Thinking in decibels
28 Jan 2014  Eric BogatinShare this page with your friends
Next, we look at some of the subtle aspects of the dB scale, like how much voltage is 10dBm and do we use 10 or 20 when describing an impedance in the dB scale.
Decibels and SParameters
Earlier, I introduced the dB (decibel) scale to measure the ratio of two powers as dB=10log(P1/P0). I explained why we use a factor of 10 when describing the ratio of powers, but a factor of 20 when describing the ratio of voltages. I also explained that Sparameters, a ratio of voltages, need a factor of 20 when converted into dB.
An S11 with a value of 20 dB means a reflected voltage signal that is 10^{20/20} = 10% the incident voltage signal. Because power scales as S11², the power reflected is 1% the incident power, which is 10^{20/10} = 1%. This is a pretty good value for most digital interconnect structures.
A value of an output power of 3 dB means that the ratio of output to input power is 10^{3/10} = 0.5. This is what most of us learn early in our engineering career: 3 dB down is a 50% drop. But, this is a 50% drop in the power level. What happens to the voltage level?
A value of 3dB is always a 50% drop in power, but the ratio of output voltage to input voltage is 10^{3/20} = 70%. When S21 is 3dB, the voltage level of the signal coming out is 70% of the voltage going in.
When we refer to the SNR (signaltonoise ratio), we usually measure it in dB. Without exception, the SNR in dB always refers to the ratio of the power between the signal and noise. When described in dB, the SNR can immediately be converted into the ratio of the signal power to the noise power by SNR_power = 10^{SNR_dB/10}.
An SNR of 20 dB means the signal power is 100x the noise power. What is the ratio of the signal voltage to noise voltage? SNR_voltage = 10^{SNR_voltage/20}. The same SNR of 20 dB, a signal power 100x the noise power, is at the same time, a signal voltage = 10x the noise voltage.
We often use the dB scale to describe cross talk. An isolation of noise to signal of 60 dB is a noise power that is 10^{60/10} = 1,000,000 times smaller than the signal power. But, the voltage noise is only 10^{60/20} = 1,000 times smaller than the signal voltage.
In some mixedsignal applications, 100dB isolation is needed between the RF receiver and the noisy digital circuits. This means the received noise voltage needs to be 0.001% of the signal voltage. This is a tiny amount of coupling and is very difficult to achieve in practice.
In some applications, impedance is measured in dB. While the reference level of impedance is always 1Ω, do we use a 10x or 20x when converting impedance in Ohms to impedance in dBΩ? Is Ohms an amplitude or a power?
The way to answer this question is by observing how impedance is derived from an Sparameter. In a 1port measurement, impedance is directly related to the ratio of (1+S11)/(1S11). In a 2port Sparameter measurement of impedance, the impedance is roughly = 25Ω×S21.
Counterintuitively, impedance is considered an amplitude and uses a factor of 20 when converting from Ohms to dB.
An impedance of 0dB is an impedance in Ohms of 10^{0/20} = 10^{0} = 1Ω. When the impedance is 10mΩ, the value in dB is 20log(0.01/1)=40 dB. Figure 2 shows the same measured impedance of a decoupling capacitor displayed in impedance and in dB.
Figure 2: The upper plot shows a measured impedance profile of a capacitor displayed in Ohms with the same measurement in the lower plot, shown in dB. 
What else have you encountered that is measured in dB and is it an amplitude or a voltage?
About the author
Eric Bogatin is Signal Integrity Evangelist at Teledyne LeCroy.
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