Timers

Timers#

Many of the sequential logic circuits we’ve discussed require the input of a regular square wave to act as a clock and synchronize the timing of all the different components. In this section, we’ll explore a few different ways of generating this square wave.

Schmitt Inverter Wave Generator#

Recall an RC circuit when charging:

Figure: A circuit with at the top feeding a resistor , which connects to a capacitor to ground. The voltage across the capacitor is labeled .

The charge at the top of the capacitor is: \(V_c(t) = V_A \left(1 - e^{-t/RC}\right)\), and the max potential is \(V_A\).

When discharging:

Figure: A circuit with a resistor connected to a capacitor to ground, with the voltage across the capacitor labeled (no source — the capacitor discharges through the resistor).

Here, \(V_c(t) = V_0 \, e^{-t/RC}\) where \(V_0\) is the initial charge that had built up on the capacitor.

We can make a clock generator by attaching a Schmitt Inverter to an RC circuit.

Figure: A circuit where a Schmitt Inverter (triangle with hysteresis symbol and output bubble) has input and output . A resistor feeds back from the output to the input node , and a capacitor connects the node to ground.

Let’s use as an example the case where the Schmitt Trigger levels are \(\frac{1}{3}V_{cc}\) and \(\frac{2}{3}V_{cc}\) and see how this generates a regular square wave.

Suppose initially \(V_i\) is low. This makes \(V_o\) high, and starts to charge the capacitor, slowly increasing \(V_i\). Once \(V_i\) rises above \(\frac{2}{3}V_{cc}\), \(V_o\) will switch to low (0 V), and the capacitor will start discharging, thus lowering \(V_i\) again. Once it goes below \(\frac{1}{3}V_{cc}\), the inverter output will switch to high again and the cycle will repeat. This process is shown in the timing diagram below.

Figure: A timing diagram. oscillates between two dashed horizontal lines at and , with exponential charging rises and discharging falls (a shark-fin / sawtooth-like waveform). Below it, is a square wave switching between and 0, with red dashed vertical lines aligning its transitions to the moments crosses the thresholds.

Now let’s determine how long the the wave spends in each of the high and low states.

How long is the capacitor charging for? I.e. how long is the output in the high state?

The charging formula assumes that the initial voltage on the capacitor is 0 V. Thus to find the time it takes to go from \(\frac{1}{3}V_{cc}\) to \(\frac{2}{3}V_{cc}\), we have to find the time to go from \(0 \to \frac{1}{3}V_{cc}\) and subtract it from the time to go from \(0 \to \frac{2}{3}V_{cc}\). Let’s define \(t_2\) as the time to charge from 0 V all the way to the upper threshold, and \(t_1\) the time it takes to go from 0V to the lower threshold. Then, \(t_{charging} = t_{2} - t_{1}\).

First let’s calculate \(t_{2}\):

\[\frac{2}{3}V_{cc} = V_{cc}\left(1 - e^{-t_{2}/RC}\right) \quad \text{[Charging Formula]}\]
\[\frac{2}{3} = 1 - e^{-t_{2}/RC}\]
\[e^{-t_{2}/RC} = \frac{1}{3}\]
\[-\frac{t_{2}}{RC} = \ln\!\left(\frac{1}{3}\right)\]
\[t_{2} = -\ln\!\left(\frac{1}{3}\right) RC\]

Similarly, for \(t_{1}\):

\[t_{1} = -\ln\!\left(\frac{2}{3}\right) RC\]

and thus:

\[t = t_{2} - t_{1}\]
\[t = \left(-\ln\!\left(\frac{1}{3}\right) + \ln\!\left(\frac{2}{3}\right)\right) RC \approx 0.693RC\]

By changing the resistance and capacitance we can control the charging time!

For discharging, we have the much easier task of determining how long it takes for the capacitor to discharge from the high threshold to the low threshold. How long is the capacitor discharging, i.e. how long does the wave generator’s output in the low state?

\[\frac{1}{3}V_{cc} = \frac{2}{3}V_{cc}\, e^{-t/RC} \quad \text{[Discharging Capacitor Eq'n]}\]
\[\frac{1}{2} = e^{-t/RC}\]
\[\ln\!\left(\frac{1}{2}\right) = -\frac{t}{RC}\]
\[t = -\ln\!\left(\frac{1}{2}\right) RC\]
\[t \approx 0.693RC\]

In this setup, the discharge time is exactly equal to the charging time! This is not true in general, but arises from our choice of threshold levels. In general, we define a quantity called the duty cycle (\(D\)) of a square wave: that is the percentage of time spent in the high state.

Figure: A square wave with braces underneath marking (the low interval) and (the high interval), and a longer brace marking period.

\[\text{D} = \frac{t_H}{T} \times 100\%\]

The 555 Timer#

Finally, we come to the 555 timer, a common IC used to generate a clock signal. The name comes from three 5 kΩ resistors that are used to set threshold voltages of comparators.

Figure: A 555 timer circuit. Externally, feeds , then , then a capacitor to ground. A dashed red box labeled "555 Timer" outlines the IC internals: feeds a voltage divider of three equal resistors in series to ground, producing nodes (between the first and second resistors) and (between the second and third). The node between and connects to the collector of a discharge transistor whose emitter goes to ground. The node between and connects to the "+" input of the upper comparator (whose "−" input is ) and to the "−" input of the lower comparator (whose "+" input is ). The upper comparator drives the R input of an SR latch; the lower comparator drives the S input. The latch's output drives the base of the discharge transistor, and its output is , drawn as a square wave.

When the output voltage \(Q\) of the SR Latch is high, \(\bar{Q}\) is low, and the control transistor is thus in cutoff mode meaning no current can flow from collector to emitter. This leads to the capacitor charging.

Once the voltage between \(R_B\) and \(C\) rises above \(V_{T+}\) (\(= \frac{2}{3}V_{cc}\)), this triggers the reset on the latch. \(Q\) switches to low and \(\bar{Q}\) to high. This allows current to flow through the transistor, and the capacitor discharges until the voltage between \(R_B\) and \(C\) falls below \(V_{T-}\) (\(= \frac{1}{3}V_{cc}\)). This sets \(Q\) to high again and the cycle starts anew.

Note that the capacitor charges through \(R_A\) & \(R_B\), but discharges solely through \(R_B\). Thus:

\[\begin{split}\boxed{\begin{aligned} t_L &\approx 0.693R_BC \\ t_H &\approx 0.693(R_A + R_B)C \end{aligned}} \quad \text{for a 555 timer}\end{split}\]

Aside: Crystal Oscillators

Instead of using RC circuits, most computers use quartz crystal oscillators as a clock, which are far more precise. The quartz crystal is deformed by an electric field. If the field is removed, the crystal snaps back to its original shape. But the movement of the crystal creates another electric field. If this is fed back to the crystal it will oscillate at a resonant frequency determined by the shape and size of the crystal.