Erik Lindberg, Lifemember IEEE,
Reader Emeritus at The Technical University of Denmark,
Kongens Lyngby, DK2800, Denmark, e-mail: erik.lindberg@ieee.org
K. Murali
Department of Physics, Anna University,
Chennai-600 025, India e-mail: kmurali@ns.annauniv.edu
Abstract —Oscillators are normally designed according to the
Modified Barkhausen Criterion i.e. the complex pole pair is moved out in RHP so that the linear circuit becomes unstable.
By means of the Mancini Phaseshift Oscillator it is demonstrated that the distortion of the oscillator may be minimized by introducing a nonlinear ”Hewlett Resistor” so that the complex pole-pair is in the RHP for small signals and in the LHP for large signals i.e. the complex pole pair of the instant linearized small signal model is moving around the imaginary axis in the complex frequency plane.
I. I
NTRODUCTION
In order to design electrical and electronic circuits we assume that the wave length of the highest frequency is much larger than the circuit so lumped elements could be used quasi stationary circuits.
Since the 1930’ies the Barkhausen Criterion [1] has been used as a starting point for the design of oscillators. The basic assumption is that the oscillator topology is a loop of a linear inverting amplifier and a linear frequency determining feedback circuit. The loop is opened and the transfer-function of the open loop circuit - the loop gain - is designed to be equal to 1 0 . The loop is then closed and we have a linear
TABLE I
(A) ideal op.amps
Sigma (real)
+0.00000e-01
+0.00000e-01
-1.00000e+04
(B) perfect op.amps
Sigma
-9.99990e-2
-9.99990e-2
-1.00000e+4
(C)
(real)
A 1 =
Omega
∞
(imag)
-1.00000e+04
+1.00000e+04
+0.00000e-01
A 1 = 100 e + 3
Omega (imag)
-9.99990e+03
+9.99990e+03
+0.00000e-01 perfect op.amps
R 2 A = 10 .
1 e + 3
Sigma (real)
+2.46521e+01
+2.46521e+01
-9.95049e+03
Omega (imag)
-9.97496e+03
+9.97496e+03
+0.00000e-01
R
R
2
2
= 10 e + 3
A 3 =
∞
Q
∞
∞
+0.50
= 10 e + 3
A 3 = 100 e + 3
Q
+50.0e+03
+50.0e+03
+0.50
R
2
= 10 .
1 e + 3
R 2 B =
∞
Q
-202.314
-202.314
+0.50
(D) R
2
= 8248 perfect op.amps
R 2 A = 10 .
1 e + 3 R 2 B = 45 e + 3
Sigma (real) Omega (imag) Q
-5.28186e+2
-5.28186e+2
-1.10670e+4
-9.97496e+3
+9.97496e+3
+0.00000e-1
+9.9
+9.9
+0.50
Very little is reported about how far out in RHP the complex pole pair should be placed.
In order to obtain a steady state oscillation with constant amplitude nonlinear components are introduced for clipping of the amplitude. A steady state oscillator is a nonlinear circuit
[2], [3]. The clipping introduce distortion so also filtering must be introduced to obtain clean sinusoidal oscillation.
Fig. 1. Mancini’s Phaseshift Oscillator.
oscillator with a complex pole-pair on the imaginary axis of the complex frequency plane. In order to secure oscillations the complex pole-pair is moved to the right half of the complex frequency-plane RHP so the linear circuit becomes unstable.
This approach is called the Modified Barkhausen Criterion .
II. M ANCINI ’ S P HASESHIFT O SCILLATOR
Figure 1 shows the Mancini Phaseshift Oscillator [4]. The circuit is a loop of three integrator circuits, two active and one passive RC integrator. The idea of this quadrature oscillator is to use the fact that the double integral of a sine wave is a negative sine wave of the same frequency and phase, in other words, the original sine wave 180 deg phase shifted. The phase of the second integrator is then inverted and applied as positive feedback to induce oscillation. Both outputs V (3) and V (6) have relatively high distortion, that can be reduced with a gainstabilizing circuit. With reference to fig. 1 - which is taken from figure 20, page 19 of [4] - node 1 is connected to an ideal 2.5V power source in the derivation of the following equations. It is assumed that the amplifiers are ideal op-amps

Fig. 3. Hewletts Wien Bridge Oscillator.
