Use PSpice for behavioral modeling of VCOs

PSpice, a member of the Spice family for PC users, is becoming a standard tool for analog and mixed analog-digital simulation. Many analog designers are familiar with the software of Design Center, Design Lab, and OrCad (www.orcad.com) PSpice and use the software in their everyday lives. This Design Idea should be helpful to those who design and simulate PLL systems at a behavioral level. A basic PLL system comprises a phase detector, a loop filter, and a VCO (voltage-controlled oscillator), connected in a negative-feedback loop. The phase detector produces a voltage that, after lowpass-filtering by the loop filter, becomes an error voltage, V_E, applied to the control input of the VCO to set the angular frequency, (ω). When V_E=0, the VCO oscillates at some initial frequency, ω₀, called the frequency offset or free-running frequency. The output frequency of the VCO is:

ω(t)=ω₀+G_VCOV_E(t). (1)

Here,



where G_VCO is the VCO's voltage-to-radian frequency gain in radians per second per volt and g_VCO is the VCO's voltage-to-frequency gain in hertz per volt. Consider a sinusoidal voltage with amplitude V_AMPL, argument θ(t), and dc offset V_OFF:

V(t)=V_AMPLsin(θ(t))+V_OFF. (3)

The argument θ(t) is the time integral of the angular frequency:

θ(t)=∫ω(t)dt+θ(0). (4)

Assuming ω is constant and substituting θ(t) in V(t) yields the popular form:

V(t)=V_AMPLsin(ωt+θ(0))+V_OFF. (5)

If ω is not constant, the general form is:

Replacing ω(t) with Equation 1 and considering θ(0)=0, you obtain the VCO's transfer function:

Equation 7 is the heart of the VCO's time-analysis modeling. Thus, you can easily perform VCO behavioral modeling, using the ABM1 part from the Analog Behavioral Modeling Symbol Library abm.slb (abm.olb in OrCad Capture). You can simply write the formula for the transfer function, adapted for PSpice. Figure 1 shows the PSpice representation of the process. Figure 2 shows a simulation example, in which V_E is the VCO's input voltage, and V_(OUT1) is the output voltage. Because most phase detectors (types I, II, and III) use a digital input signal, you can easily obtain a digital VCO-output waveform using the conditional expression above the lower box in Figure 1. The simulated curve is V_(OUT2), also shown in Figure 2. Finally, for loop-gain evaluation in ac analysis, you simply model the VCO in the frequency domain in an ideal integrating unit, using the Laplace form T_VCO(s)=G_VCO/s. Most obvious behavioral VCO models are based on a real circuit concept. Thus, they comprise many elements—for example, controllable sources, capacitors, and others. Hence, the models are complicated. This modeling is simple without involving superfluous computation resources.

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