Page 228 - Phase-Locked Loops Design, Simulation, and Applications

MIXED-SIGNAL PLL APPLICATIONS PART 1: INTEGER-N FREQUENCY
SYNTHESIZERS Ronald E. Best 138
where S (f ) is the “power spectral density of phase perturbation (jitter)” θ n1 at
θθ m
2
(modulating) frequency f ; the unit is rad /Hz. S (f ) is the power spectral density of the
nn m
m
noise signal at a frequency that is displaced by the offset f from the carrier frequency; the
m
unit is W/Hz. Finally, the unit of signal power P is W.
s
Next we are looking for a quantitative expression for the PSD of phase noise. Let’s
assume that the carrier power is P = 1 mW and its frequency is f . The noise source is
s 0
assumed to be thermal noise. Thermal noise has a power spectral density of kT—in other
words

(6.8)

with k = Boltzmann constant = 1.4 · 10 −23 Ws/K
T = absolute temperature in K (Kelvin)

At room temperature, we have T = 293 K, and the noise spectral density becomes S ( f ) =
nn m
(f ), we then get
0.41 · 10 −20 W/Hz. For S θθ,out m

Because this is an extremely small quantity, S , (f ) is mostly expressed in a logarithmic
θθ out m
scale—thus, in decibels. We therefore introduce a new variable S , (f )
θθ out m dB

(6.9)

The unit of S , (f ) is dBc/Hz, and for the current example, the result is S , (f ) =
θθ out m dB
θθ out m dB
−174 dBc/Hz. This tells us that the noise power contained within a bandwidth of 1 Hz (located
at a frequency that is offset by f from the carrier frequency) is 174 dB below the power of the
m
carrier. The letter “c” in the unit dBc signifies that S , (f ) stands for “noise power
θθ out m dB
referred to carrier power.” This value S , (f ) = −174 dBc/Hz is the absolute best result
θθ out m dB
we could get from an amplifier, since thermal noise is always present and real amplifiers
create additional noise. To get an idea of phase jitter to expect in this example, we compute its
rms value—thus, the square root of S , (f ). This yields
θθ out m

Note The unit dBc/Hz is widely used in textbooks, data sheets, and application notes on PLL
frequency synthesizers. From a mathematical point of view, however, this unit is not entirely
correct for the reasons explained in the following. As we know, S (f ) as defined in Eq.
θθ m
(6.7) represents a ratio of noise power to carrier power. If total