When timing is off in your computer, specific events don’t occur in the right order. But if you know the physics and *i*-*v* relationships of resistors and capacitors, you can create a circuit that detects pulses; then when a pulse is missing, the circuit can trigger an alarm notifying the user of a timing problem.

A digital system is controlled by a rectangular clock waveform serving as a standard timing reference. You can model a single rectangular pulse *v** _{S}*(

*t*) as the sum of two step functions, where one of the step functions is delayed and inverted, as you see in the following figure.

Here’s the equation for the rectangular pulse:

*V** _{A}* is the amplitude, which equals 5 volts,

*u*(

*t*) is a step input starting at time

*t*= 0, and

*u*(

*t – T*) is a delayed step function starting at

*T*= 40 nanoseconds.

The rectangular waveform serves as an input to the digital device modeled by the following circuit. The output *v*(*t*) will detect the clock pulse if *v*(*t*) exceeds the logic 1 threshold level of 3.80 volts.

Imagine that you have to find the zero-state response of the voltage *v*(*t*) when the time constant *RC* = 20 ns. Will this circuit detect the rectangular pulse? Here’s how to solve this problem:

Determine the equation for the zero-state response

*v*_{0}.*Zero-state*means the initial voltage across the capacitor is zero. As a result, the total response is determined by two inputs given in the rectangular waveform*v*(_{s}*t*):A positive step input with a 5-volt amplitude step function applied at

*t*= 0A negative step input with a 5-volt amplitude step function applied at

*t*= 40 ns

Here’s the formula for the zero-state response for a step input RC (resistor-capacitor) series circuit:

*V*is the amplitude of a step input, and_{A}*RC*is the time constant with*R*as the resistor value and*C*as the capacitor value.Determine the response due to the positive step input applied at

*t*= 0.The zero-state response is given by

The response

*v*(_{1}*t*) begins at time zero and reaches a final value of 5 volts after 5 time constants (5*RC**,**t*=*T*= 40 ns (two time constants equal to the width of the rectangular pulse), the response*v*_{1}(*t*) isDetermine the response due to the negative step input applied at

*t*= 40 ns.The second response starts at

*t*=*T*= 40 ns, and the response is equal and opposite to*v*_{1}(*t*) but is delayed by 40 nanoseconds. Before*t*= 40 ns,*v*_{2}(*t*) = 0:Find the total response

*v*(_{O}*t*)*.*You can find the total response by adding

*v*_{1}(*t*) and*v*_{2}(*t*):The following figure shows that the total response at

*t*= 40 ns is*v*(40 ns) = 4.32 V. For this example, the circuit will detect the clock pulse because the total response reaches a maximum of 4.32 volts, exceeding the threshold of 3.8 volts._{O}If the pulse width is narrowed down to 1 time constant, or 20 nanoseconds, the total response would reach a maximum of 3.16 volts, which is less than 3.8 volts. In that case, the pulse clock wouldn’t be detected.