# IBM Oscillator

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```					Voltage Controlled Oscillator
Introduction

One of the components of a PLL is the voltage controlled oscillator (VCO). There are several
ways in which these can be realized; LC-based and ring oscillator based. In this report we
consider the LC-based oscillator.

An excellent reference is [1].

LC Oscillator

Every LC oscillator has an inductance and capacitance connected to an active circuit. The role of
the active circuit is to provide a negative resistance to counterbalance the positive resistance
present in the passive LC circuit. The purpose of the LC circuit is to determine the frequency of
oscillation.

We model the LC oscillator as shown in Figure 1.

Figure 1 Oscillator Model        R = -R1 is the input resistance of the active circuit.

We convert this to discrete form in the usual way, using the transformation from the
s-plane to the z-plane s -> (z – 1)/T, where T is the sample rate of the digital system.
The equations are as follows.

1
I ( R  sL              ) V
sC
1  z 1
s
Tz 1
RT T 2               RT            T
I  z 1 (2                   )  z 2 (1     ) I  Vz 1 (1  z 1 )
L LC                 L            L
For R = 0

T2                   T
I  z 1 (2                )  z 2 I  Vz 1 (1  z 1 )
LC                   L

T2                                                                                            theta
cos( )  2 
LC
The last equation for, I, has poles on the unit circle with the position of the poles
determined by the equation for the cosine. This determines the frequency of oscillation.

These equations can now easily be programmed in Simulink using the SPO blocks.
Figure 2 shows the Simulink Model.

.2

Constant
1
Scope
z
Product
Step            1                             Unit Delay

z
Unit Delay1   Unit Delay2
Unit Delay4
1             1
simout
z             z
To Workspace

Unit Delay3       Product1
Constant1
1
1.9
z
Running this model shows the expected oscillation.

Figure 3 Circuit Response

Analysis

The next step is to analyze the performance of this circuit in terms of the parameters important to
its use in a PLL. These include frequency stability, phase noise, amplitude noise and others as
discussed in [1].

Matlab has many tools to perform such analyses. For example it is easy to compute the FFT of
the oscillator output to determine the spectral purity. Figure 4 shows the FTT of a wave with
32768 points.

Figure 4 FFT of Oscillator Signal, Overall and Detail

10000                                                  10000

9000                                                   9000

8000                                                   8000

7000                                                   7000

6000                                                   6000

5000
5000

4000
4000

3000
3000
2000
2000
1000
1000
0
0   0.5   1   1.5   2   2.5   3      3.5          0
4                 1648 1650 1652 1654 1656 1658 1660 1662 1664 1666 1668
x 10
Oscillator Notes

The MDL file is ibmvco2.mdl In IBM

The evaluation program is

ff = abs(fft(simout)); stem(ff,'.'); zoom xon

The plan is to study the performance of the oscillator with respect to noise and other
factors.

References

[1] “High Purity Oscullators”, E Hegazi, J Rael, A Abidi, Kluwer, 2005

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