Suspended Bicore
Wouter Brok
Eindhoven, The Netherlands.
Abstract:
First the suspended bicore is described. Although the circuitlayout of this symmetric oscillator is quite simple  three passive and two active components  its behavior is rather complex. The equations for a suspended bicore with ideal inverters and equal capacitors will be derived and with them the influence of noise on the circuit will be explained. Then the suspended bicore with different capacitors will be discussed, and the limit of this case  only one capacitor  will be explained. Finally, with knowledge of the suspended bicore, a masterslave dual bicore, which is a coupled oscillator, will be explained.
The suspended bicore:
The circuit of an Nvneuron, as introduced by M.W.Tilden, is drawn in fig. 1. This is a pulsedelaycircuit the behavior of which is described in ‘Controller for a four legged walking machine’ [1]. This Nvneuron can serve as one part of a chain in which a pulse can circulate [2], generating an oscillatory behavior of the outputvoltage of the inverters.
Figure 1: The NvNeuron.
When two Nvneurons are connected as shown in fig. 2a the circuit is called a bicore. The neurons form a ringlike structure which generates an oscillatory outputvoltage with a period determined by the capacitors and the resistors.

(1) 
In which

(2) 
with i = 1,2
r has the dimension of time and is determined by the resistance R in Ohms and the capacitance C in Farads; a is a constant, determined by the characteristics of the inverter. From now the capacitance will be assumed constant and equal for all capacitors to be used, unless mentioned otherwise. This gives no restrictions since the resistance alone is needed to vary the time constants of the circuits described.
If R1 = R2 the oscillation will have a dutycycle of 50% and thus be symmetric.


(a)

(b)

Figure 2: (a) the normal bicore, a ring like structure of two Nvneurons, and (b) the suspended bicore.
In figure 2b the schematic of a suspended bicore is shown. As one can see the resistors in figure 2a are disconnected from ground, then connected to each other and replaced by a single resistor. Doing this results in a circuit which has a high symmetry.
The suspended bicore works quite differently from the normal bicore. To gain an understanding of the suspended bicore, and later the masterslave dual bicore, it could be useful to write down a set of equations for the circuit. Prior to this the oscillation is described in terms of the different parts of one period of oscillation. For that the names for certain voltages need to be defined first (fig. 3): V11 and V10 are the input and outputvoltage of inverter U1 respectively and V21 and V20 are the input and the outputvoltage of inverter U2 respectively.
Figure 3: Direction of current I and names for voltages, which are used in the text.
Let V11 = 0 V and V21 = Vcc be the supplyvoltage. Because of the action of the inverters V10 will be equal to Vcc and V20 will be equal to 0 V. There are no voltagedifferences across the capacitors, but the voltagedifference across the resistor R is Vcc, so a current will be flowing, charging the capacitors (see fig. 4). At a certain point the voltage across the resistor is almost zero and V11 and V21 will near the thresholdvoltage of the inverters, Vcc/2.
One of the inverters will start to change state first [3], for example U1: V10 will start to go from Vcc to 0 V. Consequently, V21 will decrease as well, since it is coupled to V10 via C2, and U2 will also change state. The output of U2 is coupled via C1 to the input of U1, so this in turn will accelerate the change of state of U1. The result is that V10 = ( Vcc/2) and V20 = (3Vcc/2), but if it is assumed that the inverter only allows the inputvoltages between the boundaries set by the supplyvoltage levels then V10 = 0 V and V20 = Vcc (The assumption is quite reasonable since most inverters have a protection against too
Figure 4: The waveforms V11, V10, V21 and V20 of the suspended bicore of fig. 3 with C1 = C2
and inverters having a thresholdvoltage equal Vcc/2 for both positive and negative edges. Some
noise is assumed to be superimposed on the curves so that thresholdvoltages are reached. The
inverters are assumed to have a thresholdvoltage equal Vcc/2 for both positive and negative edges.
high or too low voltages (see fig. 5 for an example)). The described cycle will start again, with the input and the outputvoltages of U1 and U2 reversed, and at the end of it one period is completed.
Now the equations: it is assumed that no current is flowing into the inputs of the inverters (infinite inputimpedance), and that outputvoltage is not dependent on the current drawn (zero outputimpedance). Then it is possible to define one current I flowing like shown in fig. 3.
For a capacitor the relation between the current Ic and the voltage Vc across the capacitor is a timedependent function, given by:

