Because a rectangular wave such as a square wave theoretically contains an infinite number of harmonics, we can use a square wave as the basis for determining the bandwidth of a signal. If the processing circuit should pass all or an infinite number of harmonics, the rise and fall times of the square wave will be zero. As the bandwidth is decreased by rolling off or filtering out the higher frequencies, the higher harmonics are greatly attenuated. The effect this has on the square wave is that the rise and fall times of the waveform become finite and increase as more and more of the higher harmonics are filtered out. The more restricted the bandwidth, the fewer the harmonics passed, and the greater the rise and fall times. The ultimate restriction is where all the harmonics are filtered out, leaving only the fundamental sine wave (Fig. 2-63).

The concept of rise and fall times is illustrated in Fig. 2-66. The rise time tr is the time it takes the pulse voltage to rise from its 10 percent value to its 90 percent value. The fall time tf is the time it takes the voltage to drop from the 90 percent value to the 10 percent value. Pulse width t0 is usually measured at the 50 percent amplitude points on the leading (rise) and trailing (fall) edges of the pulse. A simple mathematical expression relating the rise time of a rectangular wave and the bandwidth of a circuit required to pass the wave without distortion is

**Example 2-28** A pulse train has a rise time of 6 ns. What is the minimum bandwidth to pass this

pulse train faithfully?

This is the bandwidth of the circuit required to pass a signal containing the highest frequency component in a square wave with a rise time of tr. In this expression, the bandwidth is really the upper 3-dB down the cutoff frequency of the circuit given in megahertz. The rise time of the output square wave is given in microseconds. For example, if the square wave output of an amplifier has a rise time of 10 ns (0.01 μs), the bandwidth of the circuit must be at least BW = 0.35/0.01 = 35 MHz. Rearranging the formula, you can calculate the rise time of an output signal from the circuit whose bandwidth is given: tr = 0.35/BW. For example, a circuit with a 50-MHz bandwidth will pass a square wave with a minimum rise time of tr = 0.35/50 = 0.007 μs = 7 ns. This simple relationship permits you to quickly determine the approximate bandwidth of a circuit needed to pass a rectangular waveform with a given rise time. This relationship is widely used to express the frequency response of the vertical amplifier in an oscilloscope. Oscilloscope specifications often give only a rise time figure for the vertical amplifier. An oscilloscope with a 60-MHz bandwidth would pass rectangular waveforms with rise times as short as tr = 0.35/60 = 0.00583 μs = 5.83 ns.

**Example 2-29 **A circuit has a bandwidth of 200 kHz. What is the fastest rise time this circuit will pass?

Similarly, an oscilloscope whose vertical amplifier is rated at 2 ns (0.002 μs) has a bandwidth or upper cutoff frequency of BW 5 0.35/0.002 5 175 MHz. What this means is that the vertical amplifier of the oscilloscope has a bandwidth adequate to pass a sufficient number of harmonics so that the resulting rectangular wave has a rise time of 2 ns. This does not indicate the rise time of the input square wave itself. To take this into account, you use the formula

**tri = rise time of input square wavetra = rise time of amplifiertr = composite rise time of amplifier output**

The expression can be expanded to include the effect of additional stages of amplification by simply adding the squares of the individual rise times to the above expression before taking the square root of it.

**Example 2-30 **An oscilloscope has a bandwidth of 60 MHz. The input square wave has a rise time of 15 ns. What is the rise time of the displayed square wave?

Keep in mind that the bandwidth or upper cut-off frequency derived from the rise time formula on the previous page passes only the harmonics needed to support the rise time. There are harmonics beyond this bandwidth that also contribute to unwanted emissions and noise.

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