Differential Algebraic Equations of MOS Circuits and Jump Behavior

Many nonlinear electronic circuits showing fast switching behavior exhibit jump effects which occurs when the state space of the electronic system contains a fold. This leads to difficulties during the simulation of these systems with standard circuit simulators. A method to overcome these problems is by regularization, where parasitic inductors and capacitors are added at the suitable locations. However, the transient solution will not be reliable if this regularization is not done in accordance with Tikhonov’s Theorem. A geometric approach is taken to overcome these problems by explicitly computing the state space and jump points of the circuit. Until now, work has been done in analyzing example circuits exhibiting this behavior for BJT transistors. In this work we apply these methods to MOS circuits (Schmitt trigger, flip flop and multivibrator) and present the numerical results. To analyze the circuits we use the EKV drain current model as equivalent circuit model for the MOS transistors.


Introduction
In this work our focus lies on circuits which exhibit fast switching behavior (Schmitt Trigger, flip flop and multivibrator).It is known that the derivative of the capacitor voltages and inductor currents govern the dynamics of an electronic circuit.Also, the differential equations of electronic circuits can be viewed as a flow on the state space manifold, which is represented by the algebraic constraints of the circuit.These circuits with discontinuous changes in states, which are called "jumps in state space", contain a fold in their state space manifold.The simulation of these circuits leads to a simulation failure as the circuit can adopt multiple operating points at the same time.A method to overcome this problem is to regularize the system by adding capacitors and inductors at appropriate nodes, in accordance with Tikhonov's theorem (Tikhonov et al., 1985).When the network is -regularized (Ihrig, 1975), the jump behavior can be viewed as the limit → 0 of the solutions of the singularly perturbed system (Sastry and Desoer, 1981).This method can regularize the system, but it gives erroneous transient solutions by choosing wrongly located L's and C's.Another problem is due to the widely spaced time-constants, which appear because the dynamics of a regularized circuit can be divided into a slow and a fast part, leading to the so-called "time-constant problem" of circuit simulation (Sandberg and Shichman, 1968).Hence, we adopt a geometric approach and calculate the jump points and state space explicitly.This approach has been succesfully applied to example transistor circuits involving BJTs.In this work, we apply the method to MOS circuits and calculate the state space and jump points for Schmitt Trigger, flip flop and multivibrator and show that the results confirm with the simulation results.To efficiently model the MOS circuits, the EKV drain current model has been used (Enz et al., 1995).

Geometric interpretation of jump behavior
The state space S of an electronic circuit can be interpreted as a differentiable manifold and is given by the intersection of the Ohmian O and the Kirchhoffian K space S := K ∩ O (Smale, 1972;Desoer and Wu, 1972;Chua, 1980).The dynamics of an electronic circuit then is defined on S (Mathis, 1992).This implies that we need to satisfy the following conditions: (1) S is a smooth manifold and (2) the dynamics can be created on S. The first is a typical or so-called generic condition (for a detailed discussion, see Mathis, 1992), and Published by Copernicus Publications on behalf of the URSI Landesausschuss in der Bundesrepublik Deutschland e.V.
in the following we assume S to be a smooth manifold.The second condition requires the construction of a vector field X on the smooth manifold S. Based on fundamental physical laws, the relationships between currents and voltages of capacitors and inductors are given by means of differential relations.Therefore these differential equations are formulated in i L and u C coordinate planes.Now, one has to "lift" or "pull-back" the dynamics on the state space S. Therefore, the vector field ceases to exist if the pull-back or the dynamics is degenerated, which leads to jumps in S.This degeneracy occurs if S contains a fold.A detailed discussion of degeneracy can be found in Thiessen and Mathis (2011) and Mathis (1992).
If the circuit is characterized by the following algebrodifferential equations (DAEs) in a semi explicit form: then the set of all jump points (jump-set) is characterized by (see also Nielsen and Willson Jr. (1980), Tchizawa (1984), Ichiraku (1979), Thiessen et al. (2012)).The vector x ∈ R n corresponds to the capacitor voltages and inductor currents and y ∈ R m is a vector of additional voltages and currents.Since there are circuits which exhibit a fold respectively their input voltages, we assign an additional vector z ∈ R η to independent voltage or current input sources.We treat the independent input sources as norators and assume z to be another variable in our system of equations.Therefore, the state space S of the circuit has to be extended by the number of independent sources η.Now, the dimension k of the embedding space E ∈ R k can be determined by k = n + m + η and the dimension of S by dim(S) = l = n + η.The state space S can be defined as a subspace of the E and is represented by the solution set of the algebraic equations (2).The dynamical beahvior of the circuit is represented by the differential equations (1).
The jump takes place in a subspace parallel to the space spanned by y, where y is the vector of all coordinates which are not fixed and do not conserve energy Thiessen and Mathis (2011), Thiessen et al. (2012).The corresponding "hit-set" is the intersection of the "bundle" of all jump spaces at points of the jump-set and the state space S.
To solve the equivalent circuits of the Schmitt Trigger, flip flop and multivibrator, we take this approach where we numerically calculate the jump points.For the determination of S, we interpret z as variables and by specifying l components of y, we can calculate S Thiessen et al. (2012).