(3) 
With Ic in Amperes, C in Farads and Vc in Volts
For the current I the following relations hold:
And Ohms Law for the resistor:

(4c) 
For a timeinterval, in which the inverters do not change state, V1 can be replaced by (—V11) and V2 by (—V21), since the timederivatives of V20 and V10 are zero in this particular interval.
Making use of the assumption that C1 = C2 = C, substituting (4c) in (4a) and (4b) and rewriting gives:

(5a) 

(5b) 
This is a coupled system of two first order differential equations. If it is transformed from the initial basis {V11;V21} to a new basis {V11+V21; V11V21}, the system becomes decoupled:

(6a) 

(6b) 
These equations are easily solved:

(7a) 

(7b) 
In which A and B are constants and the factors 2 are there for convenience, t0 is the time at which the exponential curve starts. From (7a) and (7b) the following equations result for a timeinterval in which the inverters do not change state:

(8a) 

(8b) 
Note that the sign of B changes every half period, because of new the boundaryconditions which determine A and B as can be seen in fig. 4.
For example, for an ideal inverter, which has a thresholdvoltage of Vcc/2 for both positive and negative edges and does not allow the input to exceed boundaries set by the supplyvoltages levels 0 V and Vcc, A = B = Vcc/2.
The suspended bicore with this ideal inverter has one very remarkable feature: it does not oscillate; the exponential curves V11 and V21 never reach the thresholdvoltage Vcc/2. This shows how very important noise and nonideal properties are. If the exponential curve has relatively small noise superimposed on it, the inverterinput will pass the thresholdvoltage and thus the circuit will oscillate. The period of oscillation however, is not constant since the circuit oscillates because of the stochastic noise. The noise and the non ideal properties of the components are not negligible in most cases.
Figure 5: Schematic diagram to illustrate how the 74HC240 has internal
diode protection against too high or too low voltages at its input.
For example: a suspended bicore made with inverters of the 74HC240. These inverters do allow the inputs to exceed supplyvoltage levels, but only to a small extend: the inputs are protected with two diodes as shown in fig. 5. They allow the input voltages in the range (—Vdiode) ... (Vcc + Vdiode), in which Vdiode is the thresholdvoltage of the diode, which is approximately 0.6 V. Also, the thresholdvoltage of the inverter at which the output starts to change is about Vcc/2 + 25 mV for the negative edges and Vcc/2  25 mV for the positive edges if Vcc = 5 V (The voltagegain of the 74HC240 is approximately 100). For the 74HC240 the constants A and B in (8a) and (8b) are: A = Vcc/2 and B = Vcc/2 + Vdiode.
Thus the noise which has an amplitude in the mVrange has an influence on the oscillationperiod. Without the noise this period is calculated to be , for Vcc = 5 V; with the noise this is shorter.
Normally C1 will not be equal to C2. This could be due to the fact that no two capacitors are exactly equal or because C1 and C2 are to be different. If C1 &Mac185; C2 then the voltages on either sides of the resistor will not converge to Vcc/2, but one side will converge to a higher voltage and the other side to a lower voltages (see the dashed curves in fig. 6a). This implies that one of the inverters will reach its thresholdvoltage sooner than it would in the case of equal capacitors. If one inverter reaches its thresholdvoltage the output will start to change and the change will influence the input of the other inverter via the capacitor in between. So that inverter will change state as well. Noise will have less influence on the circuit if the capacitors are not equal: the gradient of the inputvoltage of the inverter which initiates the change of state is bigger near the thresholdvoltage than it is in the equal capacitor case.
It is important to realize that if C1 &Mac185; C2 the dutycycle of the oscillation will still be 50%.
This more general case, in which C1&Mac185; C2 can also be described by equations. The derivation is analogue to the previous derivation with C1 = C2; the differential equations describing the circuit are:

(9a) 

(9b) 
Again this is a coupled system of two differential equations, which becomes decoupled by transforming it to another basis:
Note that the basis chosen in the previous derivation with C1 = C2 = C is a special case of this basis.

(10a) 

(10b) 
The solutions of these equations are:

(11a) 

(11b) 
In which A and B are constants and the factors (C1 + C2)/C1C2 and 2 are there for convenience; t0 is the time at which the exponential curve starts. From (11a) and (11b) the following equations result for a timeinterval in which the inverters do not change state:

(12a) 

(12b) 
It should be noted that in general both A and B are different for the two parts in which one period of oscillation can be divided.