Fig. 4. ”Hewlett Resistor” to be used instead of an incandescent lamp R
3
.
Fig. 2. Amplitude control section of Hewlett’s oscillator patent
The quality-factor Q of a complex pole is defined as:
√
Q = Sigma 2 + Omega 2 / (
−
2
×
Sigma ) (3)
It is a measure for the distance of the pole from the imaginary axis. It is seen that Q becomes
∞ for poles on the imaginary axis. Q is negative for poles in the right-half-plane RHP and positive for poles in the left-half-plane LHP .
Table I case (A) shows that the solution of the characteristic polynomial for R
1
= R
2
= R
3
= R = 10 k Ω and C
1
= C
2
= i.e.
V (1) = V (2) and V (4) = V (5) and V (1) = 0 . The order of the circuit is 3 i.e. the characteristic polynomial is a third order polynomial. The roots of the polynomial are a real root and a complex pole pair. The node equations for the nodes 2,
4 and 5 may be derived by means of Kirchhoffs current law.
From the node equations the following equation is derived:
V
3
[
G
1
G
2
( G
3
+ sC
( G
2
3
) + sC
+ sC
2
1
( G
) sC
3
2
+ sC
2
) sC
3
]
= 0 (1) where G n
= 1 /R n
.
If we observe that V (3) = 0 the numerator in equation (1) must be zero and the characteristic polynomial of the linear circuit becomes: s
3
+ s
2
G
2
C
2
+ s
G
1
G
2
C
1
C
2
+
G
1
G
2
G
3
C
1
C
2
C
3
= 0 (2)
It is difficult to find expressions for the roots of this polynomial so numerical calculations are used.
Fig. 5. FFT spectrum of Sine output V
3 of Mancini’s Phaseshift Oscillator fig. 1. R2A and R2B refer to fig. 4.

C
3
= C = 10 nF is a complex pole-pair on the imaginary axis and real pole in LHP . The frequency becomes ω = 2 πf =
1 /RC i.e.
f = ω/ (2 π ) = 1 .
591549431 kHz
Fig. 7. Power source currents I ( V R 10) , I ( V P ) , I ( V N )
Fig. 6. Static gains A
1
= V
3
/ ( V
1
−
V
2
) and A
3
= V
6
/ ( V
4
−
V
5
)
III. D ISTORTION M INIMIZATION
Figure 2 shows the comments on the amplitude control in
Hewlett’s oscillator patent [7], [8], [9]. He assumes that the circuit is linear so it is necessary to introduce a nonlinear negative feedback resistor R
3 which vary with the amplitude of the signals. He also claim that this approach could be used in other types of oscillation generators. He mention nothing about the poles of the instant small signal model.
Figure 3 shows fig.1 of Hewletts Wien Bridge Oscillator patent. Please note the resistor R
3 a tungsten lamp.
In [10] it is shown that the distortion for a negative resistance oscillator may be minimized by introducing a nonlinear feed-back resistor ”Hewlett resistor” so that the complex polepair of the time-varying small-signal model move symmetrically around the j ω axis. In the following it is shown that it is possible to reduce the distortion of the Mancini Phaseshift
Oscillator by introducing resistor R
2 resistor”, fig. 4.
as a nonlinear ”Hewlett
The cases (B), (C) and (D) of TABLE I are calculated by means of the ANP3 program [5]. Case (B) shows that the complex pole pair is very close to the imaginary axis if the amplifiers are assumed to be perfect with infinite input impedance, zero output impedance and constant gain
A = v out
/v in
= 100 e + 3 , Q = 50 k. Case (C) shows that the complex pole pair is in RHP when R
2
( Q =
−
200 ). Q becomes greater for smaller values of R
R
2
= 10 .
010 e + 3
⇒
Q =
−
2 k and R
Q =
−
33 k.
2
= 10 .
100 e
2
+ 3 e.g.
= 10 .
001 e + 3
⇒
,
Now the ”Hewlett resistor” is introduced and case (D) shows that the complex pole pair is in LHP for large signals R
2
=
8 .
