ANALOGOUS SYSTEM

An Analogous System represents one physical system using another physical system whose mathematical equations remain identical. Control engineering frequently uses analogy to study system behaviour without working directly on the original physical setup.

• Engineering systems exist in domains such as mechanical, electrical, hydraulic, and thermal.
• Mathematical modelling becomes simpler when dynamic behaviour is represented using an equivalent electrical model.
• Electrical variables like voltage and current are easy to measure, simulate, and control, which makes the electrical analogy useful in control engineering.

General Condition of Analogy :

Dynamic Equation of System A = Dynamic Equation of System B

Example Equations:

Mechanical System: F(t) = M \frac{d^2x}{dt^2} + B \frac{dx}{dt} + Kx

Electrical System : V(t) = L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{1}{C}q

Where:

• L = Inductance
• R = Resistance
• C = Capacitance
• q = Charge
• V(t) = Applied voltage

Similarity between the two equations enables the creation of an analogous model.

Need for Analogous Systems

• Simplifies complex mathematical analysis of physical systems.
• Allows electrical simulation of mechanical behaviour.
• Reduces experimental cost and development time.
• Helps in transfer function derivation and stability analysis.
• Provides a common modelling technique for multiple engineering domains.

Types of Analogous Systems

1. Force-Voltage Analogy.
2. Force-Current Analogy.

Translational Mechanical Analogous System

At Equilibrium:

Applied force = opposing force, f(t) = f_i(t) + f_D(t) + f_k(t)

f(t) = M \frac{d^2x(t)}{dt^2} + D \frac{dx(t)}{dt} + Kx(t)

f(t) = M \frac{d^2x(t)}{dt^2} + D v(t) + K \int v(t) \, dt

where,

f(t) = Applied Force.

f_i(t) = Inertia force.

f_D(t) = Damper Force.

f_K(t) = Spring Force.

Rotational Mechanical Analogous System

At Equilibrium

Applied Torque = opposing Torque, T = T_J + T_D + T_K

T = J \frac{d^2\theta(t)}{dt^2} + D \frac{d\theta(t)}{dt} + K\theta(t)

Where,

• Applied torque = T(t)
• Angular displacement = θ(t)
• Angular velocity = ω(t)

Electrical System

Series

Electrical circuits containing resistor, inductor, and capacitor form electrical equivalents of mechanical systems.

Applying Kirchhoff’s Voltage Law: V = V_1 + V_2 + V_3

V = L \frac{di}{dt} + Ri + \frac{1}{C} \int i \, dt

Using charge relation i = \frac{dq}{dt} :

V = L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{q}{C}

Parallel

I = I_1 + I_2 + I_3

I = C \frac{dV}{dt} + \frac{V}{R} + \frac{1}{L} \int V \, dt

Analogous Quantities In Force (Torque) Voltage Analogy

Analogous quantities in force (Torque) current analogy

Steps to Form an Analogous System

1. Write governing differential equation of mechanical system.
2. Choose analogy type (Force-Voltage or Force-Current).
3. Replace mechanical variables using analogy table
4. Construct equivalent electrical circuit.
5. Analyse system response using circuit analysis methods.

Advantages of Analogous System

• Write governing differential equation of mechanical system and select suitable analogy type.
• Replace mechanical variables with equivalent electrical variables using analogy relations.
• Construct equivalent electrical circuit and analyse system response using circuit analysis methods.

Disadvantages of Analogous System

• Physical meaning of variables changes when converting one energy domain into another.
• Incorrect parameter mapping leads to an inaccurate equivalent system model.
• Representation of nonlinear behaviour becomes complex and requires careful selection of analogy method.

Applications

• Analogous systems assist control system design by simplifying dynamic modelling and controller analysis.
• Vibration behaviour of mechanical systems can be analysed using equivalent electrical circuit models.
• Robotics and servo mechanisms use analogy concepts for precise motion and speed control.
• Automotive suspension systems are modelled through analogies to improve stability and ride performance.
• Aerospace and mechatronics systems apply analogous modelling for multi-domain system analysis.