In between electrical and mechanical systems there exists a fixed analogy and their exists a similarity between their equilibrium equations. Due to this, it is possible to draw an electrical system which will behave exactly similar to the given mechanical system, this is called electrical analogous of given mechanical system and vice versa. It is always advantageous to obtain electrical analogous of the given mechanical system as we are well familiar with the methods of analyzing electrical network than mechanical systems.
There are two methods of obtaining electrical analogous networks, namely
1) Force – Voltage analogy i.e. Direct Analogy.
2) Force – Current Analogy i.e. Inverse Analogy.
1.1 Mechanical Systems
Consider simple mechanical system as shown in the Fig.1.
Due to the applied force, mass M will display by an amount x(t) in the direction of the force f(t) as shown in the Fig.1.
According to Newton's law of motion, applied force will cause displacement x(t) in spring, acceleration to mas M against frictional force having constant B.
Where, a = acceleration, v = velocity
This is equilibrium equation for the given system.
Now we will try to derive analogous electrical network.
1.2 Force Voltage Analogy (Loop Analysis)
In this method, to the force in mechanical system, voltage is assumed to be analogous one. Accordingly we will try to derive other analogous terms. Consider electric network as shown in the Fig.2.
The equation according to Kirchhoff's law can be written as
Taking Laplace,
But we cannot compare F(s) and V(s)unless we bring them into same form.
For this we will use current as rate of flow of charge.
Replacing in above equation,
Comparing equations for F(s) and V(s) it is clear that,
i) Inductance 'L' is analogous to mass M
ii) Resistance'R' is analogous to friction B.
iii) Reciprocal of capacitor i.e. 1/C is analogous to spring of constant K.
Note : As x is equivalent to current I, while solving example use x →∫idt.
1.3 Force Current Analogy (Node Analysis)
In this method, current is treated as analogous quantity to force in the mechanical system. So force shown in replaced by a current source in the system shown in the Fig. 3.
The equation according to Kirchhoff's current law for above system is,
Let node voltage be V,
Taking Laplace,
But to get this equation in the similar form as that of F(s) we will use,
Substituting in equation for I(s)
Comparing equations for F(s) and I(s) it is clear that,
i) Capacitor 'C' is analogous to mass M.
ii) Reciprocal of resistance 1/R is analogous to frictional constant B.
iii) Reciprocal of inductance I/L is analogous to spring constant K.
Note : As x is equivalent to voltage v, while solving problems use x →∫vdt.
Note : The element which are in series in F - V analogy, get connected in parallel in F – I analogous network and which are in parallel in F – V analogy, get connected in series in F – I analogous network.