135
Original scientific paper
Extending Leeson’s Equation
Matjaž Vidmar
Univerza v Ljubljani, Fakulteta za Elektrotehniko, Ljubljana, Slovenia
Abstract: The oscillator phase noise is one of the key limitations in several fields of electronics. An electronic oscillator phase noise 
is usually described by the Leeson’s equation. Since the latter is frequently misinterpreted and misused, a complete derivation of the 
Leeson’s equation in modern form is given first. Second, effects of flicker noise and active-device bias are accounted for. Next the 
complete spectrum of an electronic oscillator is derived extending the result of the Leeson’s equation into a Lorentzian spectral line. 
Finally the spectrum of more complex oscillators including delay lines is calculated, like opto-electronic oscillators.
Keywords: phase noise; Leeson’s equation; oscillator bias; Lorentzian line; opto-electronic oscillator
Razširitev Leesonove Enačbe
Izvleček: Fazni šum oscilatorja je ena ključnih omejitev v številnih področjih elektronike. Fazni šum elektronskega oscilatorja običajno 
opisuje Leesonova enačba. Ker je slednja pogosto slabo razumljena in napačno uporabljena, bo najprej opisana celotna izpeljava 
Leesonove enačbe. V drugem koraku je nujna obravnava učinkov šuma 1/f  in nastavitve delovne točke aktivnega gradnika. Sledi 
celovita izpeljava spektra elektronskega oscilatorja, ki rezultat Leesonove enačbe razširi v Lorentzovo spektralno črto. Končno se izpelje 
spekter bolj kompliciranih oscilatorjev, kot so to opto-elektronski oscilatorji.
Ključne besede: fazni šum; Leesonova enačba; delovna točka oscilatorja; Lorentzova črta; opto-elektronski oscilator
* Corresponding Author’s e-mail: matjaz.vidmar@fe.uni-lj.si
Journal of Microelectronics, 
Electronic Components and Materials
Vol. 51, No. 2(2021), 135 – 146
https://doi.org/10.33180/InfMIDEM2021.205
1 Introduction
Towards the end of the 19
th
 century, the Hertz experi-
ments connected two areas of physics, namely elec-
tricity and optics. While radio communications started 
with filtered noise from spark gaps, the latter were 
quickly replaced by much more efficient vacuum-tube 
electronic oscillators, invented independently by Arm-
strong and Meissner around 1912.
Electronic oscillators were so successful that their spec-
trum was considered an infinitely narrow spectral line 
at relatively low radio frequencies f<30MHz in the first 
half of the 20
th
 century. Their spectral line was only 
broadened by external causes like unfiltered supply, 
load pull, temperature drift and/or vacuum-tube aging.
On the other hand, in optics it was quickly discovered 
that spectral lines of different light sources were not in-
finitely narrow. The optical line width ∆λ
o
 or ∆f could be 
measured with (relatively simple) interferometers and 
expressed as longitudinal coherence length d in free 
space c
0
:
 
2
00
0
λ
ΔΔ λ
c
d
f
≈≈    (1)
Unfortunately the amplitude dynamic range of simple 
optical instruments was quite limited.
In the second half of the 20
th
 century, both the fre-
quency resolution of radio measurements as well as 
the amplitude dynamic range of optical measurements 
improved by several orders of magnitude. Both keep 
improving as the user requests keep increasing. Last 
but not least, the spectrum gap between radio and 
optics is shrinking as radio frequencies are increasing 
towards the terahertz region and optical wavelengths 
are increasing towards the far-infrared region.
One of the most important contributions is the deriva-
tion of the oscillator noise spectrum by David Leeson in 
1966 [1]. The same derivation is applicable to (relatively 
low) radio-frequency electronic oscillators as well as to 
lasers. In electronics, high-performance oscillators are 

136
M. Vidmar; Informacije Midem, Vol. 51, No. 2(2021), 135 – 146
followed by buffer stages that may add their own noise. 
Electronic limiters may reduce the amplitude noise but 
they have no effect on the phase noise.
The design of a performing radio-frequency oscillator 
is complex. Besides basic radio-frequency design the 
knowledge of different noise contributions is required 
as well as the knowledge of feedback theory. Due to 
this complexity the Leeson’s equation is frequently mis-
understood, misused and even degraded to an “empiri-
cal” equation by some sources. The term phase noise 
only starts appearing in equipment specifications as 
well as in text books in the 21
st
 century as it is becoming 
the limiting parameter for increasingly complex modu-
lation schemes at ever increasing carrier frequencies.
2 Electronic oscillator
An electronic oscillator includes an amplifier with a 
voltage gain A and a feedback network with a voltage 
transfer function H(ω). The feedback network is usually 
a frequency-selective resonator to define the output 
spectrum of the oscillator:
Figure 1: Electronic oscillator.
For the circuit to oscillate, the Barkhausen criterion ap-
plies:
 
