# Converting from normalized SFDR (dBHz^(2/3)) to real SFDR (dB)

SFDR is frequently written in the units of dBHz^(2/3), particularly for fiber optic links. Fiber optic links can often have such high bandwidth, that assuming a bandwidth in SFDR is unhelpful or misleading. Normalizing to 1Hz therefore became a standard practice. The units of SFDR for a real system with a bandwidth are dB.

Now consider that the real system has a specific bandwidth. The real SFDR can be calculated using the following formulas:
SFDR_real = SFDR_1Hz – (2/3)*10*log10(BW)

Here are a few examples.

# Where do the units of SFDR “dB·Hz^(2/3)” come from?

The units of spurious-free dynamic range (SFDR) are dB·Hz^(2/3). The units can be a source of confusion. The short answer is that it is a product of ratios between power levels (dBm) and noise power spectral density (dBm/Hz). The units of dBHz^(2/3) are for SFDR normalized to a 1Hz bandwidth. For the real SFDR of a system, the units are in dB.

If we look at a plot of the equivalent input noise (EIN), the fundamental tone, OIP3 (output intercept point of the third order distortion), and IMD3 (intermodulation distortion of the third order), a ratio of 2/3 exists between OIP3 and SFDR. This can be recognized from the basic geometry, given that the slop of the fundamental is 1 and the slope of IMD3 is 3.

Now, we need to look at the units of both OIP3 and EIN. The units of OIP3 are dBm and the units of the equivalent input noise (a noise power spectral density) are dBm/Hz.

SFDR = (2/3)*(OIP3 – EIN)

[SFDR] = (2/3) * ( [dBm] – [dBm/Hz] )

Now, remember that in logarithmic operations, division is equal to subtracting the denominator from the numerator. and therefore:

[dBm/Hz] = [dBm] – 10*log_10([Hz])

Note that the [Hz] term is still in logarithmic scale. We can use dBHz to denote the logarithmic scale in Hertz.

[dBm/Hz] = [dBm] – [dBHz]

Substituting this into the SFDR unit calculation:

[SFDR] = (2/3) * ( [dBm] – ( [dBm] – [dBHz] )

This simplifies to:

[SFDR] = (2/3) * ( [dBm] – [dBm] + [dBHz] )

Remember that the difference between two power levels is [dB].

[SFDR] = (2/3) * ( [dB] + [dBHz] )_

The units of [dB] + [dBHz] is [dBHz], as we know from the same logarithmic relation used above for [dBm] and [dB].

[SFDR] = (2/3) * [dBHz]

Now, remember that this is a lkogarithmic operation, and a number multiplying a logarithm can be taken as an exponent in the inside of the logarithm.Therefore, we can express Hz again explicitly in logarithm scale, and move the (2/3) into the logarithm.

(2/3) * [dBHz] = (2/3) * 10*log_10([Hz]) = 10*log_10([Hz]^(2/3))

We can return our units back to the dB scale now, giving us the true units for SFDR: dBHz^(2/3):

[SFDR] = [dBHz^(2/3)]

# Microstrip Stub Low-Pass Filter (10 GHz)

This is a 10 GHz Stub Low-pass filter, made using ADS.

First, build the component using the ADS DesignGuide/Smart Component Passive Circuit tool.

This is the original, equation-based simulation.

This is the substrate used for the Low-pass filter.

This is the Momentum simulation of the layout component.

This is the layout component for the 10 GHz Stub Low-pass filter component.

# Microstrip Lange Coupler (5 GHz)

The Lange Microstrip (quadrature) coupler is known for it’s low loss, wide bandwidth and compact layout. Similar to other couplers, it consists of an isolated port, through port and coupled port.

You can build a microstrip Lange coupler using the DesignGuide tool in ADS:

These are the results for the equation-based simulation. These results admittedly look considerably better.

This is the substrate used:

These are the results for the momentum simulation. Admittedly, some tuning would improve this considerable.

And here is the layout component:

# 027/100 Shunt Reactance on Smith Chart

Example 3.5-2A: Measure the effect of susceptance on Smith Chart impedance matching.

First, build the circuit and run the Smith Chart Matching tool.

A shunt capacitor moves in a clockwise direction across the smith chart tool:

Also note that a shunt inductor moves counter-clockwise across the smith chart tool:

# Microstrip Double-Stub Load Matching

The following matches a 50 Ohm line to a 100 Ohm load at 10 GHz using a double-stub component. This was designed using the ADS passive circuit DesignGuide tool. This method is a great alternative to using the Smith Chart matching tool for lumped elements if you need a microstrip line for matching.

Circuit:

Substrate:

Momentum simulation result (can be tuned to center better at 10 GHz):

Layout component:

# 026/100 Lumped Element Smith Chart Movements: Series Inductor

Example 3.5-1: Measure the amount of movement caused by the reactance added to the circuit below. Measure the change from the starting point to the end point on the Smith Chart.

The circuit simulated gives the following result:

Recall that the circuit without a series inductor had the following result:

Through this simulation, it is shown that adding a series inductor causes the smith chart diagram plot to move in a clockwise direction.

Note the change using the Smith Chart matching tool: