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Is impedance matching as important when working with digital circuits as it is with analog circuits?
December 31, 2007

Impedance matching is just as important for digital electronics as it is for analog electronics. Remember that the real world is analog. At the hardware level, digital signals are analog signals with sharp edges.

Two issues arise with impedance mismatches in analog circuits: reflections and inability to drive power. Reflections occur whenever an electromagnetic wave on a transmission line reaches an abrupt change in surge impedance. Power transfer between a source and load is thus best when the source impedance matches the load impedance.

A signal circuit always includes a source, transmission line, and load. Each has its own characteristic impedance.

Every transmission line connects a source and a load. Suppose we have an RF source (say, a 300 MHz system clock) with an output impedance of R_s connected to a transmission line having a surge impedance of R_t, and terminated by a load resistance R_L. Let’s say for starters that all three impedances are 100 Ω and that the RF source is trying to drive a 10 V amplitude. We’ll use root-mean-squared (RMS) amplitudes to make the power calculations easier.

Since the impedances all match, it’s easy to see what happens. Ten volts into 100 Ω gives a current of 100 mA and a power of 1.0 W traveling down the transmission line. When it reaches the end, all that power goes into heating the load because Kirchoff’s law for current tells us that all the current in the transmission line must flow through the load, and P = I²R.

If, however, we lower the load impedance to 50 Ω, something nasty comes out of the woodshed. The current still must be the same in the transmission line and in the load, but now Ohm’s law tells us that the voltage across the load can only be 5 V.

What happens to the other 5 V? It goes into creating a 5 V wave reflected back into the transmission line. The two waves are at the same frequency, but 180º out of phase at the load so that they sum to a 5 V amplitude (10 V + (-5 V) = 5 V) across the load. The ratio of the reflected wave amplitude to that of the forward wave is called the voltage standing wave ratio (VSWR).

When the load impedance doesn’t match the line impedance, a reflected wave appears, reducing the power transmitted through the line.

The power absorbed by the load equals the voltage across the load squared divided by the load impedance, which comes out to 0.5 W. What happened to the other half Watt? It was reflected back to the source, where it ended up heating the final amplifier.

In the old vacuum-tube radio days, we sometimes tuned our transmitter antennas by watching the visible “plate” of the final amplifier. If there was a significant mismatch, the plate would glow red. We’d just tune the antenna until the plate stopped glowing!

Vacuum tubes could take that abuse. Transistors in modern transmitter final amplifiers can’t. They turn from power transistors into lumps of amorphous silicon.

What if we increased the load impedance in our example circuit to 200 Ω? Again, the current would be the same throughout the circuit by Kirchoff’s law, only this time we can’t drive 100 mA into the load because that would take 20 V, and we have only 10 V.

This time, it’s the current that drops—to 50 mA. The only way to make that happen is to reflect a current wave back along the transmission line. Again, this wave carries half of the power back to the source, where it again dissipates in the final amplifier.

It is important to recognize that these problems with power transmission are frequency independent. They, in fact, apply unchanged all the way down to dc. Certainly, they affect all digital electronics. Impedance mismatches can therefore affect the logic levels at the receiving end. If too much voltage is lost in transmission, flip-flops just flop, and logic gates become fences.

Note, however, that when the load impedance exceeds the source impedance, full voltage appears across the load. If that’s all you care about, feel free to jack the load impedance as high as you please without worrying about mismatches.

I do this all the time to minimize voltage drops in low-frequency signal leads. At frequencies below a few hundred kilohertz, reflections aren’t a problem because the line length is short compared to the signal wavelength: there’s no room for phase shifts. I’ll often terminate the source with a low-noise resistor matching the output impedance, then use twisted pair or even coaxial cable (to eliminate EMI) for a transmission line. Finally, I'll couple that transmission line directly to the input of a super-high-impedance voltage amplifier. The amplifier input impedance is reflected directly back across the transmission line, and appears as a high resistive impedance in parallel with the relatively low source-termination resistance. The source thinks there’s nothing there, but the output resistor!

When there’s a mismatch at the transmission line’s source end as well as at the load end, however, reflections get reflected, leading to standing waves and resonances in the transmission line—if it’s long enough. These effects start becoming important when the transmission line becomes longer than about an eighth of the signal’s wavelength. Resonances occur whenever the transmission line is an integral number of half wavelengths long. 

Impedance mismatches at both ends of a transmission line create standing waves that resonate whenever the line length equals an integer number of half wavelengths.

Fourier components of square waves have significant (20% or more) amplitude out to the fourth harmonic (fundamental frequency X 5). The fourth harmonic of our 300 MHz clock rate is 1.5 GHz.

Approximating the speed of signals in the transmission line as the 3E8 m/s speed of light in vacuum (it's about 95% of that in most transmission lines) gives a wavelength for the clock’s fundamental as 1 m, and for the fourth harmonic as 20 cm. Any mismatched transmission line longer than one eighth of that (2.5 cm = 1 in) will show reflection effects. These effects will distort digital pulses’ leading and trailing edges, causing jitter, synchronization problems, and even data loss.

Network-standard specifications include impedance tolerances for connectors, cables, and the electronics at both ends. As long as you are using high-quality commercial off the shelf (COTS) cables, connectors, and transmit/receive electronics at both ends, you can expect to have trouble-free digital-signal communications. If, however, you insist on making your own, don’t forget to match your impedances carefully. Otherwise, you’re likely to find yourself with data transmission problems you’ll never chase down.

Posted by Charlie Masi on December 31, 2007 | Comments (2)