A rotating load is connected to a field-controlled DC...

A rotating load is connected to a field-controlled DC electric motor through a gear system. The motor is assumed to be linear. A test results in the output load reaching a speed of $1 \mathrm{rad} / \mathrm{s}$ within $0.5 \mathrm{~s}$ when a constant $80 \mathrm{~V}$ is applied to the motor terminals. The output steady-state speed is $2.4 \mathrm{rad} / \mathrm{s}$. Determine the transfer function $\theta(s) / V_f(s)$ of the motor, in $\mathrm{rad} / \mathrm{V}$. The inductance of the field may be assumed to be negligible (see Figure 2.18). Also, note that the application of $80 \mathrm{~V}$ to the motor terminals is a step input of $80 \mathrm{~V}$ in magnitude.

Key Concepts

Laplace Transform and System Integration

The Laplace transform converts differential equations that describe time-domain system behavior into algebraic equations in the s-domain. This methodology simplifies the analysis of dynamic systems, and when the output variable is an integrated quantity (like angular displacement being the integral of speed), the integration is represented as division by s in the transfer function.

Assumption of Negligible Inductance

This simplification assumes that the inductance effects in the field circuit are small enough to be ignored, reducing the order of the system dynamics. By eliminating the inductance term, the analysis focuses on the dominant factors (resistance and back EMF) that govern the immediate motor response to an applied voltage.

Time Constant

The time constant is a key parameter in first-order system dynamics, indicating the rate at which the system approaches its steady state after a disturbance. In a first-order response, it is the duration taken for the output to reach about 63% of its final value following a step input, and it directly influences the transient response in the transfer function.

DC Motor Dynamics

This concept covers the fundamental equations that describe the behavior of a DC motor, linking the applied voltage to torque production and ultimately to mechanical motion. It involves both the electrical side (voltage, current, resistance, back EMF) and the mechanical side (inertia, friction, load), providing the basis for analyzing the motor's response to various inputs.

Transfer Function

A transfer function represents the relationship between the output and the input of a linear time-invariant system in the Laplace domain. It encapsulates the system's dynamic behavior by relating the transformed output (such as angular displacement) to the transformed input (such as applied voltage), allowing for an analysis of both steady-state and transient responses.

Step Response Analysis

This concept involves evaluating how a system reacts over time when subjected to a step input. By analyzing the resulting output curve, important parameters such as the steady-state gain and time constant can be extracted, which in turn are critical for constructing the system’s transfer function.

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