Suspended Bicore.- Wouter Brok

Suspended Bicore.

Abstract:
First the suspended bicore is described. Although the circuit-layout of this symmetric oscillator is
quite simple - three passive and two active components - its behaviour 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 master-slave dual bicore, which is a coupled
oscillator, will be explained.
The suspended bicore:
The circuit of an Nv-neuron, as introduced by M.W.Tilden, is drawn in fig. 1. This is a pulse-delaycircuit
the behaviour of which is described in ‘Controller for a four legged walking machine’ [1]. This
Nv-neuron can serve as one part of a chain in which a pulse can circulate [2], generating a oscillatory
behaviour of the output-voltage of the inverters.

Figure 1: The Nv-Neuron.
When two Nv-neurons are connected as shown in fig. 2a the circuit is called a bicore. The neurons
form a ring-like structure which generates an oscillatory output-voltage with a period determined by
the capacitors and the resistors.

t 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 duty-cycle of 50% and thus be symmetric.

Figure 2: (a) the normal bicore, a ring like structure of two Nv-neurons, 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 master-slave 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 output-voltage of inverter U1 respectively and V21 and V20 are the input- and the
output-voltage 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 supply-voltage. Because of the action of the inverters V10 will be
equal to Vcc and V20 will be equal to 0 V. There are no voltage-differences across the capacitors, but
the voltage-difference 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 threshold-voltage 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 input-voltages between the boundaries set by the supply-voltage levels then V10 = 0 V
and V20 = Vcc (The assumption is quite reasonable since most inverters have a protection against too
high or too low voltages (see fig. 5 for an example)). The described cycle will start again, with the
input- and the output-voltages 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
input-impedance), and that output-voltage 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:

Figure 4: The waveforms V11, V10, V21 and V20 of the suspended bicore of fig. 3 with C1 = C2 and
inverters having a threshold-voltage equal Vcc/2 for both positive and negative edges. Some noise
is assumed to be superimposed on the curves so that threshold-voltages are reached. The inverters are
assumed to have a threshold-voltage equal Vcc/2 for both positive and negative edges.

For a time-interval, in which the inverters do not change state, V1 can be replaced by (–V11) and V2 by
(–V21), since the time-derivatives 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:

For example, for an ideal inverter, which has a threshold-voltage of Vcc/2 for both positive and
negative edges and does not allow the input to exceed boundaries set by the supply-voltages 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 threshold-voltage Vcc/2. This shows how very
important noise and non-ideal properties are. If the exponential curve has relatively small noise
superimposed on it, the inverter-input will pass the threshold-voltage 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.

For example: a suspended bicore made with inverters of the 74HC240. These inverters do allow the
inputs to exceed supply-voltage 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 threshold-voltage of the diode, which is approximately 0.6 V. Also, the threshold-voltage
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 voltage-gain 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 mV-range has an influence on the oscillation-period.
Without the noise this period is calculated to be 4.8 × RC , 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 ¹ 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
threshold-voltage sooner than it would in the case of equal capacitors. If one inverter reaches its
threshold-voltage 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 input-voltage of the inverter
which initiates the change of state is bigger near the threshold-voltage than it is in the equal capacitor
case.
It is important to realise that if C1 ¹ C2 the duty-cycle of the oscillation will still be 50%.
This more general case, in which C1 ¹ 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:

The Master-Slave 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 master-slave 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 master-slave 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.

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 coupling-resistors 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:

However, at Dt after the start of the exponential curve it will reach the threshold-voltage 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 coupling-resistors. This state of rest continues until the master
inverters reverse states, which will create voltage-differences across the coupling-resistors. 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:

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 slave-voltages (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
coupling-resistor will change sign and V31 will decrease. Again the master-output reverses so V31 rises;
this will happen until V31 reaches the threshold-voltage of the slave-inverter. The slave-inverter
changes state, and via the capacitor the change causes the other slave-inverter to change state as well.
The cycle will start again, only with the voltage-levels reversed.
Conclusion
The suspended bicore of fig. 3 shows a very interesting behaviour in response to noise: if the inverters
are ideal and the capacitors equal, the oscillation-period of the circuit is very sensitive to noise. This
sensitivity is decreased by using inverters which have a threshold-voltage, which is not exactly in the
middle of maximum and minimum input-voltages, 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 input-voltages of the inverters have a larger gradient near the threshold-voltage of the inverters.
Consequently the relatively small noise that is superimposed on these exponential curves will have
less influence on the exact inversion-moment, and thus less influence on the period of oscillation.

If the circuit is expanded to a master-slave 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
oscillation-period of the slave is smaller than the oscillation-frequency of the master. The phasedifference
between the master and the slave can be controlled by varying the oscillation-frequency 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 master-slave dual bicore is as driving-circuit for a two-motor 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 motor-noise is coupled back into the
master-slave dual bicore via an appropriate filter. This way both the frequency of oscillation and the
phase-difference between the master and the slave will adapt to the properties which influence the
level of noise, like the load on the motor. The oscillator-circuit itself can be designed to have a certain
sensitivity to this noise.
It should be noted that in this text the output-impedance of the inverters is assumed to be zero.
Generally this however is not the case and will have an effect on the behaviour of the circuit if a load
is applied.