(a)

(b)

Figure 6: (a) Waveforms of the suspended bicore with unequal capacitors; C1 < C2. (b)
Waveforms of the suspended bicore with only C1; instead of C2 there is a shortcircuit 
this is the limit of the case with unequal capacitors of (a). The inverters are assumed to
have a thresholdvoltage equal Vcc/2 for both positive and negative edges.
To illustrate how the values of A and B can be determined it is useful to examine the suspended bicore with ideal inverters: the inverters have a thresholdvoltage Vcc/2 and do not allow the inputs to exceed the boundaries set by the supplyvoltage levels 0 V and Vcc.
Let V11 be equal to 0 V and V21 be equal to Vcc (see fig. 6a). From equation (12a) and (12b) immediately follows for (t — t0) = 0:

and 

(13) 
It should be noted that either A or B needs to be negative. Furthermore V11 and V21 will converge to A further in time and both V11 and V21 need to stay in the interval set by the inverterinputs: 0 V and Vcc. It is evident that A > 0 and that B < 0. From (13) then results:

and 

(14) 
In the second half of one period of oscillation C1 and C2 are replaced by each other and B will have the opposite sign.
If one of the capacitors is left out, (replaced by a short circuit), the resulting circuit can be seen as a limit of the suspended bicore with unequal capacitors, for which the equations (12a) and (12b) have been derived.
The oscillation of this circuit with one capacitor is shown in fig. 6b and can be understood as follows: recalling equation (3) it is clear that a short circuit can be interpreted as an infinite capacitance  no matter what value I has, there is no change in the voltage across the short circuit. Let C2 in fig. 3 be replaced by a short circuit, then in the limit of C2 to infinity equation (12b) reduces to:

(15) 
To see what happens to equation (12a) it is important to realize that:
so equation (12a) reduces to:

(16) 
This is also the equation describing a single Nvneuron in a ringlike structure.
The MasterSlave Dual Bicore:
When the normally grounded sides of the resistors of a normal bicore are connected to the outputs of a suspended bicore, a circuit is formed which is called masterslave dual bicore (fig. 7). The master is the suspended bicore, which oscillates unaffected by the slave. The slave however is affected in its oscillation by the master via the coupling resistors R3 and R4.
To explain the action of the masterslave dual bicore, names for certain voltages are defined as shown in fig. 7: V10, V11, V20 and V21 as already defined for the suspended bicore in fig. 3 and V30, V31, V40 and V41 for the voltages of the outputs and the inputs of the inverter U3 and U4.
Figure 7: The MasterSlave Dual Bicore, with the names for the voltages which are used in the text.
Let V10 = Vcc, V20 = 0 V, V31 = 0 V and V41 = Vcc as at t = 0 in fig. 8. Then the voltages across the capacitors C3 and C4 are zero, but the voltages across the couplingresistors R3 and R4 are not, so currents will be flowing through these resistors. For example V31: this voltage will exponentially rise from 0 V to Vcc as in the second graph of fig. 8, according to the relation

(17) 
However, at Dt after the start of the exponential curve it will reach the thresholdvoltage of inverter U3 and the inverter will start to change state: V30 will start to go from Vcc to 0 V. Doing this it takes V41 down with it, because of the coupling via C4. In turn V40 will go from 0 V to Vcc, speeding up the changing of state of U3 via C3. When this has happened the slave is at rest: no voltages across the capacitors and no voltages across the couplingresistors. This state of rest continues until the master inverters reverse states, which will create voltagedifferences across the couplingresistors. A current will start to flow and the process starts all over, with reversed voltage levels.
When the circuit oscillates in this manner the frequency of oscillation of the slave is equal to and determined by the frequency of oscillation of the master: the slave waits for the master during the horizontal parts of the graphs of V31 and V41 of fig. 8. The phase difference Dj between V20 and V30 (between the master and the slave) is determined by the time Dt and by the frequency of oscillation:

(18) 
In which Dt can be calculated from (17): for the circuit with inverters having a thresholdvoltage equal Vcc/2 and an inputvoltage bounded in the interval between 0 V and Vcc, the timedelay Dt is:

(19) 
In which R = R3 = R4, the resistance of the couplingresistors in Ohms and C = C3 = C4 the capacitance of the slave capacitors in Farads.
It should be noted that for the phase difference between V20 and V40, p needs to be added to Dj (V40 is in antiphase with V30).
By varying the frequency of the master (for example by changing R in fig. 7) both the frequency of the slave and the phase difference Dj between the master and the slave are varied.
Figure 8: The waveforms V10, V31, V30, V20, V41 and V40 of the masterslave
dual bicore of fig. 7. In the second graph V31(t) the exponential curves are dashed
to give an idea of how the whole curve looks like. The time delay between the
master and the slave is Dt. The inverters are assumed to have a thresholdvoltage
equal Vcc/2 for both positive and negative edges.
The frequency of oscillation of the slave can differ from the frequency of oscillation of the master: this happens if half of the period of the master is shorter than the Dt caused by the exponential curves of the slavevoltages (if Dj in equation (18) is bigger than p). Contrary to the previous case, in which the slave had to wait for the master, the slave is now too slow for the master. An oscillation like this is shown in fig. 9: the slave starts with an exponential rise of V31, due to a voltage across the couplingresistor. However, at a certain moment the output of the master will reverse so the voltage across the couplingresistor will change sign and V31 will decrease. Again the masteroutput reverses so V31 rises; this will happen until V31 reaches the thresholdvoltage of the slaveinverter. The slaveinverter changes state, and via the capacitor the change causes the other slaveinverter to change state as well. The cycle will start again, only with the voltagelevels reversed.
Conclusion:
The suspended bicore of fig. 3 shows a very interesting behavior in response to noise: if the inverters are ideal and the capacitors equal, the oscillationperiod of the circuit is very sensitive to noise. This sensitivity is decreased by using inverters which have a thresholdvoltage, which is not exactly in the middle of maximum and minimum inputvoltages, or by using capacitors which are not equal. These two ways of decreasing the sensitivity of the circuit are based on the fact that the exponential curves of the inputvoltages of the inverters have a larger gradient near the thresholdvoltage of the inverters. Consequently the relatively small noise that is superimposed on these exponential curves will have less influence on the exact inversionmoment, and thus less influence on the period of oscillation.
Figure 9: The waveforms V10, V31, and V30, of the masterslave dual bicore of fig. 7, when
the period of oscillation of the master is to fast for the slave to follow. The waveforms V20,
V41 and V40 are not shown: they are the same, but inverted around Vcc/2 if both coupling
resistors are equal.
If the circuit is expanded to a masterslave dual bicore (fig. 7) a coupled oscillator results. For such an oscillator the frequency of oscillation of the master is equal to that of the slave if the natural oscillationperiod of the slave is smaller than the oscillationfrequency of the master. The phasedifference between the master and the slave can be controlled by varying the oscillationfrequency of the master alone. This however can only be done to a certain extend since for a frequency which is too high the slave is not able to follow the master anymore: its frequency will then be smaller than the frequency of the master.
A possible application for the masterslave dual bicore is as drivingcircuit for a twomotor walking robot such as the one described in ‘Controller for a four legged walking machine’ [1] and ‘Coupled Oscillators and Walking Control’ [4]. The circuit could also be used as a central pattern generator (CPG) in more advanced mobile robots which are inspired by biological systems.
The controller for the robot could be designed so that for example motornoise is coupled back into the masterslave dual bicore via an appropriate filter. This way both the frequency of oscillation and the phasedifference between the master and the slave will adapt to the properties which influence the level of noise, like the load on the motor. The oscillatorcircuit itself can be designed to have a certain sensitivity to this noise.
It should be noted that in this text the outputimpedance of the inverters is assumed to be zero. Generally this however is not the case and will have an effect on the behavior of the circuit if a load is applied.
Literature
[1] 
S. Still and M.W. Tilden: ‘Controller for a four legged walking machine’, in: ‘Neuromorphic Systems: Engineering Silicon from Neurobiology’, editors: L. S. Smith and A. Hamilton, publisher: World Scientific 
[2] 
B. Haslacher and M.W. Tilden: ‘Living machines. Robotics and Autonomous Systems: The Biology and Technology of Intelligent Autonomous Agents’, editor: L. Steels, publisher: Elsevier Publishers 1995. 
[3] 
Wilf Rigter, Personal Communications. 
[4] 
S. Still and M.W. Tilden: ‘Coupled Oscillators and Walking Control: A Hardware Implementation of a Distributed Motor System’, in: ‘Proceedings of the 26th Goettingen Neurobiology Conference 1998’, vol.2, editors: N. Elsner and R. Wehner, p.262. 
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