Modelling the MOS equivalent circuit
It is known that the MOS drain current follows a square law and is a function of the gate-source and the drain-source volt-ages and goes to zero below V th .It is seen, that below the threshold voltage the current-voltage characteristic is exponential and is called as sub-threshold current and the behavior is as follows where U T is the thermal voltage, κ the non-ideality factor and I S is the saturation current.Since we are dealing with circuits that are switching from cutoff to saturation, we need a model that holds good for all regions of operation and does not exhibit the jump in the current function itself, as seen in the square law case.Hence, we use the EKV drain current equation, which is valid in all regimes.The following equation shows the EKV current characteristic.
where κ is a variable and is adjusted according to the MOS under consideration.We can see that when V gs is a significant value, the exponent dominates inside the logarithm and hence we can approximate ln(1+e x ) ln(e x ) = x.Upon using this approximation we get If the gate source potential is a value comparable or less than V t , then we can approximate ln(1 + e x ) e x .With this approximation we get Figure 1 illustrates how the EKV equation closely resembles the square law curve as well as the sub-threshold current in their respective regimes.The threshold voltage used for the analysis is V th = 1.6 V. We compare the EKV model with the actual MOS (BSS123) that is going to be used for the subsequent analysis.From the parameters of the BSS123 MOS, we calculate the constants that need to be used in the EKV model.It gives us the following empirical drain current equation, which can be used to simulate the circuits.
where a = 0.0013 and b = 10.7250.Figure 2 shows the drain current versus the gate source voltage characteristic of the EKV approximation and the BSS123 for κ = 1.8.

Example 1: Schmitt Trigger circuit
In this section we analyze the Schmitt Trigger circuit from a geometric point of view.The design parameters of the cir- where a = 0.0013 and b = 10.7250.Fig. 2 shows the drain current versus the gate source voltage characteristic of the EKV approximation and the BSS123 for κ = 1.8.

Example 1: Schmitt Trigger Circuit
In this section we analyze the Schmitt Trigger circuit from a geometric point of view.The design parameters of the circuit are We neglect the gate source capacitance during the calculation of the jump points.These capacitances are used to overcome the singular points and they aid in the simula-  .) is the EKV equation as described in eq put voltages are then found out as a funct U gs2 .The state space of the circuit is given tersection of the solution sets of these two where the constants p and k are p = 1 + 1 R e = 300 , U o = 9 V and V T = 1.6 V. We neglect the gate source capacitance during the calculation of the jump points.These capacitances are used to regularize the circuit and therefore enable the simulation of the circuits with a common circuit simulator.In our approach, addition of these regularization capacitances C T is not necessary.Equation ( 9) gives us the state space description of the system.U in is the input and U gs1 and U gs2 are the gate source voltages and are set as the state variables.f (.) is the EKV equation as described in Eq. ( 8).The output voltages are then found out as a function of U gs1 , U gs2 .The state space of the circuit is given by the intersection of the solution sets of these two equations: P. Sarangapani, T. Thiessen: Fast Switching Behaviour in MOS Circuits: A Geometric Approach 3 where a = 0.0013 and b = 10.7250.Fig. 2 shows the drain current versus the gate source voltage characteristic of the EKV approximation and the BSS123 for κ = 1.8.