248 e + 3 , ( Q = +10 ) i.e. it is not necessary to have a symmetric placement of the complex pole pairs.
Figure 5 shows the ”fft” spectrum of the Sine output V
3 for the cases (B), (C) and (D) calculated by means of PSpice
[6], with the following options: ”.tran 0 1000e-3 100e-3 1e-4
; uic” and ”+ RELTOL = 1e-6”. Distortion is increased when the complex pole pair is moved to RHP , case (C). Distortion is minimized when the ”Hewlett resistor” is introduced, case
(D).

Figure 6 shows the static gains of the op-amps
A
1
= V
3
/ ( V
1
−
V
2
) and A
3
= V
6
/ ( V
4
−
V
5
) in the three cases. It is obvious that the distortion is minimized when the ”Hewlett Resistor” is introduced, case (D).
Figure 7 shows the power source currents. It is seen how the currents I ( V P ) and I ( V N ) become almost constant in case (D).
IV. C ONCLUSION
The starting point for the design of an oscillator is the
Barkhausen Criterion which result in a linear circuit with a complex pole-pair on the imaginary axis. In order to guarantee oscillations the complex pole pair is moved to the right half of the complex frequency plane RHP so that the linear circuit becomes unstable, the Modified Barkhausen Criterion . The amplitudes of signals will go to infinity,
∞
. When the oscillator is built the amplifiers used are nonlinear. The amplitudes of the steady state signals will be limited and distortion is observed.
In his Wien Bridge oscillator William Hewlett used a tube amplifier operating in the linear range so he introduced a nonlinear resistor, a tungsten lamp, as the necessary nonlinearity for steady state oscillations. Very little distortion was observed. Apparently the instant linearized resistor gave rise to a complex pole pair in RHP for small signals and in LHP for large signals.
Here we demonstrate that a ”Hewlett Resistor” based on two resistors and two diodes may be used to minimize the distortion of Mancini’s Quadrature Oscillator. The complex pole-pair of the instant small signal model of the circuit is in the RHP for small signals (Q = -200) and in the LHP for large signals (Q = +10).
R EFERENCES
[1] H. Barkhausen, Lehrbuch der Elektronen-Rohre, 3.Band, ”R¨uckkopplung”, Verlag S.Hirzel, 1935.
[2] L.O.Chua, C.A. Desoer and E.S. Kuh, ”Linear and nonlinear circuits”,
McGraw-Hill Inc, New York, 1987, p.426.
[3] Raj Senani, D. R. Bhaskar, V. K. Singh, R. K. Sharma, ”Sinusoidal oscillators and waveform genertors using modern electronic circuit building blocks”, Springer (2015).
[4] Ron Mancini and Richard Palmer, ”8.5 Quadrature Oscillator, page 18”,
Texas Instruments, TI-SLOA060 Application Report - March 2001.
[5] Erik Lindberg, ”ANP3 & NAP2 - A package for circuits and systems simulation”, pp. 686-700 in R.A. Adey (edt.) Engineering Software
II, Proceedings of the Second International Conference on Engineering
Software, held at Imperial College of Science and Technology, CML
Publications, London, March 1981.
[6] Cadence, PSpice, 16.6-S060(v16-6-112FA) Oct.13.2015, 01.14.18
[7] Hewlett, William R. (1939). ”A New Type Resistance-Capacity Oscillator”, M.S.Thesis, Stanford University, Palo Alto, California.
[8] W.R.Hewlett’s Oscillator Patent http://www.hp.com/hpinfo/abouthp/histnfacts
/museum/earlyinstruments/0002/other/0002patent.pdf
[9] Bauer, Brunton (1949). ”Design Notes on the Resistance-Capacity Oscillator Circuit”, Parts I and II, Hewlett-Packard Journal, Volume I, Nos 3 and 4, (November-December 1949).
[10] Erik Lindberg, K. Murali and Arunas Tamasevicius, ”On Negative Resistance Oscillators as Modified Multi-vibrators”, ECCTD 2015, European
Conference on Circuit Theory and Design, Trondheim, Norway, Aug.
2015.

Barkhausen stability criterion - Wikipedia, the free encyclopedia
From Wikipedia, the free encyclopedia
In electronics, the
Barkhausen stability criterion
is a mathematical condition to determine when a linear electronic circuit will oscillate.
