This C5 tuning fork will vibrate at its damped natural frequency
Those familiar with oscillators are most likely to think in terms of a simple harmonic oscillator, like a pendulum or a mass on a spring. These systems are conceptually simple, but their mathematical models fail to account for realistic properties in these systems. Any oscillator has some important physical parameters, and the system’s response when driven will not match the response seen when allowed to oscillator freely.
In electronics, different circuits will function as oscillators, where the voltage and current exhibit a periodic response in time. Just like mechanical oscillators, oscillator circuits can exhibit resonance under the right conditions. Things become mathematically confusing for some designers in that the real response of an oscillator is defined in terms of three different frequencies. Let’s clear up the difference between the resonant frequency vs. natural frequency in purely linear oscillator circuits.
Damped Oscillation Frequency vs. Natural Frequency in Undriven Oscillators
Although we can quantify a natural frequency in mechanical and electrical harmonic oscillators, the system never really oscillates at the natural frequency. This is because, in an ideal model for an oscillator, we like to ignore the effect of damping so that we can understand some basic aspects of the system. In a mechanical oscillator, this means we briefly ignore friction or any other mechanism that dissipates kinetic energy. In a circuit, this means we omit circuit elements that dissipate power as heat, i.e., the circuit only contains capacitive and inductive elements.
When an undriven, undamped oscillator is displaced from equilibrium, the system will oscillate at its natural frequency. However, real oscillator circuits always contain some damping; in an LC circuit, the conductors have some small amount of resistance, which provides damping in the circuit. This is also true in mechanical oscillators; there is always some source of damping that converts kinetic energy to heat, which is why a swinging pendulum eventually slows to a stop.
The effect of damping leads to two phenomena in undriven oscillators that are allowed to oscillate naturally when displaced from equilibrium:
The oscillation decays over time. Damping in an oscillator circuit occurs because some electrical energy (i.e., the kinetic energy of flowing charges) is lost as heat. This causes the amplitude of the oscillation to decay over time.
The damped oscillation frequency does not equal the natural frequency. Damping causes the frequency of the damped oscillation to be slightly less than the natural frequency. The damped oscillation frequency is defined in the equation below:
The oscillation frequency of a damped, undriven oscillator
Eventually, when the damping rate is equal to the natural frequency, there is no transient oscillation, meaning the voltage and current in the circuit just decay back to equilibrium; this is known as critical damping. As the damping rate keeps increasing above the natural frequency, the time required for the voltage and current to fall back to equilibrium becomes longer.
If you were to perform a transient analysis of an oscillator circuit and measure the oscillation frequency, you are not measuring the natural frequency. You are actually measuring the damped oscillation frequency defined in the above equation. You can then extract the damping rate by plotting the natural logarithm of decay data (shown using red dots in the graph below) in the response waveform over time; the negative of the slope of this line is equal to the damping rate.
The oscillation frequency of a damped, undriven oscillator
In the above graph, the successive maxima are marked with red dots, and the logarithm of these electric current data are plotted in the right graph. From the regression line, we see that the damping rate in this circuit is 0.76 per sec. The damped oscillation rate can be determined between two consecutive maxima in the left graph and has a value of 3.929 rad per sec. Once you know the damping rate and the damped oscillation frequency, you can easily calculate the natural frequency using the above equation. In this simulation, the natural frequency is 4 rad per sec. You can also see from the exponential decay curve that the initial current was 1 A.
Resonant Frequency vs. Natural Frequency in Driven Oscillators
When an oscillator circuit is driven with a periodic signal, the current and voltage will oscillate at the same repetition rate as the driving signal. However, the waveforms will not match perfectly because an oscillator circuit’s transfer function will distort these signals; in other words, the oscillator circuit also acts like a filter/amplifier (more on this below). To see how different oscillator circuits can behave, it helps to consider a mechanical oscillator that is only driven with a sinusoidal signal.
Resonance is a phenomenon that results when an oscillator is driven with a periodic signal with a specific frequency, known as the resonant frequency. In a driven oscillator without damping, the resonant frequency is equal to the natural frequency. This is always the case in undamped oscillators, but it is not always the case in damped oscillators. Real driven oscillators have damping, and the resonant frequency is not always equal to the natural frequency. For a typical driven damped mechanical oscillator, the resonant frequency is defined in the following equation:
Resonant frequency vs natural frequency of a driven damped mechanical oscillator
Note that resonance can only occur when the natural frequency is greater than the damping rate, multiplied by the square root of 2. If the damping is too large, then resonance cannot occur.
What about the case of small damping? In the limit where the damping constant is zero, the resonant frequency equals the natural frequency and there is no dissipation of energy in the circuit. As a result, when an undamped oscillator is driven exactly at its natural frequency, the amplitude of the resulting oscillation will (theoretically) diverge to infinity at a linear rate. In a real circuit, nonlinear effects will eventually take over at high voltage/current, which can cause the response to saturate, or will cause the circuit to burn out.
There is a certain range of frequencies in between which the mechanical oscillator will not exhibit resonance when driven, but it will still exhibit a decaying oscillation when displaced from equilibrium. This decaying oscillation will still occur with the damped oscillation frequency defined in the first equation above. Going back to a mechanical oscillator, we have:
Case where resonance is eliminated, but there can still be a damped oscillation
Note that the above conditions we discussed for a mechanical oscillator also apply to an RL circuit with a parallel capacitor.
Oscillator Transfer Functions
The damping in a circuit will define the circuit’s transfer function, which is usually described in terms of its bandwidth. When the oscillator is driven with a sinusoidal signal, the output will also be sinusoidal. However, when the oscillator is driven with a non-sinusoidal periodic signal (e.g., a sawtooth wave, frequency modulated signal, stream of clock pulses, or other repeating analog waveform), then the resulting voltage and current waveforms in the oscillator may not resemble the driving signal. You can extract the transfer function from a frequency sweep by applying a sinusoidal source to your oscillator circuit. Examples that show these transfer functions in the frequency domain for different damping rates for a mechanical oscillator are shown below.
Amplitude curves for a mechanical oscillator as a function of driving frequency
Note that these curves are normalized with respect to the natural frequency. Again, certain RLC circuits will have similar curves, while others (e.g., the series RLC circuit) will have curves that always peak at the natural frequency, i.e., resonant frequency = natural frequency.
The graphs above for a mechanical oscillator show how the peak in each curve (which corresponds to the resonant frequency) moves towards the natural frequency as the damping rate decreases. Each of these curves can be thought of as a transfer function. Determining how each of these curves affects an arbitrary analog driving signal in the time domain requires working with Fourier transforms or Laplace transforms, which is a bit beyond the scope of this article.
No matter what type of oscillator circuit you are designing, you can easily distinguish the resonant frequency vs. natural frequency of your circuit with OrCAD PSpice Simulator from Cadence. This powerful package includes analysis tools that take data directly from your circuit schematics and determine the behavior of your most complex PCB designs.
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