()
0
1 AHω ⋅=     (2)
The Barkhausen criterion is an equation with complex 
numbers defining both the phase and the magnitude 
of the feedback. The circuit can only oscillate at the 
frequency ω
0
 where the feedback phase is zero or an 
integer multiple of 2π. The amplifier should provide 
enough gainto start the oscillation. During steady os-
cillation, saturation will eventually decrease the ampli-
fier gain A to satisfy the Barkhausen criterion.
Some feedback networks may generate complex re-
sults. A laser may oscillate at many different modes at 
the same time. Some electronic circuits may satisfy the 
Barkhausen criterion at zero frequency. Such circuits do 
not oscillate but act as bi-stables. A flip-flop intention-
ally driven into a meta-stable state will quickly settle 
into one of its two stable states.
Some form of noise is always present in all circuits. In 
electronic circuits operating in the radio-frequency 
range, the main contribution is thermal noise. No mat-
ter how small, noise will always significantly affect the 
output spectrum of an oscillator as shown later in the 
derivation of the Leeson’s equation.
In the case of a class A amplifier, noise actually starts 
the oscillation:
Figure 2: Oscillator start.
With some excess gain, the oscillation amplitude will 
initially grow exponentially out of noise. As the oscil-
lation amplitude increases, the amplifier will be driven 
into saturation. The excess gain shrinks and finally 
reaches the Barkhausen criterion during steady oscil-
lation.
Some oscillators use a class C amplifier. Such oscilla-
tors can not start out of noise, but need a start pulse. 
Unfortunately, after reaching steady oscillation, class C 
amplifiers add even more noise than class A amplifiers. 
The gain in class C is lower, there is much less control 
over the device bias and due to the heavily non-linear 
operation, class C amplifiers efficiently up-convert low-
frequency noise to the desired oscillator frequency.
3 Leeson’s equation
The Leeson’s equation [1] describes how noise propa-
gates through the circuit of an oscillator. The derivation 
below refers to Fig 1:
 
()
Nout NinN out
UUAH U ω =+ ⋅⋅
   (3)
 





















	 





 

























	






137
can be rearranged to:
 
() 1
Nin
Nout
U
U
AHω
=
−⋅
    (4)
A simple resonator with a lumped capacitor C and a 
lumped inductor L with losses R' provides the follow-
ing transfer function of the feedback:
 
()
1
'
in
in out
R
H
Rj LR R
jC
ω
ω
ω
=
++ ++
  (5)
During steady oscillation the Barkhausen criterion 
simplifies the transfer function for small signal s 
U
Nout
≪U
0
(ω
0
) compared to the carrier to:
 
()
1
R
AH
Rj L
jC
ω
ω
ω
∑
⋅=
∑+ +
  (6)
where the sum of resistors denotes:
 
'
in out
RR RR ∑= ++    (7)
The transfer function can be further simplified by intro-
ducing the loaded quality Q
L
 of the resonator:
 
0
L
L
Q
R
ω
=
∑
     (8)
and the frequency offset from the carrier ω
0
:
 
0
1
LC
ωωωω Δ=−= −   (9)
into:
 
()
0
1
12
L
AH
jQ
ω
ω
ω
⋅≈
Δ
+
                (10)
resulting in:
 
0
1
1
12
Nin
Nout
L
U
U
jQ
ω
ω
≈→
−
Δ
+
 
0
1
2
Nout Nin
L
UU
jQ
ω
ω

→≈ ⋅+

Δ 
               (11)
Dealing with noise is easier with average signal powers 
P
j
=α |U
j
|
2
 rather than voltages. The resulting propaga-
tion of noise power is:
 
2
0
1
2
Nout Nin
L
PP
Q
ω
ω


 ≈⋅ +

Δ 


                 (12)
In engineering it is also preferred to replace angular 
frequencies ω
j
 = 2πf
j
 with ordinary frequencies:
 
2
0
1
2
Nout Nin
L
f
PP
Qf


 ≈⋅ +

Δ 


                 (13)
All derivations in this paper are made considering just 
one side-band of the symmetrical noise spectrum on 
both sides of the carrier U
0
(ω
0
) or U
0
(f
0
). If a single side-
band is observed, there is no distinction between am-
plitude noise and phase noise.
When both upper and lower side-bands are summed, 
the resulting noise signal has both an in-phase com-
ponent and a quadrature component with respect to 
the carrier. Due to the random nature of noise, both the 
in-phase component and the quadrature component 
are of equal magnitude. The in-phase component adds 
a random amplitude modulation to the carrier, also 
called amplitude noise. The quadrature component 
adds a random phase modulation to the carrier, also 
called phase noise.
The original Leeson’s derivation [1] as well as many oth-
er theoretical papers include both noise side-bands, 
frequently denoted as S(ω) or S(f ). On the other hand, 
single side-band noise is required in many practical cal-
culations. Care should be taken since both side bands 
have twice the power of a single side band.
The oscillator noise includes both amplitude noise and 
phase noise. Both have equal power:
 
2
0
1
22 2
Nout Nin
NA N
L
PP f
PP
Qf
φ


 == ≈⋅ +

Δ 


      (14)
Since the amplitude noise P
NA
 can be removed easily 
with an electronic limiter, only the phase-noise power 
P
Nф
 is interesting.
In electronics, noise is usually referred to the input of 
an amplifier although it can only be measured on its 
output. Therefore for compatibility all quantities onare 
referred to the amplifier input. The thermal-noise spec-
tral density dP
Nin
/df at the amplifier input is equal to the 
M. Vidmar; Informacije Midem, Vol. 51, No. 2(2021), 135 – 146

138
sum of the temperatures of all noise sources multiplied 
by the Boltzmann constant k
B
 ≈ 1.38  10
-23
 J/K:
 
