Time Constant
From BEAM Robotics Wiki
- A term used in reference to the temporal characterisics of a resistor-capacitor (RC) pair; specifically the period of time required for the potential across a capacitor, being charged through a resistor, to reach a specific value in volts.
- In general electronics, the voltage value typically used is equal to 65% of the potential across the entire RC circuit. In this case the time constant can be easily calculated using the formula: T=R*C, where...
- In reference to BEAM circuits (Nvs and/or Nu) the voltage value used is equal to the switching threshold of the inverter being used for the circuit in question. Calculating the time constant of a BEAM circuit can be a bit more involved.
[edit] Practical Formulas for BEAM Circuits
The following is from an exchange between Wilf Rigter and J Wolfgang Goerlich that took place on the BEAM Group on Yahoo
It has been edited for flow and to include several corrections that Wilf made regarding the original material.
A 74HC240 type microcore (using reset diodes) and a grounded 74HC240 Bicore have a pulse duration T=0.65RC.
... At Vcc=5V, the input switching threshold of a 74HC14 is about 1.9V and the pulse duration calculation can be simplified to T=RC
The timing is different with 74HC/74AC logic for which the threshold is about 2.5V and 74HCT/HCT logic for which the threshold is 1.6V.
The grounded Nv is an RC differentiator connected to a logic inverter input with a switching threshold of Vth. With a grounded Nv one side of the input capacitor is rapidly driven positive by a 74HC output to Vcc. At that point the initial voltage across the cap is assumed to be 0V and as the other side of the cap is connected through R to 0V, the initial voltage across the resistor is Vcc. As the voltage across the capacitor (Vc) starts to increase exponentially to Vcc, the voltage across the grounded resistors (Vr) at the inverter input decreases correspondingly to 0V. At time T, when Vr crosses the switching threshold Vth, the Nv output changes state. Therefore the Nv output pulse duration T is dependent on R, C, the limit voltages (Vmax and Vmin) and the threshold voltage (Vth).
(1) T = RC log e (1/((Vth-Vmin)/(Vmax-Vmin)))
Since the maximum voltage Vmax=Vcc while Vmin= 0V, the simplified expression for calculating the Nv pulse duration is:
(2) T = RC log e (1/(Vth/Vcc))
To simplify the expression further for Vcc = 5V, the grounded Nv pulse duration T for each logic type is:
(3) (For a 74HC14 with a threshold equal to 1.9V, T=R x C x ln x (1/(1.9V/5V)) = 0.97RC
(4) For a 74HC240 with a threshold equal to 2.6V, T=R x C x ln x (1/(2.6V/5V)) = 0.65RC
(5) For a 74HCT240 with a threshold equal to 1.6V, T=R x C x ln x (1/(1.6V/5V)) = 1.14RC
The ratio of Vth to Vcc is nearly constant for 74HC/AC logic over a wide range of Vcc, so the expression 0.65RC applies and the pulse duration does not vary with (slow) changes in power supply voltage.
I have tested a few 74HC14 chips and some (but not all) exhibit constant pulsewidth with supply variation.
For the only 74HCT logic chip I tested, the ratio of Vth to Vcc was not constant with Vcc, so for a given Vcc, the threshold voltage Vth or simply the pulse duration should be empirically determined.
Figure 2 has two parallel resistor]s R1 and R2 which are both connected to 0V and expression (2) can be used with R=R total
Figure 3, is a more general case of parallel resistors. For the purpose of timing the resistance R is equal to the parallel resistance
(6) R= Rtotal = 1/ (1/R1 +1/R2)
but R1 and R2 also form a voltage divider with a midpoint voltage of
(7) Vmin = Vcc x (R2 / (R1+R2))
Now the maximum capacitor voltage (Vc) is not Vcc - 0V but Vcc-Vmin.
If Vmin is greater than Vth, the Nv will obviously never timeout. So the minimum ratio of resistor values must ensure that R1 > ((Vcc / Vth)-1) x R2..
The expression in (1) can now be used for Figure 3.
Let's use some common values
Vcc=5V
Vth=2.6V (74HC240)
R1= 10M
R2=4.3M
C=0.33uF
When coupling resistor R1 is driven by a low frequency bicore output between 0V and Vcc, two different pulse durations (T1 and T2) are generated
Solving for T1 with R1 connected to ground, the circuit is like Figure 2
(6) R=3M
(4) T1 = 0.65 x 1
T1 = 0.65 seconds
Solving for T2 with R1 connected to Vcc, the circuit is like Figure 3
(6) R=3M
(7) Vmin=1.7V
(1) T2 = RC log e (1/((2.6V-V1.7V)/(5V-1.7V)))
T2 = 1 x ln x (1/(0.9V/3.3V))
T2= 1.3 seconds (exactly 2 x T1)
Note that for Nvs with resistors connected to Vcc, the threshold voltage Vth will be with respect to Vcc except that for 74AC/HC devices which are nearly symmetrical, Vth remains about the same (~Vcc/2).
Also note that the TC for suspended bicores can be similarly calculated.
