- 2.1 The Time Domain
- 2.2 Sine Waves in the Frequency Domain
- 2.3 Shorter Time to a Solution in the Frequency Domain
- 2.4 Sine Wave Features
- 2.5 The Fourier Transform
- 2.6 The Spectrum of a Repetitive Signal
- 2.7 The Spectrum of an Ideal Square Wave
- 2.8 From the Frequency Domain to the Time Domain
- 2.9 Effect of Bandwidth on Rise Time
- 2.10 Bandwidth and Rise Time
- 2.11 What Does Significant Mean?
- 2.12 Bandwidth of Real Signals
- 2.13 Bandwidth and Clock Frequency
- 2.14 Bandwidth of a Measurement
- 2.15 Bandwidth of a Model
- 2.16 Bandwidth of an Interconnect
- 2.17 The Bottom Line

## 2.16 Bandwidth of an Interconnect

The bandwidth of an interconnect refers to the highest sine-wave-frequency component that can be transmitted by the interconnect without significant loss. What does *significant* mean? In some applications, a transmitted signal that is within 95% of the incident signal is considered too small to be useful. In other cases, a transmitted signal that is less than 10% of the incident signal is considered usable. In long-distance cable-TV systems, the receivers can use signals that have only 1% of the original power. Obviously, the notion of how much transmitted signal is significant is very dependent on the application and the particular specification. In reality, the bandwidth of an interconnect is the highest sine-wave frequency at which the interconnect still meets the performance specification for the application.

The bandwidth of an interconnect can be measured in either the time domain or the frequency domain. In general, we have to be careful interpreting the results if the source impedance is different than the characteristic impedance of the line, due to the complication of multiple reflections.

Measuring the bandwidth of an interconnect in the frequency domain is very straightforward. A network analyzer is used to generate sine waves of various frequencies. It injects the sine waves in the front of the interconnect and measures how much of each sine wave comes out at the far end. It is basically measuring the transfer function of the interconnect, and the interconnect is acting like a filter. This is also sometimes referred to as the *insertion loss* of the interconnect. The interpretation is simple when the interconnect is 50 Ohms, matched to the network analyzer's impedance.

For example, Figure 2-19 shows the measured transmitted amplitude of sine waves through a 4-inch length of a 50-Ohm transmission line in FR4. The measurement bandwidth is 20 GHz in this case. The 3-dB bandwidth of the interconnect is seen to be about 8 GHz. This means that if we send in a sine wave at 8 GHz, at least 70% of the amplitude of the 8-GHz sine wave would appear at the far end. More than likely, if the interconnect bandwidth were 8 GHz, nearly 100% of a 1-GHz sine wave would be transmitted to the far end of the same interconnect.

Figure 2-19 Measured transmitted amplitude of different sine-wave signals through a 4-inch-long transmission line made in FR4. The 3 dB bandwidth is seen to be about 8 GHz for this cross section and material properties. Measured with a GigaTest Labs Probe Station.

The interpretation of the bandwidth of an interconnect is the approximation that if an ideal square wave were transmitted through this interconnect, each sine-wave component would be transmitted, with those components lower than 8 GHz having roughly the same amplitude coming out as they did going in. But the amplitude of those components above 8 GHz would be reduced to insignificance.

A signal that might have a rise time of 1 psec going into the interconnect would have a rise time of 0.35/8 GHz = 0.043 nsec or 43 psec when it came out. The interconnect will degrade the rise time.

If the bandwidth of an interconnect is 1 GHz, the fastest edge it can transmit is 350 psec. This is sometimes referred to as its intrinsic rise time. If a signal with a 350-psec edge enters the interconnect, what will be the rise time coming out? This is a subtle question. The rise time exiting the interconnect can be approximated by:

#### Equation 2-6

where:

- RT
_{out}= the 10–90 rise time of the output signal - RT
_{in}= the 10–90 rise time of the input signal - RT
_{interconnect}= the intrinsic 10–90 rise time of the interconnect

This assumes that both the incident spectra and the response of the interconnect correspond to a Gaussian-shaped rise time.

For example, in the case of this 4-inch-long interconnect, if a signal with a rise time of 50 psec were input, the rise time of the transmitted signal would be:

#### Equation 2-7

This is an increase of about 17 psec in the rise time of the transmitted waveform compared to the incident rise time.

In Figure 2-20, we show the measured time-domain response of the same 4-inch-long, 50-Ohm interconnect that was measured in the frequency domain above. The input waveform has been time shifted to lie directly at the start of the measured output waveform.

Figure 2-20 Measured input and transmitted signal through a 4-inch long, 50-Ohm transmission line in FR4 showing the rise-time degradation. The input rise time is 50 psec. The predicted output rise time is 67 psec based on the measured bandwidth of the interconnect. Measured with a GigaTest Labs Probe Station.

The rise time of the waveform going into the PCB trace is 50 psec. The measured 10–90 rise time of the output waveform is about 80 psec. However, this is somewhat distorted by the long roll to stabilize at the top, characteristic of the behavior of lossy lines. The extra delay at the 70% point is about 15 psec, which is very close to what our approximation above predicted.

If a 1-nsec rise-time signal enters an interconnect with an intrinsic rise time of 0.1 nsec, the rise time of the signal transmitted would be about sqrt(1 nsec^{2} + 0.1 nsec^{2}), or 1.005 nsec, which is still basically 1 nsec. The interconnect would not affect the rise time. However, if the interconnect intrinsic rise time were 0.5 nsec, the output rise time would be 1.1 nsec, and would start to have a significant impact.

It is important to keep in mind that this is a rule of thumb and it should not be used for design sign-off. It should be used only for a rough estimate or to identify a goal. If the bandwidth of an interconnect is within a factor of two of the bandwidth of the signal, it would probably be important to perform an analysis of how the interconnect affected the entire signal's spectrum.