()
Nin
Bj BRA
dP
kT kTT
df
=⋅ ∑=⋅+                (15)
The resonator temperature T
R
 ≫ T
0
 = 290 K may be 
much higher than the reference temperature in the 
case of resonators using active circuits. The noise tem-
perature of a passive resonator is usually close to the 
reference (room) temperature T
R
 ≈ T
0
 = 290 K. In this 
case the thermal-noise spectral density can be rewrit-
ten using the amplifier noise figure F (in linear units!):
 
0
Nin
B
dP
kTF
df
≈⋅ ⋅                  (16)
Note that the amplifier noise figure F will be higher in 
saturation (steady oscillation) than in linear operation!
The phase-noise spectral density of an oscillator be-
comes:
 
2
0
0
1
1
22
N
B
L
dP
f
kTF
df Qf
φ


 =⋅+⋅

Δ 


                (17)
Since the oscillator output is amplified, limited and/or 
attenuated, the important quantity is the phase-noise 
spectral density relative to the oscillator output power 
P
0
:
 
()
0
1
N
dP
Lf
Pd f
φ
Δ= ⋅                  (18)
The relative phase-noise spectral density is denoted by 
the symbol L(Δf ) and has units [Hz
-1
] in the Leeson’s 
equation:
 
()
2
00
0
Δ1
2Δ 2
B
L
fk TF
Lf
Qf P


 =+ ⋅




                (19)
Due to the extremely wide dynamic range of L(Δf ) it 
is common to use logarithmic units, namely decibels 
relative to the carrier per unit bandwidth or [dBc/Hz]:
 
()
[]
()
10
dBc/Hz
10log 1Hz Lf Lf  Δ= Δ⋅

                (20)
Unfortunately many popular sources like [2] forget to 
multiply L(Δf ) in linear units with the unit bandwidth  
1 Hz, degrading the Leeson’s equation to an empirical 
equation.
As an example, the spectrum of a typical oscillator is 
computed on Fig. 3 using the Leeson’s equation. The 
carrier power is selected as P
0
 = 0.1 mW typical at the 
input of a small-signal RF transistor. The noise figure 
degradation is comparable to the gain compression 
due to saturation, therefore F = 10 dB is a reasonable 
choice. The most important parameter of an oscillator, 
the loaded quality of the resonator is selected Q
L
 = 10 
corresponding to a varactor-tuned microstrip resona-
tor at f
0
 = 3 GHz:
Figure 3: Oscillator spectrum.
The propagation of noise through an oscillator in-
creases the phase noise close to the desired carrier 
well above the thermal noise. Since the two noise side-
bands are symmetric, it makes sense to observe a sin-
gle side band in detail using a logarithmic scale for the 
frequency offset ∆f  from the carrier as shown on Fig. 4:
Figure 4: SSB phase-noise spectrum.
At frequency offsets |∆f | > f
0
/(2Q
L
) larger than the Lee-
son’s frequency, the oscillator has little effect on the 
noise spectral density. Other circuits like buffer ampli-
fiers, limiters and/or attenuators add their own thermal 



















  

 











M. Vidmar; Informacije Midem, Vol. 51, No. 2(2021), 135 – 146

139
noise. If required, this thermal noise can easily be fil-
tered away using resonators with a similar Q
L
 as used 
in the oscillator itself.
At frequency offsets |∆f |< f
0 
/(2Q
L
) smaller than the 
Leeson’s frequency, the predominant noise is the oscil-
lator phase noise. Other circuits like amplifiers, limiters 
and/or attenuators have little effect on the phase-noise 
spectral density. The oscillator phase noise can NOT be 
filtered away using resonators with a similar Q
L
 as used 
in the oscillator itself.
Since the oscillator phase-noise is the interesting quan-
tity, a simplified Leeson’s equation neglecting thermal 
noise is frequently used:
 
()
2
00
0
8
B
L
fk TF
Lf
Qf P

Δ≈ ⋅

Δ 
                 (21)
The result of the simplified Leeson’s equation is shown 
as a dotted extension on Fig. 4. There is a significant 
difference from the full equation only at large offsets 
|∆f |> f
0 
/(2Q
L 
)≈150 MHz in the example shown on Fig. 
3 and Fig. 4.
The Leeson’s equation was derived assuming that the 
noise amplitude U
Nout
 ≪ U
0
 (ω
0
) is much smaller than 
the desired-carrier amplitude. This assumption no 
longer holds at small offsets ∆f. The Leeson’s equation 
only holds when the relative phase-noise spectral den-
sity is much smaller than the L(∆f ) ≪∆f 
-1
 limit shown 
with a dotted line on Fif. 4. In practice, the result on Fig. 
4 is only valid at offsets above |∆f |>1 kHz.
The relative phase-noise density at very small offsets ∆f 
is usually not very important in practical electronic os-
cillators. It is much more important in laser oscillators. 
A corrected derivation of the Leeson’s equation for very 
small offsets ∆f will be presented later.
4 Effects of phase noise
Phase noise was first noted as residual frequency mod-
ulation in analog radio links. The unwanted random 
frequency deviation (root-mean-square value) can be 
calculated as:
 
()
2
2
MAX
MIN
f
f
f
fL fdf σ =Δ ΔΔ
∫
                (22)
The frequency limits f
MIN
 and f
MAX
 of the integral are the 
band limits of the analog base-band modulation signal.
In QAM radio links, phase noise randomly rotates the 
constellation of the modulation. The unwanted ran-
dom angle of rotation (root-mean-square value) can be 
calculated as:
 
