# Frequency Deviation In LC-Oscillators Using IC Amplifiers

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 Jochen Bauer Editor D  Articles: 111 Schem.: 0 Pict.: 0 24.Mar.16 21:05 Count of Thanks: 9 While integrated circuit (IC) amplifiers are mostly used with RC and crystal oscillators, they can also be used as a feedback device for LC oscillators. However, two basic problems of practical LC oscillators, the development of parasitic oscillations with frequencies above the intended oscillation frequency range under certain conditions and a deviation of the actual oscillation frequency from the natural frequency $\omega_0=\sqrt{\frac{1}{LC}}$     of the LC tank often become more pronounced when using integrated circuit amplifiers instead of one single active component. This is mostly because integrated circuit amplifiers often exhibit an appre-ciable delay between input and output signal even at relatively low frequencies. In the attached PDF document an extensive investigation of both, parasitic oscillations and frequency deviation, using a generic non-ideal differential amplifier LC-oscillator can be found. In the remainder of this post I would like to give a summary for the reader less interested in the math- ematical details: We shall base our analysis on a generic LC oscillator using a differential voltage amplifier as shown in the following diagram. The LC tank is modeled by an ideal inductor L, an ideal capacitor C and a parallel loss resistance RP that subsumes the losses occurring in the inductor and the capacitor. The feedback device "FB" is a differential amplifier, delivering it's output voltage Uf back into the LC tank via Rf. In the ideal case, such a differential amplifier exhibits neither a delay time between input and output nor is it's amplification behavior frequency dependent. However, integrated circuit amplifiers, due to the relatively large number of internal transistor systems typically have an appreciable delay between input and output signal even at relatively low frequencies, resulting in a phase shift that can easily reach values of 360° or beyond. The latter will be the case if the delay time exceeds the period of the input signal.   A basic understanding of the behavior of the oscillator with phase shifted feedback can be obtained by going below the oscillation threshold and analyzing the response of the feedback device when driven externally by a small sinusoidal input voltage using complex phasors. From this linear analysis, we obtain the phase shift regions where the feedback device delivers positive feedback into the LC tank causing oscillations if the amplification factor of the feedback device is sufficiently high. Those phase shift regions are: 0°<φ<90°,  270°<φ<450°,  630°<φ<810°  and so on... Oscillations occurring for phase shifts of 270° and above are typically undesired parasitic oscillations well above the desired oscillation frequency. Introducing a non-linear feedback device model with delay and gain roll-off at higher frequencies we are able to do a time domain analysis using differential equations. The model in use has a phase shift and gain frequency dependency that is shown in the following diagram. Here, a0 is the small signal DC gain (ω=0). Note that the parameters of the model have been chosen such that the above phase shift behavior re- sembles the phase shift behavior of an LM311 voltage comperator. We shall now focus on the occur- rence of parasitic oscillations above the primary (intended) oscillation frequency range for different values of the DC gain a0. In the following diagram, the oscillation amplitude (if oscillations occur) has been plotted over the natural frequency ω0 of the LC tank for different values of a0. (Note that a0=200000 is the typical DC gain of a LM311 voltage comperator). Obviously, for a high gain of a0=200000 (red curve) the circuit will oscillate not only when the natural frequency of the LC tank is below approximately 1.4MHz but also when the natural frequency is in a range around 6MHz. In practical oscillator circuits with the intended natural frequency of the LC tank set below 1.4MHz this may cause undesired oscillations near parasitic resonant frequencies of the circuit around 6MHz. This unfavorable behavior persists even down to a0=10 (green curve) although the amplitude of the oscillations around 6MHz now becomes somewhat smaller than the amplitude of the oscillations below approximately 1100kHz. Finally, at a0=5 (blue curve) the oscillator is left with only one oscillation frequency range ending at approximately 800kHz. This rules out parasitic oscillations above the in- tended frequency range and can be considered a stable oscillator design. The reader may have noticed that for a0=200000 even in the frequency range around 3MHz where the linear analysis predicts no oscillations at all, the amplitude curve indicates some residual oscillations. A deeper investigation reveals non-sinusoidal oscillations with frequencies jittering between the primary and parasitic oscillation frequency range. This is clearly a non-linear phenomenon of the circuit. We shall now turn our attention to the deviation of the oscillation frequency ωosc from the natural frequency ω0 of the LC tank. From a linear analysis of the circuit we see that this is due to a virtual parallel reactance appearing across the LC tank due to feedback. Here, we are interested in the fre- quency deviation occurring in the primary oscillation range, where in the linear analysis, 0°<φ<90° and hence the actual oscillation frequency is below ω0 due to the appearance of a virtual capacitance. The frequency deviation of the oscillator for different Q-factors of the LC tank is shown in the diagram below.   Attachments: Full Article as PDF (393 KB) This article was edited 24.Mar.16 21:13 by Jochen Bauer .