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On the Modified Barkhausen Criterion Erik Lindberg, Lifemember IEEE, K. Murali Reader Emeritus at The Technical University of Denmark, Department of Physics, Anna University, Kongens Lyngby, DK2800, Denmark, Chennai-600 025, India e-mail: [email protected] e-mail: [email protected] Abstract—Oscillators are normally designed according to the TABLE I Modified Barkhausen Criterion i.e. the complex pole pair is moved out in RHP so that the linear circuit becomes unstable. (A) R2 = 10e + 3 1 1 By means of the Mancini Phaseshift Oscillator it is demonstrated ideal op.amps A1 = A3 = Sigma (real) Omega (imag) Q that the distortion of the oscillator may be minimized by 1 introducing a nonlinear ”Hewlett Resistor” so that the complex +0.00000e-01 -1.00000e+04 +0.00000e-01 +1.00000e+04 1 pole-pair is in the RHP for small signals and in the LHP for -1.00000e+04 +0.00000e-01 +0.50 large signals i.e. the complex pole pair of the instant linearized (B) R = 10e + 3 small signal model is moving around the imaginary axis in the 2 perfect op.amps A1 = 100e + 3 A3 = 100e + 3 complex frequency plane. Sigma (real) Omega (imag) Q -9.99990e-2 -9.99990e+03 +50.0e+03 I. INTRODUCTION -9.99990e-2 +9.99990e+03 +50.0e+03 In order to design electrical and electronic circuits we -1.00000e+4 +0.00000e-01 +0.50 (C) R2 = 10:1e + 3 assume that the wave length of the highest frequency is much perfect op.amps R2A = 10:1e + 3 R2B = 1 larger than the circuit so lumped elements could be used - Sigma (real) Omega (imag) Q quasi stationary circuits. +2.46521e+01 -9.97496e+03 -202.314 Since the 1930’ies the Barkhausen Criterion [1] has been +2.46521e+01 +9.97496e+03 -202.314 -9.95049e+03 +0.00000e-01 +0.50 used as a starting point for the design of oscillators. The (D) R2 = 8248 basic assumption is that the oscillator topology is a loop of a perfect op.amps R2A = 10:1e + 3 R2B = 45e + 3 linear inverting amplifier and a linear frequency determining Sigma (real) Omega (imag) Q feedback circuit. The loop is opened and the transfer-function -5.28186e+2 -9.97496e+3 +9.9 -5.28186e+2 +9.97496e+3 +9.9 of the open loop circuit - the loop gain - is designed to be -1.10670e+4 +0.00000e-1 +0.50 equal to 16 0. The loop is then closed and we have a linear 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. II. MANCINI’S PHASESHIFT OSCILLATOR 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 Fig. 1. Mancini’s Phaseshift Oscillator. of the second integrator is then inverted and applied as positive feedback to induce oscillation. Both outputs V (3) and V (6) oscillator with a complex pole-pair on the imaginary axis of have relatively high distortion, that can be reduced with a gain- the complex frequency plane. In order to secure oscillations stabilizing circuit. With reference to fig. 1 - which is taken the complex pole-pair is moved to the right half of the complex from figure 20, page 19 of [4] - node 1 is connected to an frequency-plane RHP so the linear circuit becomes unstable. ideal 2.5V power source in the derivation of the following This approach is called the Modified Barkhausen Criterion. 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 R3. The quality-factor Q of a complex pole is defined as: p Q = Sigma2 + Omega2=(−2 × Sigma) (3) It is a measure for the distance of the pole from the imaginary axis. It is seen that Q becomes 1 for poles on the imaginary Fig. 2. Amplitude control section of Hewlett’s oscillator patent axis. Q is negative for poles in the right-half-plane RHP and positive for poles in the left-half-plane LHP . i.e. V (1) = V (2) and V (4) = V (5) and V (1) = 0. The order Table I case (A) shows that the solution of the characteristic of the circuit is 3 i.e. the characteristic polynomial is a third polynomial for R1 = R2 = R3 = R = 10kΩ and C1 = C2 = 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: [ ] G1G2(G3 + sC3) + sC1(G2 + sC2)sC3 V3 = 0 (1) (G2 + sC2)sC3 where Gn = 1=Rn. If we observe that V (3) =6 0 the numerator in equation (1) must be zero and the characteristic polynomial of the linear circuit becomes: G G G G G G s3 + s2 2 + s 1 2 + 1 2 3 = 0 (2) C2 C1C2 C1C2C3 It is difficult to find expressions for the roots of this polynomial Fig. 5. FFT spectrum of Sine output V3 of Mancini’s Phaseshift Oscillator so numerical calculations are used. fig. 1. R2A and R2B refer to fig. 4. C3 = C = 10nF is a complex pole-pair on the imaginary axis and real pole in LHP . The frequency becomes ! = 2πf = 1=RC i.e. f = !=(2π) = 1:591549431kHz Fig. 7. Power source currents I(VR10);I(VP );I(VN) The cases (B), (C) and (D) of TABLE I are calculated by means of the ANP3 program [5]. Case (B) shows that A = V =(V − V ) A = V =(V − V ) Fig. 6. Static gains 1 3 1 2 and 3 6 4 5 the complex pole pair is very close to the imaginary axis if the amplifiers are assumed to be perfect with infinite III. DISTORTION MINIMIZATION input impedance, zero output impedance and constant gain Figure 2 shows the comments on the amplitude control in A = vout=vin = 100e + 3, Q = 50k. Case (C) shows that Hewlett’s oscillator patent [7], [8], [9]. He assumes that the the complex pole pair is in RHP when R2 = 10:100e + 3, − circuit is linear so it is necessary to introduce a nonlinear (Q = 200). Q becomes greater for smaller values of R2 e.g. ) − ) negative feedback resistor R3 which vary with the amplitude R2 = 10:010e + 3 Q = 2k and R2 = 10:001e + 3 of the signals. He also claim that this approach could be used Q = −33k. in other types of oscillation generators. He mention nothing Now the ”Hewlett resistor” is introduced and case (D) shows about the poles of the instant small signal model. that the complex pole pair is in LHP for large signals R2 = Figure 3 shows fig.1 of Hewletts Wien Bridge Oscillator 8:248e + 3,(Q = +10) i.e. it is not necessary to have a patent. Please note the resistor R3 a tungsten lamp. symmetric placement of the complex pole pairs. In [10] it is shown that the distortion for a negative resis- Figure 5 shows the ”fft” spectrum of the Sine output V3 tance oscillator may be minimized by introducing a nonlinear for the cases (B), (C) and (D) calculated by means of PSpice feed-back resistor ”Hewlett resistor” so that the complex pole- [6], with the following options: ”.tran 0 1000e-3 100e-3 1e-4 pair of the time-varying small-signal model move symmetri- ; uic” and ”+ RELTOL = 1e-6”. Distortion is increased when cally around the j! axis. In the following it is shown that it the complex pole pair is moved to RHP , case (C). Distortion is possible to reduce the distortion of the Mancini Phaseshift is minimized when the ”Hewlett resistor” is introduced, case Oscillator by introducing resistor R2 as a nonlinear ”Hewlett (D). resistor”, fig. 4. Figure 6 shows the static gains of the op-amps A1 = V3=(V1 − V2) and A3 = V6=(V4 − V5) 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(VP ) and I(VN) become almost constant in case (D). IV. CONCLUSION 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, 1. When the oscil- lator 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 nonlin- earity for steady state oscillations.
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