() σ2 ΔΔ
modulation
carrier recovery
B
B
Lf df
φ
−
=
∫
                (23)
Any phase noise above ∆f > B
modulation
 is filtered away by 
the channel filter in the receiver. Further it is assumed 
that the carrier-recovery circuit of the receiver is able 
to track slow frequency and/or phase changes below 
∆f < B
carier - recovery
.
In digital communications, phase noise manifests itself 
as clock jitter. The unwanted clock jitter (root-mean-
square value) can be calculated as:
 
()
00
1
2
2
MAX
clockr ecovery
f
t
B
Lf df
f
φ
σ
σ
ωπ
−
== ΔΔ
∫
            (24)
Limiting the bandwidth of the clock, the upper limit 
f
MAX
 < f
0
 is less than the clock frequency. Further it is 
assumed that the clock-recovery circuit of the receiver 
is able to track slow frequency and/or phase changes 
below ∆f < B
clock - recovery
.
Finally in all radio communications, phase noise causes 
interference to neighbor channels. The interference 
power can be calculated as:
 
()
2
1
0
f
i
f
PP Lf df
Δ
Δ
=⋅ ΔΔ
∫
                  (25)
The frequency limits ∆f
1
 and ∆f
2
 of the integral are the 
frequency offsets of the interfered channel from the in-
terfering carrier P
0
(f
0 
).
Note that all of the above-mentioned integrals start 
from an offset ∆f > 0 larger than zero. Radio equipment 
is usually designed to work with relatively clean sources 
where the phase-noise power P
Nф
 ≪ P
0
 is much smaller 
than the carrier power and the Leeson’s equation is 
valid thanks to L(∆f ) ≪ ∆f 
-1
 in the region of interest.
5 Active-device noise
Besides thermal noise, active devices also add flicker 
noise to the amplified signal. Flicker noise is usually 
described as an increase of the radio-frequency noise 
figure F into a frequency-dependent noise figure F'(f ):
M. Vidmar; Informacije Midem, Vol. 51, No. 2(2021), 135 – 146

140
 
() '1
C
f
Ff F
f

=⋅+


                  (26)
The parameter describing flicker noise is the corner fre-
quency f
C
. The latter depends on the device technol-
ogy [3]. In general, surface devices have higher current 
densities and more structure defects than bulk devices. 
Surface semiconductor devices like a silicon MOSFET, a 
GaAs MESFET or a GaAlAs HEMT may have the corner 
frequency in the range f
C
≈1…10 MHz. Bulk semicon-
ductor devices like a silicon BJT or a silicon JFET may 
have the corner frequency in the range f
C 
≈ 1…10 kHz.
Although a HEMT may produce slightly less noise at 
radio frequencies than a BJT, a HEMT is significantly 
noisier at low frequencies than a BJT as shown on Fig. 5:
Figure 5: Active device noise figure.
In an oscillator, the active device operates in saturation 
while producing steady oscillations. The nonlinear ef-
fects associated with saturation up-convert the low-
frequency flicker noise into noise side bands very close 
to the carrier radio frequency. High-performance radio-
frequency (microwave) oscillators therefore use silicon 
bipolar transistors due to their lower flicker noise.
The additional up-converted flicker noise can be built 
into the Leeson’s equation describing the increase the 
oscillator phase noise at small offsets |∆f |<f
C
:
 
()
2
00
0
11
22
BC
L
fk TF f
Lf
Qf Pf

 
 Δ=+⋅ ⋅+
 
ΔΔ  


    (27)
The phase noise of the same oscillator example as 
shown earlier including flicker noise is shown on Fig 6:
Calculations including flicker noise may not be simple. 
Calculating the flicker-noise power P
N
 from equation (26):
 
1
MAX
MIN
f
C
NB
f
f
Pk Fd f
f

=⋅ ⋅+


∫
                 (28)
may give an infinite result:
 
0
lim1
MAX
MIN
MIN
f
C
B
f
f
f
kF df
f
→

⋅⋅+→ ∞


∫
                (29)
suggesting that further limitations apply to (26) at very 
low frequencies.
Further it is necessary to understand that the flicker-
noise corner frequency f
C
 in equation (26) is different 
from the f
C
 in equation (27)! Between the two quanti-
ties there is a frequency conversion that may be more 
or less efficient depending on parameters that are NOT 
described by the Leeson’s equation!
The phase noise of an oscillator depends heavily on the 
bias and DC decoupling circuits. Since the impedance 
parameters [Z
ij
] of a bipolar transistor depend mainly 
on the DC currents through the device, the currents 
through the RF amplifier transistor have to be regulat-
ed as constant as possible with a bias circuit like that on 
Fig. 7 [4]. Keeping the impedance parameters [Z
ij
] con-
stant attenuates the up-conversion of low-frequency 
flicker noise to the RF carrier frequency:








	
Figure 6: Phase noise including flicker noise.
  