[1][2][3] It was put forth in 1921 by German physicist Heinrich Georg
Barkhausen (1881–1956).
[4] It is widely used in the design of electronic oscillators, and also in the design of general negative feedback circuits such as op amps, to prevent them from oscillating.
https://en.wikipedia.org/wiki/Barkhausen_stability_criterion
1 Limitations
2 Criterion
3 Erroneous version
4 See also
5 References
Barkhausen's criterion applies to linear circuits with a feedback loop. Therefore it cannot be applied to one port negative resistance active elements like tunnel diode oscillators.
Block diagram of a feedback oscillator circuit to which the Barkhausen criterion applies. It consists of an amplifying element A whose output v o
is fed back into its input v f
through a feedback network
β (j ω ) .
1 af 3
It states that if A is the gain of the amplifying element in the circuit and β ( j ω ) is the transfer function of the feedback path, so β A is the loop gain around the feedback loop of the circuit, the circuit will sustain steady-state oscillations only at frequencies for which:
1. The loop gain is equal to unity in absolute magnitude, that is,
2. The phase shift around the loop is zero or an integer multiple of 2 π :
and
Barkhausen's criterion is a necessary condition for oscillation but not a sufficient condition: some circuits satisfy the criterion but do not oscillate.
[5]
Similarly, the Nyquist stability criterion also indicates instability but is silent about oscillation. Apparently there is not a compact formulation of an
28-06-2016 11:14

Barkhausen stability criterion - Wikipedia, the free encyclopedia oscillation criterion that is both necessary and sufficient.
[6]
Barkhausen's original "formula for self-excitation", intended for determining the oscillation frequencies of the feedback loop, involved an equality sign: | β A | = 1. At the time conditionally-stable nonlinear systems were poorly understood; it was widely believed that this gave the boundary between stability (| β A | < 1) and instability (| β A | ≥ 1), and this erroneous version found its way into the literature.
[7] However, stable oscillations only occur at frequencies for which equality holds.
https://en.wikipedia.org/wiki/Barkhausen_stability_criterion
2 af 3
Nyquist stability criterion
1. Basu, Dipak (2000). Dictionary of Pure and Applied Physics . CRC Press. pp. 34–35. ISBN 1420050222.
2. Rhea, Randall W. (2010). Discrete Oscillator Design: Linear, Nonlinear, Transient, and Noise Domains .
Artech House. p. 3. ISBN 1608070484.
3. Carter, Bruce; Ron Mancini (2009). Op Amps for Everyone, 3rd Ed.
Newnes. pp. 342–343. ISBN 0080949487.
4. Barkhausen, H. (1935). Lehrbuch der Elektronen-Röhren und ihrer technischen Anwendungen [ Textbook of
Electron Tubes and their Technical Applications ] (in German) 3 . Leipzig: S. Hirzel. ASIN B0019TQ4AQ.
OCLC 682467377.
5. Lindberg, Erik (26–28 May 2010). "The Barkhausen Criterion (Observation ?)" (PDF) . Proceedings of 18th
IEEE Workshop on Nonlinear Dynamics of Electronic Systems (NDES2010), Dresden, Germany . Inst. of
Electrical and Electronic Engineers. pp. 15–18. Retrieved 2 February 2013. discusses reasons for this.
(Warning: large 56MB download)
6. von Wangenheim, Lutz (2010), "On the Barkhausen and Nyquist stability criteria", Analog Integrated Circuits and Signal Processing (Springer Science+Business Media, LLC)
66
(1): 139–141, doi:10.1007/s10470-010-9506-4, ISSN 1573-1979. Received: 17 June 2010 / Revised: 2 July 2010 / Accepted:
5 July 2010.
7. Lundberg, Kent (2002-11-14). "Barkhausen Stability Criterion". Kent Lundberg . MIT. Archived from the original on 7 October 2008. Retrieved 2008-11-16.
To find the loop gain, the feedback loop is considered broken at some point and the output v o
for a given input v i
is calculated:
:
Retrieved from "https://en.wikipedia.org/w/index.php?title=Barkhausen_stability_criterion&oldid=723051155"
28-06-2016 11:14