October 9, 1997

PSpice tunes oscillator circuits

Lutz von Wangenheim, Department of Electrical Engineering, Hochschule Bremen, Germany

A previous Design Idea (Reference 1) proposes a simple technique to extend the frequency range of a tungsten-lamp-stabilized crystal oscillator (Figure 1). The idea involves taking into account a simplified first-order expression for the frequency-dependent gain of the op amp. After some mathematical manipulations, you can see that an additional inductance (L_C=0.4 µH) in series with R_C balances the bridge. In this case, the impedance, Z_B, is purely resistive (R_B=430 ohms).

However, this method has one drawback: Before calculating the proper L_C value, you must know the characteristic values for the stabilization circuitry (R_X and L_X of the tungsten lamp), for the crystal (ohmic resistance R_Y at resonance), and for the amplifier (dc gain A₀ and pole frequency f_P). Moreover, circuit simulations based on a realistic three-pole macromodel of the AD8041 op amp reveal a tendency toward parasitic oscillations at approximately 100 MHz, with a gain margin of only 0.5 dB. The compensating inductor significantly increases loop gain, which causes the oscillations. This situation worsens if you also take into consideration the parasitic capacitances at the op-amp input terminals, which you must do for frequencies around 100 MHz. In this case, the stability margin further reduces to nearly zero.

An alternative compensation scheme allows more stable circuit operation. Also, a simple procedure using a simulation program, such as PSpice, makes it easy to calculate the compensation circuitry. Because you can obtain or create Spice-compatible macromodels for all of the circuit elements (quartz crystal, amplifier, and tungsten lamp), you can consider even their nonideal parameters during optimization.

You can apply this optimization method to all kinds of oscillator circuits. The technique is especially useful for compensating the resonant-frequency displacement due to parasitic-amplifier phase shifts in circuits with low phase slopes. Of course, the simulation results correlate closely with real circuits only if the macromodels properly reflect the characteristics of the real devices.

The "substitution theorem" is the basis of this optimization technique. According to this theorem, if and only if the overall loop gain is unity, you can substitute a signal source for a branch between two nodes of a circuit that have feedback without changing currents and voltages within the circuit. Because the unity-loop-gain condition is identical to the well-known oscillation criteria, you can exploit the theorem to design oscillator circuits. Thus, this idea applies the theorem in its reverse direction: If you replace a circuit element between two nodes with a sinusoidal stimulus of voltage V_Z and frequency F_Z, the current-to-voltage ratio, I_Z/V_Z, equals a complex conductance that you can use instead of the stimulus to fulfill the oscillation condition at frequency F_Z.

You can determine this conductance simply by running an ac analysis at the desired frequency of oscillation. An ohmic resistor, R_P, and a parallel capacitor, C_P, realize the real and imaginary parts, respectively, of the conductance, G_P. The corresponding values are as follows, where RE and IMG denote the respective real and imaginary parts of I_Z,:

G_P=1/R_P=RE(I_Z)/V_Z, (1) and

C_P=IMG(I_Z)/(V_Z·2 pi F_Z). (2)

Note that if you want to use a series instead of a parallel connection, you need to evaluate the real and imaginary parts of the voltage-to-current ratio, V_Z/I_Z, to find series elements R_S and C_S.

You can use the proposed optimization procedure to determine the complex impedance, Z_B, between nodes 1 and 2 in the circuit in Figure 1, which compensates for the effect of the amplifier's nonideal characteristics on overall circuit performance. Note that the technique takes into account all amplifier parameters, such as input and output impedances and frequency characteristics, as far as they are part of the macromodel.

First, replace Z_B with an ac voltage source V_Z=1V, and then perform an ac analysis at one frequency (F_Z=5 MHz). Choose all other components just as in Reference 1: R_A=220 ohms, R_C=75 ohms, R_X=34 ohms, L_X=100 nH, and a quartz crystal with a series resonance at 5 MHz and R_S=50 ohms (Q approximately equals 1E4). According to Equations 1 and 2, the real and imaginary parts of current I_Z through V_Z lead to the parallel values of R_P and C_P, respectively, which constitute impedance Z_B.

The simulation program can perform these calculations and list the results as part of the output file. In addition to the description of the oscillator circuit, the simulation netlist should contain two voltage-controlled sources in conjunction with a corresponding print statement:

VZ 1 2 AC 1
E_GP GP 0 VALUE={I(VZ)
E_CP CP 0 VALUE={I(VZ)/(2*PI*FZ)
.PARAM PI=3.1416 FZ=5E6
.PRINT AC VR(GP) VI(CP)
.AC LIN 1 5E6 5E6

The .OUT file resulting from this ac analysis contains the following two lines:

FREQ VR(GP) VI(CP)
5E+06 2.216E-03 2.591E-11

Thus, the component values are R_P=1/G_P=451.26 ohms, and C_P=25.9 pF. Using these values for Z_B in Figure 1, another simulation run confirms that the loop gain at 5 MHz is unity and that a sufficient stability margin of approximately 16 dB exists at higher frequencies. (DI #2097)

Reference

1. Schleicher, Israel, "Simple compensation extends oscillator's range," EDN, Dec 5, 1996, pg 130.

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