 














M. Vidmar; Informacije Midem, Vol. 51, No. 2(2021), 135 – 146

141
Flicker noise is not the only concern while designing the 
bias network of an oscillator. Reactive components like RF 
chokes (inductors) may introduce additional unwanted 
modes of the resonator H(ω). Therefore resistors R
5
 and 
R
6
 are usually used to apply the DC bias in oscillators.
Besides the RF feedback there is yet another feedback 
circuit built into every electronic oscillator. Gain reduc-
tion at saturation during steady oscillation is governed 
by this additional feedback (bottom graph on Fig. 2). 
A poorly-designed bias network will make this low-
frequency feedback unstable causing self quenching 
of the oscillator. While self quenching may simplify a 
super-regenerative receiver compared to the original 
Armstrong design [9], it has a catastrophic effect on the 
oscillator spectrum.
The gain-reduction feedback already has one pole due 
to the RF energy stored in the resonator H(ω), rectified 
by the nonlinear effects of the saturation of the active 
device and added to the DC bias of the latter. Addition-
al poles are added by the RF bypass capacitors C
1
 and 
C
2
 and by the DC-bias decoupling capacitors C
3
 and 
C
4
. Unless the component values on Fig. 7 are selected 
carefully, the oscillator will be self-quenching. Even if 
the oscillator is not self-quenching, a poor phase mar-
gin of the bias feedback may cause a significant in-
crease of the oscillator phase noise.
If varactors are used to tune the oscillator (VCO) [6], the 
phase noise is degraded further. First, varactors decrease 
the Q
L
 of the resonator due to their series resistance. Sec-
ond, the tuning voltage may introduce additional noise. 
Even the noise voltage introduced by the resistors acting 
as RF chokes to tune the varactors is not insignificant.
6 Spectral-line width
The Leeson’s equation (19) is unable to describe the fre-
quency spectrum of an oscillator very close to its central 
frequency ω
0
 or f
0
 when the condition L(∆f ) ≪ ∆f 
-1
 is 
no longer fulfilled. Although there are several compre-
hensive papers on this topic like [5], [6], a simplified 
derivation is given here.
Analyzing Fig. 1, the feedback gain has to be slightly 
less than unity during steady oscillation, since some 
noise is being added all of the time. Accordingly, the 
original Barkhausen criterion (2) has to be modified to:
 
()
0
1 AHω ⋅= −ε                  (30)
where the gain decrease is described by the very small, 
but non-zero quantity 0<ϵ ≪1. The feedback transfer 
function (10) is modified to:
 
()
0
1
12
L
AH
jQ
ω
ω
ω
−
⋅=
Δ
+
ε
                 (31)
resulting in equation (11) extended to:
 
()
0
1
1
1
12
NinN in
Nout
L
UU
U
AH
jQ
ω
ω
ω
=≈ →
−
−⋅
−
Δ
+
ε
 
0
0
12
2
L
Nout Nin
L
jQ
UU
jQ
ω
ω
ω
ω
Δ
+
→≈ ⋅
Δ
−ε
   (32)
At frequency offsets |∆f |>f
0
/(2Q
L
) larger than the Lee-
son’s frequency, the oscillator has little effect on the 
noise while other circuits add their own noise. It therefore 
makes sense to evaluate (32) at small offsets |∆f |<f
0
/(2Q
L
) 
only. Considering |j2Q
L
∆ω
0
/ω
0
| ≪1, equation (32) sim-
plifies to:
 
0
2
Nin
Nout
L
U
U
jQ
ω
ω
≈
Δ
−ε
                  (33)
Replacing noise voltages with average powers, replac-
ing angular frequencies with ordinary frequencies and 
considering the phase noise only:









 




 

	


















Figure 7: Oscillator bias circuit.
M. Vidmar; Informacije Midem, Vol. 51, No. 2(2021), 135 – 146

142
 
2
2
0
/2
2
2
Nout Nin
N
L
PP
P
f
Q
f
φ
=≈
 Δ
+


ε
                (34)
Introducing the thermal-noise spectral density (15) or 
(16) and the spectral-line half width:
 
0
2
HW
L
f
f
Q
=
ε
                   (35)
the simplified Leeson’s equation (21) evolves into a Lor-
entzian spectral line:
 
()
2
00
22
0
1
8
B
LH W
fk TF
Lf
Qf fP

Δ= ⋅⋅

Δ+ 
               (36)
The missing quantities f
HW
 or ϵ can be calculated by 
summing the whole relative spectrum power consider-
ing ∆f = f – f
0
:
 
()
0
1
f
Lf df
∞
−
ΔΔ =
∫
                  (37)
In all practical cases the integral start may be replaced 
by – ∞, the error being smaller than neglecting far-
away thermal noise:
 
2
00
22
0
1
8
B
LH W
fk TF
df
Qf fP
∞
−∞

⋅⋅ Δ=

Δ+ 
∫
 
2
00
0
1
arctan
8
f
B
LH WH W
f
fk TF f
QP ff
Δ= ∞
Δ= −∞
  Δ
=⋅ ⋅=




 
2
00
0
1
8
B
LH W
fk TF
QP f
π 
=⋅ ⋅≈


  (38)
The spectral-line half width is obtained as:
 
2
00
0
π
8
B
HW
L
fk TF
f
QP

≈⋅ ⋅


                  (39)
The small correction of the Barkhausen criterion is:
 
00
0
4
B
L
fkTF
QP
π
≈ ε                  (40)
Analyzing the same oscillator example with f
0 
= 3 GHz, 
Q
L
 = 10, P
0
 = 0.1 mW and F = 10 dB as on Fig. 3 and 
Fig. 4, a spectral-line half width of f
HW
≈14Hz is obtained. 
The corresponding correction of the Barkhausen crite-
rion is small indeed ϵ≈10
-7
.
One side band of the calculated spectrum L(∆f) (solid 
line) is compared to the original Leeson’s equation 
(dotted extensions) on Fig. 8 in logarithmic scale:
Figure 8: Lorentzian spectral line.
  