Example 1: Schmitt Trigger Circuit
In this section we analyze the Schmitt Trigger circuit from a geometric point of view.The design parameters of the circuit are 6V.We neglect the gate source capacitance during the calculation of the jump points.These capacitances are used to overcome the singular points and they aid in the simulation of circuits with circuit simulators.In our approach, addition of these regularization capacitances C T is not necessary.Eq.9 gives us the state space description of the system.U in is the input and U gs1 and U gs2 are the gate source voltages and are set as the state variables. .The state space of the circuit is given by the intersection of the solution sets of these two equations: where the constants p and k are where a = 0.0013 and b = 10.7250.Fig. 2 shows the drain current versus the gate source voltage characteristic of the EKV approximation and the BSS123 for κ = 1.8.

Example 1: Schmitt Trigger Circuit
In this section we analyze the Schmitt Trigger circuit from a geometric point of view.The design parameters of the circuit are 6V.We neglect the gate source capacitance during the calculation of the jump points.These capacitances are used to overcome the singular points and they aid in the simulation of circuits with circuit simulators.In our approach, addition of these regularization capacitances C T is not necessary.Eq.9 gives us the state space description of the system.U in is the input and U gs1 and U gs2 are the gate source voltages and are set as the state variables.) is the EKV equation as described in eq.8.The output voltages are then found out as a function of U gs1 , U gs2 .The state space of the circuit is given by the intersection of the solution sets of these two equations: where the constants p and k are where the constants p and k are The system of equations is solved using Newton-Raphson method where the range of U gs1 is defined.To find the fold in the state space we need to choose a proper coordinate system.Since U in is fixed, this state cannot jump.This implies that we need to look at the U in − U out2 curve for a fold.This circuit is analyzed similar to the previous case.This circuit is analytically similar to the Schmitt Trigger, hence the results should be similar to the ones obtained there.The following are the design parameters of the circuit: The equations governing the circuit are where U in is the input, U out2 is the output and U gs1 ,U gs2 are the gate source voltages of the MOS.The phase space and the jump points are obtained by declaring U gs1 between two predefined values and solving for U gs2 for that corresponding value of U gs1 .The jump condition here turns out to be 1 The jump points of the output are shown in Fig. 6.

Example 3: Multivibrator
In the earlier sections the systems of equations were only algebraic ones.Here, for the multivibrator, we get a semi explicit DAE system.The device parameters chosen for this circuit are: 26mA.The equations governing this circuit are: outputs for the same input indicating singularity.To calculate the point where the output transition occurs, we calculate the jump points using the method as stated in Eq. ( 3).This gives us To solve this equation we assume that U gs1 varies between two predefined values and find U gs2 .The intersections of the solution set of Eq. ( 12) and the state space are defined as the jump points.The jump points of the output are shown in Fig. 4 as the intersection of both curves.

Example 2: Flip Flop circuit
The flip flop circuit is analyzed similar to the previous case.This circuit is analytically similar to the Schmitt Trigger, hence the results should be similar to the ones obtained there.The following are the design parameters of the circuit: where the constants k, p, q are 4 P. Sarangapani, T. Thiessen: Fast Switching Behaviour in MOS Circuits: A Geometric Approach The system of equations is solved using Newton-Raphson method where the range of U gs1 is defined.
To find the fold in the state space we need to choose a proper coordinate system.Since U in is fixed, this state cannot jump.This implies that we need to look at the U in − U out2 curve for a fold.Fig. 4 indeed shows a fold and as expected, there are multiple outputs for the same input indicating singularity.To calculate the point where the output transition occurs, we calculate the jump points using the method as stated in eq.3.This gives us To solve this equation we assume that U gs1 varies between two predefined values and find U gs2 .The points intersecting with the solution set of the jump condition and the state space, are defined as the jump points.The jump points of the output are shown in Fig. 4.This circuit is analyzed similar to the previous case.This circuit is analytically similar to the Schmitt Trigger, hence the results should be similar to the ones obtained there.The following are the design parameters of the circuit: The equations governing the circuit are Fig. 5: Flip Flop Circuit where the constants k, p, q are where U in is the input, U out2 is the output and U gs1 ,U gs2 are the gate source voltages of the MOS.The phase space and the jump points are obtained by declaring U gs1 between two predefined values and solving for U gs2 for that corresponding value of U gs1 .The jump condition here turns out to be 1 The jump points of the output are shown in Fig. 6.

Example 3: Multivibrator
In the earlier sections the systems of equations were only algebraic ones.Here, for the multivibrator, we get a semi explicit DAE system.The device parameters chosen for this circuit are: 26mA.The equations governing this circuit are: U in is the independent input voltage, U out2 is the output and U gs1 , U gs2 are the gate source voltages of the MOS.The state space and the jump points are obtained by declaring U gs1 between two predefined values and solving for U gs2 for that corresponding value of U gs1 .The determinant criterion here turns out to be 1 The jump points of the output are shown in Fig. 6 as the intersection of both curves.

Example 3: multivibrator
In the earlier sections the systems of equations were only algebraic ones.Here, for the multivibrator, we get a semi explicit DAE system.The device parameters chosen for this circuit are: The equations governing this circuit are: The above equations can be written in a matrix form as The state space of the circuit is given by the intersection of the surfaces S 1 and S 2 , where S 1 is the solution set of g 1 (U gs1 ,U gs2 ,U c ) = 0 and S 2 is the solution set of g 2 (U gs1 ,U gs2 ,U c ) = 0. Fig. 8 shows the intersection of the two surfaces.The intersection curve (in blue) is the state space of the circuit.The state space has to be plotted in a coordinate system, where one of the quantity does not jump whereas the other two show jump behavior.Hence the state space was plotted in U gs1 − U gs2 − U c coordinate system.The jump criterion for this circuit is given as Upon solving this with Newton-Raphson method we Fig. 8: State space of multivibrator circuit get the jump points as shown in Fig. 9.The jump points are shown in the U gs1 −U gs2 coordinate system as jump occurs only for these two voltages.Fig. 10, on the other hand shows the jump behaviour in the U gs1 −U gs2 −U c coordinate system.We can see that jump occurs in the direction where the capacitance potential U c is conserved.This happens as the capacitance potential cannot jump instantaneously.

Conclusion
Analysis of electronic circuits that contain a fold in their state space with common circuit simulators like SPICE sometimes gives errors due to time constant problems.The analysis of these circuits require regularization, which is achieved by adding capacitors and inductors at appropriate nodes.If the regularization is not done in accordance with Tikhonov's theorem, the transient solutions will not be reliable.With our approach regularization is no longer necessary as it is pos- The above equations can be written in a matrix form as The state space of the circuit is given by the intersection of the surfaces S 1 and S 2 , where S 1 is the solution set of g 1 (U gs1 ,U gs2 ,U c ) = 0 and S 2 is the solution set of g 2 (U gs1 ,U gs2 ,U c ) = 0. Fig. 8 shows the intersection of the two surfaces.The intersection curve (in blue) is the state space of the circuit.The state space has to be plotted in a coordinate system, where one of the quantity does not jump whereas the other two show jump behavior.Hence the state space was plotted in U gs1 − U gs2 − U c coordinate system.The jump criterion for this circuit is given as Upon solving this with Newton-Raphson method we

Conclusion
Analysis of electronic circuits that contain a fold in their state space with common circuit simulators like SPICE sometimes gives errors due to time constant problems.The analysis of these circuits require regularization, which is achieved by adding capacitors and inductors at appropriate nodes.If the regularization is not done in accordance with Tikhonov's theorem, the transient solutions will not be reliable.With our approach regularization is no longer necessary as it is pos- The above equations can be written in a matrix form as The state space of the circuit is given by the intersection of the surfaces S 1 and S 2 , where S 1 is the solution set of h 1 (U gs1 ,U gs2 ,U c ) = 0 and S 2 is the solution set of h 2 (U gs1 ,U gs2 ,U c ) = 0. Figure 8 shows the intersection of the two surfaces.The intersection curve (in blue) is the state space of the circuit.The state space has to be plotted in a coordinate system, where one of the quantity does not jump whereas the other two show jump behavior.Hence the state The above equations can be written in a matrix form as The state space of the circuit is given by the intersection of the surfaces S 1 and S 2 , where S 1 is the solution Upon solving this with Newton-Raphson method we Fig. 8: State space of multivibrator circuit get the jump points as shown in Fig. 9.The jump points are shown in the U gs1 −U gs2 coordinate system as jump occurs only for these two voltages.Fig. 10, on the other hand shows the jump behaviour in the U gs1 −U gs2 −U c coordinate system.We can see that jump occurs in the direction where the capacitance potential U c is conserved.This happens as the capacitance potential cannot jump instantaneously.

