x I r2 I r = I 2 y I u I r = I x I v I r = I y KCL: I r = I r1 +I r2

61 Static Translinear Circuit Synthesis: Vector Normalizer 19 TLP: I r1 I r = I 2 x I r2 I r = I 2 y I u I r = I x I v I r = I y KCL: I r = I r1 +I r2

62 Static Translinear Circuit Synthesis: Vector Normalizer 19 TLP: I r1 I r = I 2 x I r2 I r = I 2 y I u I r = I x I v I r = I y KCL: I r = I r1 +I r2

63 Static Translinear Circuit Synthesis: Vector Normalizer 19 TLP: I r1 I r = I 2 x I r2 I r = I 2 y I u I r = I x I v I r = I y KCL: I r = I r1 +I r2

64 Static Translinear Circuit Synthesis: Vector Normalizer 19 TLP: I r1 I r = I 2 x I r2 I r = I 2 y I u I r = I x I v I r = I y KCL: I r = I r1 +I r2

65 Dynamic Translinear Circuit Synthesis: Output Structures 20 noninverting inverting I n = I τ e (V n V 0 )/U T I n V n = I n U T I n = I τ e κ(v 0 V n )/U T I n V n = κi n U T

66 Dynamic Translinear Circuit Synthesis: First-Order LPF 21 Synthesize a first-order low-pass filter described by τ dy dt + y = x, where x>0. We represent each signal as a ratio of a signal current to the unit current: Substituting these into the ODE, we obtain x I x and y I y. τ d dt ( Iy ) + I y = I x = τ di y dt + I y = I x.

67 Dynamic Translinear Circuit Synthesis: First-Order LPF 21 Synthesize a first-order low-pass filter described by τ dy dt + y = x, where x>0. We represent each signal as a ratio of a signal current to the unit current: Substituting these into the ODE, we obtain x I x and y I y. τ d dt ( Iy ) + I y = I x = τ di y dt + I y = I x.

68 Dynamic Translinear Circuit Synthesis: First-Order LPF 21 Synthesize a first-order low-pass filter described by τ dy dt + y = x, where x>0. We represent each signal as a ratio of a signal current to the unit current: Substituting these into the ODE, we obtain x I x and y I y. τ d dt ( Iy ) + I y = I x = τ di y dt + I y = I x.

69 Dynamic Translinear Circuit Synthesis: First-Order LPF 21 Synthesize a first-order low-pass filter described by τ dy dt + y = x, where x>0. We represent each signal as a ratio of a signal current to the unit current: Substituting these into the ODE, we obtain x I x and y I y. τ d dt ( Iy ) + I y = I x = τ di y dt + I y = I x.

70 Dynamic Translinear Circuit Synthesis: First-Order LPF 21 To implement the time derivative, we introduce a log-compressed voltage state variable, V y. Using the chain rule, we can express the preceding equation as τ I y V y dv y dt + I y = I x = τ ( κ ) dvy I y U T dt + I y = I x = κτ U T dv y dt +1=I x I y = κτ CU T }{{} 1/I τ C dv y +1 = }{{ dt} I c I x I y = I c +1= I x = I τ I c = I τi x. I τ I y I }{{} y I p

71 Dynamic Translinear Circuit Synthesis: First-Order LPF 21 To implement the time derivative, we introduce a log-compressed voltage state variable, V y. Using the chain rule, we can express the preceding equation as τ I y V y dv y dt + I y = I x = τ ( κ ) dvy I y U T dt + I y = I x = κτ U T dv y dt +1=I x I y = κτ CU T }{{} 1/I τ C dv y +1 = }{{ dt} I c I x I y = I c +1= I x = I τ I c = I τi x. I τ I y I }{{} y I p

72 Dynamic Translinear Circuit Synthesis: First-Order LPF 21 To implement the time derivative, we introduce a log-compressed voltage state variable, V y. Using the chain rule, we can express the preceding equation as τ I y V y dv y dt + I y = I x = τ ( κ ) dvy I y U T dt + I y = I x = κτ U T dv y dt +1=I x I y = κτ CU T }{{} 1/I τ C dv y +1 = }{{ dt} I c I x I y = I c +1= I x = I τ I c = I τi x. I τ I y I }{{} y I p

73 Dynamic Translinear Circuit Synthesis: First-Order LPF 21 To implement the time derivative, we introduce a log-compressed voltage state variable, V y. Using the chain rule, we can express the preceding equation as τ I y V y dv y dt + I y = I x = τ ( κ ) dvy I y U T dt + I y = I x = κτ U T dv y dt +1=I x I y = κτ CU T }{{} 1/I τ C dv y +1 = }{{ dt} I c I x I y = I c +1= I x = I τ I c = I τi x. I τ I y I }{{} y I p

74 Dynamic Translinear Circuit Synthesis: First-Order LPF 21 To implement the time derivative, we introduce a log-compressed voltage state variable, V y. Using the chain rule, we can express the preceding equation as τ I y V y dv y dt + I y = I x = τ ( κ ) dvy I y U T dt + I y = I x = κτ U T dv y dt +1=I x I y = κτ CU T }{{} 1/I τ C dv y +1 = }{{ dt} I c I x I y = I c +1= I x = I τ I c = I τi x. I τ I y I }{{} y I p

75 Dynamic Translinear Circuit Synthesis: First-Order LPF 21 To implement the time derivative, we introduce a log-compressed voltage state variable, V y. Using the chain rule, we can express the preceding equation as τ I y V y dv y dt + I y = I x = τ ( κ ) dvy I y U T dt + I y = I x = κτ U T dv y dt +1=I x I y = κτ CU T }{{} 1/I τ C dv y +1 = }{{ dt} I c I x I y = I c +1= I x = I τ I c = I τi x. I τ I y I }{{} y I p

76 Dynamic Translinear Circuit Synthesis: First-Order LPF 21 To implement the time derivative, we introduce a log-compressed voltage state variable, V y. Using the chain rule, we can express the preceding equation as τ I y V y dv y dt + I y = I x = τ ( κ ) dvy I y U T dt + I y = I x = κτ U T dv y dt +1=I x I y = κτ CU T }{{} 1/I τ C dv y +1 = }{{ dt} I c I x I y = I c +1= I x = I τ I c = I τi x. I τ I y I }{{} y I p

77 Dynamic Translinear Circuit Synthesis: First-Order LPF 21 To implement the time derivative, we introduce a log-compressed voltage state variable, V y. Using the chain rule, we can express the preceding equation as τ I y V y dv y dt + I y = I x = τ ( κ ) dvy I y U T dt + I y = I x = κτ U T dv y dt +1=I x I y = κτ CU T }{{} 1/I τ C dv y +1 = }{{ dt} I c I x I y = I c +1= I x = I τ I c = I τi x. I τ I y I }{{} y I p

78 Dynamic Translinear Circuit Synthesis: First-Order LPF 21 TLP: I p I y = I x I τ KCL: I c + I p = I τ

79 Dynamic Translinear Circuit Synthesis: First-Order LPF 21 TLP: I p I y = I x I τ KCL: I c + I p = I τ

80 Dynamic Translinear Circuit Synthesis: First-Order LPF 21 TLP: I p I y = I x I τ KCL: I c + I p = I τ