 











M. Vidmar; Informacije Midem, Vol. 51, No. 2(2021), 135 – 146
The result of the original Leeson’s equation is plotted 
with a dotted line on the same graph as well as the ∆f 
-1
 
limit. Note that at small offsets the spectrum L(∆f ) flat-
tens thus avoiding the ∆f 
-1
 limit.
Besides thermal noise, additional noise like flicker noise 
further broadens the spectral line. The calculation is 
more difficult since the low-frequency flicker-noise 
spectrum is not up-converted by a single carrier fre-
quency but by the oscillator signal itself with non-zero 
spectral width.
In most cases the spectral-line half width remains much 
narrower f
HW
 ≪ B
recovery
 than the carrier or clock recovery 
circuits in radio equipment. In all these frequent cases 
the result of the original Leeson’s equation is sufficient.
7 Delay-line oscillators
The most important parameter in the Leeson’s equation 
is the loaded quality Q
L
 of the resonator. Unfortunately 
electrical resonators in the radio-frequency range do 
not achieve very high values of Q
L
. Mechanical reso-
nators like quartz crystals are frequently used in high-
performance radio oscillators. Electrical resonators may 
achieve very high values of Q
L
 in the optical-frequency 
range. Lasers may produce relatively very narrow spec-
tral lines. Unfortunately dividing optical frequencies 
down to radio frequencies is not practical yet.
Delay lines may act as resonators in oscillator circuits. 
Their equivalent Q
LD
 is directly proportional to the de-

143
lay τ
D
 and increases linearly with frequency:
 
0 LD D
Qf πτ =                   (41)
Unfortunately delay lines may fulfill the Barkhausen 
criterion (2) at many different frequencies causing a 
laser to oscillate on many different modes. Lasers may 
use frequency-selective mirrors or gain medium to de-
crease the number of modes.
A similar approach may be used to design radio-fre-
quency oscillators using either acoustic (BAW or SAW) 
delay lines or opto-electronic delay lines [7]. The latter 
look promising due to the low loss and wide bandwidth 
of optical fibers. The basic design of an opto-electronic 
oscillator is shown on Fig. 9. The desired mode of oscil-
lation is selected by an additional electric (microwave) 
resonator:
Figure 9: Opto-electronic oscillator.
The Barkhausen criterion (2) can be rewritten for the 
circuit on Fig. 9 as:
 
() ()
10 20
1
RD
AH AH ωω ⋅⋅ ⋅=                 (42)
If the electric resonator is tuned precisely to the desired 
mode of the delay line, the voltage transfer function of 
the latter can be written as:
 
() e
D
j
D
Ha
ωτ
ω
−Δ
=⋅
                  (43)
For small signals and small offsets:
 
() ()
12
0
e
12
D
j
RD
LR
AH AH
jQ
ωτ
ωω
ω
ω
−Δ
⋅⋅ ⋅=
Δ
+
         (44)
The noise-voltage transfer function becomes:
 
0
e
1
12
D
Nin
Nout j
LR
U
U
jQ
ωτ
ω
ω
−Δ
≈
−
Δ
+
                 (45)
The corresponding phase-noise average power is:
                (46)
Finally the extended Leeson’s equation for the opto-
electronic oscillator shown on Fig. 9 becomes:
                (47)
The largest contribution to ΣT
j
 comes from the opto-
electronic delay line that may include flicker noise:
 
1
C
jD
f
TT
f

∑≈⋅+

Δ 
                  (48)
In an opto-electronic oscillator as on Fig. 9 the most 
vulnerable point in the circuit is the photo-diode out-
put. Here the signal power P
0
 is the lowest and the rela-
tive phase-noise spectral density L(∆f ) is calculated. 
Saturation will likely be achieved in A
2
 since optical 
modulators require substantial amounts of RF drive 
power. The output L(∆f )
out
 is taken after all amplifica-
tion and filtering:
 
()
()
2
0
12
out
LR
Lf
Lf
f
Q
f
Δ
Δ≈
 Δ
+


                 (49)
The analytical result for L(∆f )
out
 is fitted to the well-
documented experimental data from [8]. The latter 
describes a microwave f
0
 = 3 GHz opto-electronic oscil-
lator with the delay line made from l≈15km of optical 
fiber resulting in a delay of τ
D
≈75μs corresponding to a 
Q
LD
≈7∙10
5
. Mode selection is performed by an addition-
al microwave dielectric resonator with the Q
LR
≈8300.
The opto-electronic delay line noise temperature may be 
rather high due to several reasons: relative intensity noise 
(RIN) of the laser, optical reflections including Rayleigh 
scattering converting optical phase noise into amplitude 
noise and inefficient broadband impedance matching of 
the photodiode. The opto-electronic delay line noise tem-
perature is found as expected around T
D
≈2∙10
5 
K. What 
really matters is the ratio T
D
/P
0
 and the latter can be meas-
ured conveniently at the output of a PIN-FET module.
Flicker noise comes at least in part from the built-in 
HEMT amplifiers. Due to the required relatively high 


 







	








 	









	

 	







  

  


