Mechanical vs Electrical Oscillators: A Comparison

This article compares mechanical and electrical oscillators, highlighting their differences. We will also derive the equation of an electrical oscillator equivalent to a harmonic oscillator.

Introduction

Oscillators are fundamental components in most communication systems and test & measurement equipment. Their specifications and tolerances directly impact the performance of these systems. Several key parameters are considered during oscillator design, including:

• Frequency sensitivity
• Frequency stability (short-term and long-term drifts)
• Oscillator noise
• Amplitude stability

Mechanical Oscillator (Harmonic Oscillator)

• Figure 1 depicts a simple mechanical oscillator, such as a mass attached to a spring. This structure is similar to a pendulum and is also known as a harmonic oscillator.
• When a body attached to a spring is displaced from its equilibrium position, the spring exerts a restoring force, pulling it back towards equilibrium. This force causes the system to oscillate or undergo periodic motion.
• The relationship between period (T) and frequency (f) of oscillation is: $f = \frac{1}{T}$.
• Angular frequency (ω) is approximately $2\pi$ times the frequency: $w = 2\pi f$.
• When the restoring force is directly proportional to the displacement from equilibrium, the motion is called simple harmonic motion, and the oscillator is a Harmonic Oscillator.

Electrical Oscillator

• Figure 2 shows the electrical analogy of a mechanical oscillator. This is an example of an electronic or electrical oscillator. Different types exist, including RC oscillators, LC oscillators, and crystal oscillators.
• An LC circuit consists of an inductor (L) and a capacitor (C). Initially, the capacitor is charged, inducing a current in the inductor. Subsequently, the current and voltage oscillate harmonically according to the following equations: $V = V_{max} \cdot cos(wt + \Phi)$ $I = I_{max} \cdot cos(wt + \Phi + 90)$

Deriving the Differential Equation for an Electrical Oscillator

Let’s derive the differential equation that represents an electrical oscillator:

• The capacitor discharges (or charges) through the inductor. Therefore, the current (I) in the inductor and the charge (Q) on the capacitor are related as: $I(t) = \frac{dQ}{dt}$.

• Consequently, the voltage across the inductor is related to the second time derivative of the charge as: $V_L(t) = -L \cdot \frac{dI}{dt} = -L \cdot \frac{d^2Q}{dt^2}$.

• In an LC circuit, the voltage across the inductor must be the same as the voltage across the capacitor. Hence, $V_L = V_C = \frac{Q}{C}$.

• Comparing the last two equations, we get: $-L \cdot \frac{d^2Q}{dt^2} = V_L = V_C = \frac{1}{C} \cdot Q(t)$.

• This second-order differential equation for charge has the form of a harmonic oscillator.