Conclusion
Analysis of electronic circuits that contain a fold in their state space with common circuit simulators like SPICE sometimes gives errors due to time constant problems.The analysis of these circuits require regularization, which is achieved by adding capacitors and inductors at appropriate nodes.If the regularization is not done in accordance with Tikhonov's theorem, the transient solutions will not be reliable.With our approach regularization is no longer necessary as it is pos- space was plotted in U gs1 −U gs2 −U c coordinate system.The determinant criterion for this circuit is given as Upon solving this with Newton-Raphson method we get the jump points as shown in Fig. 9.The jump points are shown in the U gs1 − U gs2 coordinate system as the intersection of both curves.For verifying our results we regularized the system of equations by adding regularization capacitances parallel to U gs1 and U gs2 .The transient behavior of this regularized circuit can be seen in Fig. 10 (red line).We can see that the fast transition occurs in the U gs1 and U gs2 space, where the capacitance potential U c is hold mostly constant.

Conclusions
The simulation of electronic circuits that contain a fold in their state space with common circuit simulators like SPICE sometimes gives errors due to time constant problems.The analysis of these circuits require regularization, which is achieved by adding capacitors and inductors at appropriate nodes.If the regularization is not done in accordance with Tikhonov's Theorem, the transient solutions will not be reliable.With our approach, regularization is no longer necessary, as it is possible to detect whether the manifold of the circuit's state space has a fold beforehand.The jump points, therefore help us to identify the points of transition easily.We have shown numerical results of applying the geometric concepts to three MOS circuits.Therefore, the MOS drain current was modelled using the EKV equation for robust results.sible to detect whether the manifold of the circuit's state space has a fold beforehand.The jump points, therefore help us to identify the points of transition easily.We have shown numerical results of applying the geometric concepts to three MOS circuits.The MOS drain current is modeled using the EKV equation for robust results.sible to detect whether the manifold of the circuit's state space has a fold beforehand.The jump points, therefore help us to identify the points of transition easily.We have shown numerical results of applying the geometric concepts to three MOS circuits.The MOS drain current is modeled using the EKV equation for robust results.

Fig. 1 .
Fig. 1.Comparison of the EKV model with the square law and sub-threshold current.

Fig. 1 :
Fig. 1: Comparison of the EKV model with the square law and sub-threshold current

Fig. 2 .
Fig. 2. Comparison of the EKV model with the BSS123 MOS.

Fig. 2 :
Fig. 2: Comparison of the EKV model with the BSS123 MOS Figure 4 indeed shows a fold and as expected, there are multiple www.adv-radio-sci.net/10/327/2012/Adv.Radio Sci., 10, 327-332, 2012 intersecting with the solution set of the jump condition and the state space, are defined as the jump points.The jump points of the output are shown in Fig.4.

Fig. 8 :
Fig. 8: State space of multivibrator circuit get the jump points as shown in Fig.9.The jump points are shown in the U gs1 −U gs2 coordinate system as jump occurs only for these two voltages.Fig.10, on the other hand shows the jump behaviour in the U gs1 −U gs2 −U c coordinate system.We can see that jump occurs in the direction where the capacitance potential U c is conserved.This happens as the capacitance potential cannot jump instantaneously.

Fig. 8 .
Fig. 8. State space as intersection of S 1 and S 2 .

Fig. 10 .
Fig. 10.Transient solution (red) of the regularized circuit in the U gs1 −U gs2−U c coordinate system; state space of non-regularized circuit (blue).