M. Vidmar; Informacije Midem, Vol. 51, No. 2(2021), 135 – 146

144
electronic gain, several amplifier stages are connected 
in series. If broadband amplifiers are used, flicker noise 
may originate in the first stage, it is amplified by the 
intermediate stage and it is up-converted by the last 
stage. Due to the high noise contribution from the op-
to-electronic delay line, the overall flicker-noise corner 
frequency is found around f
C
≈5 kHz.
The fitted analytical result for L(∆f )
out
 on Fig. 10 shows 
the unwanted side modes at the correct frequen-
cies. However, the peak magnitudes of the unwanted 
modes are about 15 dB stronger than the measured 
values. This may be due to an insufficient resolution of 
the phase-noise test setup:
Figure 10: Simulated OEO phase noise.
The well-documented experimental data from [8] addi-
tionally includes results with a Q-multiplier circuit. The 
latter increases the loaded quality of the microwave 
mode-selection filter to about Q
LR
≈75000 thus improv-
ing the rejection of unwanted modes. Since a Q mul-
tiplier is an active filter, the system noise temperature 
increases to about T
D
≈5∙10
5 
K.
The fitted analytical result for L(∆f )
out
 including the Q 
multiplier is shown on Fig. 11. The unwanted-mode 
magnitudes are reduced and their line widths are 
broader. Both frequencies and magnitudes are very 
close to the measured values in [8]:
Figure 11: Simulated OEO with Q multiplier.
Finally, a parabolic approximation of the close-in re-
sponse of a single microwave resonator suggests that 
the unwanted mode rejection is proportional to (Q
LR
)
4
. 
For a Q-multiplication factor m≈8 as described in [8], 
the unwanted-mode rejection improvement is expect-
ed as 10log
10 
m
4 
≈36 dB. The difference between Fig. 
10 and Fig. 11, corrected for the change in T
D
, comes 
much closer to this value than the measured data pub-
lished in [8], again suggesting an insufficient resolution 
of the phase-noise test setup.
8 Avoiding UV & DC catastrophes
When natural laws are extended from a few laboratory 
measurements up to the whole frequency spectrum, 
problems usually arise at both extremes: when the 
frequency approaches infinity f→∞ and when the fre-
quency approaches zero f→0. One of the most famous 
problems in physics was the ultraviolet catastrophe 
predicted from the Rayleigh-Jeans law for black-body 
thermal radiation [10], suggesting infinite radiated 
power. The more accurate Planck’s law solved the prob-
lem a few years later.
The same problem also applies to phase noise. What 
happens with the relative phase noise density at both 
extremes L(∆f→∞) (UV catastrophe) and L(∆f→0) (DC 
catastrophe)? The answer is not simple since L(∆ f ) 
may achieve very differing shapes and magnitudes. To 
the best of my knowledge, the limitations of different 
equations for phase noise that may produce non-phys-
ical results are identified by introducing the ∆f 
-1
 limit 
for the first time in this article.
The Leeson’s equation for electronic oscillators is usu-
ally derived from the Johnson noise. In electronics, the 
Johnson-noise spectral density is usually considered 
frequency-independent as shown in equation (15), 





















  






















  

M. Vidmar; Informacije Midem, Vol. 51, No. 2(2021), 135 – 146

145
since it is derived form the Rayleigh-Jeans law. At room 
temperatures, the Rayleigh-Jeans law becomes inaccu-
rate at infrared frequencies. At cryogenic temperatures, 
the Rayleigh-Jeans law becomes inaccurate already at 
microwave-radio frequencies.
If equation (15) is rewritten to include the complete 
Planck’s law, the resulting thermal-noise spectral density 
also depends on the Planck constant h≈6.626∙10
-34  
Js:
 
W
Hz
e1
B
N
hf
kT
dP hf
df

=


−
                  (50)
The Johnson noise is just an approximation for low fre-
quencies:
 
()
0
lim
e1
B
N
BB hf
f
kT
dP hf
hf kT kT
df
→


≈=


−
           (51)
The complete equation should be considered at fre-
quencies above f≈k
B
T/h≈6 THz at a room temperature 
T≈290 K. The electronic noise decays even sooner 
since the gain bandwidth of electronic devices is about 
three orders of magnitude smaller. Therefore there are 
at least two valid and independent reasons to avoid the 
UV catastrophe.
Flicker noise is usually modeled as 1/f noise in equation 
(26). The latter suggests an infinite amount of power 
(DC catastrophe) even in a simple amplifier without 
feedback (29). If the spectral noise density at very low 
frequencies or the total noise power is required, a bet-
ter model than 1/f should be used for flicker noise. In 
order to separate different effects, flicker noise will not 
be considered in the following discussion.
Another DC catastrophe may originate in the simple 
derivation of the Leeson’s equation for an oscillator 
(19) or (21). At small offsets ∆f→0, the noise power is 
no longer small compared to the carrier power. A com-
plete derivation of the spectral line (36) avoids this DC 
catastrophe.
In order to explain different effects, the same result from 
Fig. 8 is plotted on much broader scales on Fig. 12. On 
the latter, the frequency spans from bi-weekly 10
-6 
Hz 
up to soft X rays 10
18 
Hz. The amplitude range spans an 
incredible 350 dB:
Figure 12: Catastrophes explained.
The ∆f 
-1
 limit corresponds to an infinite amount of 
power over the whole spectrum. Considering both 
side-bands of a single octave f<∆f<2 f, the ∆f 
-1
 limit 
produces a finite amount, just slightly too much rela-
tive noise power:
 
