# Electrical Analogies of Mechanical Systems - Control Systems

## What are Electrical Analogies of Mechanical Systems?

When two systems are said to be analogous if they satisfy the following two conditions.

• The two systems are physically different
• Differential equation modeling of these two systems are same

While, Electrical systems and mechanical systems are two different systems. But there are two types of electrical analogies are there in translational mechanical systems which are force voltage analogy and force current analogy.

## Force Voltage Analogy

The mathematical equations of translational mechanical system are compared with mesh equations of the electrical system.

Let’s see the below mentioned translational mechanical system as mentioned in the following figure.

The force balanced equation for this system is

See the following electrical system as mentioned in the following figure. This circuit contains a resistor, an inductor and a capacitor. All these electrical elements are connected in a series. The input voltage applied to this circuit is [Math Processing Error]V volts and the current flowing through the circuit is [Math Processing Error]i Amps.

Mesh equation for this circuit is

If you compare the Equation 1 and Equation 3, then you will get the analogous quantities of the translational mechanical system and electrical system. Below mentioned table shows these analogous quantities.

At the same time there is a torque voltage analogy for rotational mechanical systems. Let’s discuss about this analogy.

## Torque Voltage Analogy

If you can compare the mathematical equations of rotational mechanical systems in this analogy with mesh equations of the electrical system.

Rotational mechanical system is described in the below mentioned figure.

The torque balanced equation is

We will get the analogous quantities of rotational mechanical system and electrical system by comparing Equation 4 and Equation 3. Let’s see following table which shows these analogous quantities.

Rotational Mechanical System Electrical System
Torque(T) Voltage(V)
Moment of Inertia(J) Inductance(L)
Rotational friction coefficient(B) Resistance(R)
Torsional spring constant(K) Reciprocal of Capacitance$\left(\frac{1}{c}\right)$
Angular Displacement(θ) Charge(q)
Angular Velocity(ω) Current(i)

## Force Current Analogy

Here, the mathematical equations of the translational mechanical system are compared with each other when the nodal equations of the electrical system are in force current analogy.

Now see the following electrical system as mentioned in the following figure. This circuit contains a current source, resistor, inductor and capacitor and these are connected in parallel.

The nodal equation is

If you compare the Equation 1 and Equation 6, then we will get the analogous quantities of the translational mechanical system and electrical system. The following table shows these analogous quantities.

Similarly, there is a torque current analogy for rotational mechanical systems. Let us now discuss this analogy.

## Torque Current Analogy

Torque current analogy compares the mathematical equations of the rotational mechanical system with the nodal mesh equations of the electrical system.

Here you can compare the Equation 4 and Equation 6 and we will get the analogous quantities of rotational mechanical system and electrical system. The following table shows these analogous quantities.

Here we discussed the electrical analogies of the mechanical systems. These analogies are used to analyze the non-electrical system like mechanical system from analogous electrical system.