Serious students of mathematics, physics, and engineering know that there are a number of mathematical coincidences that pop up in different fields. The mathematical analysis applied to one type of problem in one field can be applied to a completely different problem in another field, typically because the equations that govern these problems are nearly identical.
Electronics and mechanics are not without their correspondence. Simple harmonic motion of a mechanical body and a harmonic response in a circuit can be described with the same equations of motion. Understanding this aids a systems designer in understanding how circuits respond to real oscillating drivers: a circuit may be driven with a periodic source, but that does not mean the resulting motion will be simple.
Ideal Simple Harmonic Motion in Circuits
In the ideal case, and within the context of electronics design, simple harmonic motion refers to driving a circuit with a sinusoidal source. Linear circuits, meaning any circuit where the current and voltage (or derivatives of either) in different elements are directly proportional, will output a sinusoidal signal. This is a very important consideration as it allows the behavior of a circuit to be analyzed using Ohm’s law and Kirchoff’s laws.
All oscillators have a natural frequency, which refers to the frequency at which they would tend to oscillate if initially displaced from equilibrium. In the case of an undamped pendulum, the natural frequency depends on the length of the pendulum and the acceleration due to gravity. When the pendulum is initially displaced from equilibrium, it will naturally oscillate about the equilibrium position.
The electronic analog of a pendulum is an LC circuit, meaning a circuit that contains a capacitor and an inductor in series. In this case, when the circuit is initially displaced from equilibrium, meaning the capacitor initially holds some charge. Once the circuit is shorted, the current in the circuit will begin oscillating at its natural frequency, just as in the case of a mechanical oscillator.
Charging and discharging LC circuit, leading to simple harmonic motion during discharge
Real Oscillations in a PCB Are Not Simple
Simple harmonic motion can be produced in a circuit when driven with an AC signal, and resonance can occur when the circuit is driven at a frequency that matches the resonance frequency. Mathematically, the current in the circuit will increase linearly over time and will eventually reach infinity. In a real circuit, even in a passive linear circuit, nonlinear effects will set in as the current grows. This will cause the current to saturate near some maximum value.
In a PCB, every circuit will contain some capacitance and inductance due to the geometry of conductors in the circuit. These capacitance and inductance values are parasitic, but they should be included in circuit models in your board. The presence of these parasitic circuit elements means that each circuit in a PCB has some resonance frequency and can exhibit ringing when driven with digital pulses. This is problematic in transmission lines and illustrates the reason for using termination networks to suppress reflections and ringing.
Damping in RLC Circuits
Unfortunately, real systems are not free of damping. In a mechanical oscillator, damping arises due to friction and drag. In an electrical oscillator, such as the LC circuit shown above, the wires have some resistance, which will dissipate power as the current oscillates. This causes the current in the circuit to drop to zero over time at an exponential rate, and the rate of damping is equal to half the total resistance in the circuit divided by the inductance. The presence of damping will cause the resonance frequency to shift away from the natural frequency, and it will limit the response of the circuit to some maximum value at resonance.
The typical method for examining the behavior of a circuit in the presence of parasitics is to model the circuit as an RLC network. The exact placement of the parasitic capacitor and inductor in the circuit will depend on the geometry of conductors and placement of components in the circuit.
It is important to note that even components themselves can have some parasitic resistance, capacitance, and inductance. For example, a real capacitor is not perfect and is actually modeled as its own RLC network. This means that components like capacitors actually exhibit a phenomenon called self-resonance. This self-resonance phenomenon can cause EMI problems with a capacitor is driven at its resonance frequency. It also means that a capacitor can exhibit ringing during discharge if the resistance in series with the capacitor is too low, meaning the response will be underdamped.
Effect of damping on the current in an RLC circuit
Arbitrary Periodic Driving and Nonlinear Circuits
By “arbitrary” and “periodic”, one intends to mean “any signal that has repetitive shape and is not sinusoidal”. Real oscillators in electronic circuits can be driven with an arbitrary waveform. Examples include a stream of repeating digital pulses, a stream of Gaussian or Lorentzian pulses, sawtooth or triangle waves, or any other repetitive shape you can imagine.
Furthermore, not all circuit elements are linear. Instead, the relationship between current and voltage (or their derivatives) is not a straight line. A perfect example is a diode, where the output current is an exponential function of the input voltage. Another example is a transistor, where the output saturates once the input voltages become large. When these types of circuits are driven with a sinusoidal source, output from the circuit will not be sinusoidal, and there could be a phase shift that accumulates, depending on the capacitance and inductance in the circuit.
Arbitrary periodic (non-sinusoidal) sources will produce a periodic response in the circuit, but the response will not always have the same shape as the driver. There are a number of reasons for this. First, passives like inductors and capacitors exhibit a transient response that depends on the shape of an arbitrary driving pulse. Nonlinear circuit elements also affect the shape of the output pulse when the input driver is at a high signal level. As a useful example, consider the output from an inverting amplifier driven with a high voltage sinusoidal source. At high driving strength, the output will saturate as a square wave that is inverted compared to the input signal.
Any equivalent RLC network in a PCB that is driven with digital pulses can exhibit ringing if the circuit is underdamped. In general, any RLC circuit driven with a series of digital pulses will exhibit a transient response in the current in the circuit. This response can be either underdamped, perfectly damped, or overdamped, depending on the series resistance in the RLC network.
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