2
1
0
22 ln2 1.386
f
N
f
P
fdf
P
−
=Δ Δ= ≈
∫
               (52)
In order to comply with equation (37), the relative 
phase-noise spectral density L(∆f ) may approach the 
∆f 
-1
 limit over less than an octave and drop to zero else-
where else. 
A Lorentzian spectral line approaches the ∆f 
-1
 limit to 
-8 dB at a frequency offset ∆f = f
HW
(≈14 Hz) (39). At 
smaller offsets ∆f ≪f
HW
 the Lorentzian spectral line is flat 
with frequency L(∆f ) ≈ α. At larger offsets ∆f ≫f
HW
 the 
Lorentzian spectral line decays as L(∆f ) ≈α∙∆f 
-2
 with 
increasing offset. At both smaller and larger offsets ∆f, 
the Lorentzian spectral line diverges far below the ∆f 
-1
 
limit thus avoiding both UV and DC catastrophes.
The result of the Leeson’s equation (dotted line) matches 
the Lorentzian spectral line (solid line) over the usually-
interesting offset range and stays well below the ∆f 
-1
 
limit. At very small offsets ∆f→0, the Leeson’s result 
grows as L(∆f ) ≈α∙∆f 
-2
 with decreasing offset, eventu-
ally exceeding the ∆f 
-1
 limit and causing a DC catastro-
phe. At very large offsets ∆f→∞, the Leeson’s result is 
flat with frequency L(∆f) ≈ α, eventually exceeding the 
∆f 
-1
 limit and causing an UV catastrophe.
As long as the relative phase-noise spectral density L(∆f ) 
is a monotonically decreasing function, it should avoid 
the ∆f 
-1
 limit in a similar way to the Lorentzian spectral 
line. More complex spectra L(∆f ) like that shown on Fig. 
10 may even exceed the ∆f 
-1
 limit over very narrow off-
  

  















	

 	
M. Vidmar; Informacije Midem, Vol. 51, No. 2(2021), 135 – 146

146
set ranges (much less than an octave) without causing 
catastrophes analyzing a likely useless oscillator.
In any case, comparing the magnitude and slope of 
L(∆f ) to the ∆f 
-1
 limit quickly tells whether a certain 
equation for L(∆f ) with certain parameters provides 
useful results or not over the desired offset range. The 
ratio L(∆f )/∆f 
-1
= ∆f ∙ L (∆f ) tells whether the phase 
noise at the specified offset ∆f is much smaller or com-
parable to the whole signal power.
9 Conclusions
The Leeson’s equation for relative phase-noise spec-
tral density is frequently misunderstood and misused 
even in commercial simulation software. Therefore a 
complete derivation is made first to understand the 
limitations of the different forms of the same equation. 
While derivations produce results in linear units [Hz 
-1
], 
logarithmic units [dBc/Hz] (20) are used elsewhere in-
cluding the graphs in this article.
The complete Leeson’s equation (19) is frequently sim-
plified to (21), since wide-band thermal noise origi-
nates elsewhere and not just in the oscillator.
Flicker noise is usually built in the Leeson’s equation 
like (27), but its exact magnitude actually depends on 
factors not included in the Leeson’s equation, like the 
design of active-device bias networks. Last but not 
least, the simple 1/f approximation of flicker noise may 
produce non-physical, infinite results in some cases.
The original Leeson’s derivation is valid for small noise 
signals only. The result is only valid in the offset range 
when L(∆f ) ≪∆f 
-1
. When L(∆f ) approaches or even 
exceeds the ∆f 
-1
 limit, non-physical results are usually 
obtained. In the latter case a complete derivation of the 
oscillator spectrum has to be performed including the 
shape of the main spectral line of non-zero width. Flat 
thermal noise produces a Lorentzian spectrum (36).
Finally, the Leeson’s equation is extended to delay-line 
oscillators and in particular to opto-electronic oscilla-
tors. The extended equation (47) is fitted to experimen-
tal data showing potential problems of the latter.
As a conclusion of all of the above findings, an electron-
ic oscillator is just a Q multiplier amplifying and filtering 
its own noise. The Q-multiplication factor is very large 
m ≈ ϵ 
-1
 resulting in a very small, but non-zero spectral-
line half width f
HW 
> 0. Besides bandwidth differences 
of many orders of magnitude, an electronic oscillator 
produces a similar signal to the spark radio transmitter 
or filtered white light in optics.
10 Conflict of Interest
The author declares no conflict of interest.
The founding sponsors had no role in the design of the 
study; in the collection, analyses, or interpretation of 
data; in the writing of the manuscript, and in the deci-
sion to publish the results.
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tor Noise Spectrum” , Proceedings of the IEEE 54 (2), 
February 1966, pp. 329–330, 
 https://doi.org/10.1109/PROC.1966.4682
2. Wikipedia, “Leeson’s equation” https://
en.wikipedia.org/wiki/Leeson%27s_equation 
[Accessed: 08-Feb-2021]
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Quasilinear Oscillators”, https://arxiv.org/abs/
physics/0502143v1 [Accessed: 17-Feb-2021]
8. L. Bogataj, M. Vidmar, B. Batagelj,“Opto-Electronic Oscilla-
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[Accessed: 24-Apr-2021]
Arrived: 18. 02. 2021
Accepted: 20. 05. 2021
M. Vidmar; Informacije Midem, Vol. 51, No. 2(2021), 135 – 146
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