Modelling, Simulation and Engineering Analysis

Introduction

Engineering projects require more than a physical prototype or a collection of experimental results. A successful dissertation must explain why a system behaves as it does, predict its response under conditions not yet tested, compare design alternatives, and justify decisions using evidence. Mathematical modelling, simulation, numerical methods, and sensitivity analysis provide the central tools for doing this.

A model is a deliberate representation of a real system. It may be a set of circuit equations, a differential equation, a block diagram, a state-space representation, or a computational algorithm. Simulation is the process of using a model to calculate or observe system behaviour. Engineering analysis interprets those results in relation to performance requirements, constraints, uncertainty, and practical implementation.

The usefulness of a model depends not on its complexity alone, but on whether it is sufficiently accurate for its intended purpose. A detailed electromagnetic model may be appropriate for analysing motor torque ripple, whereas a simplified equivalent circuit may be better for studying system-level energy consumption. The modeller must therefore balance accuracy, computational cost, transparency, and availability of data.

Learning Objectives

After studying this chapter, a student should be able to:

1. Distinguish physical systems, mathematical models, simulations, and validation evidence.
2. Formulate circuit and dynamic-system models using accepted electrical-engineering principles.
3. Select appropriate numerical methods and simulation tools.
4. Assess stability, accuracy, convergence, and sensitivity.
5. Compare alternatives using technically defensible criteria.
6. Recognise model limitations, uncertainty, reproducibility, and ethical responsibilities.
7. Present simulation results as evidence supporting dissertation conclusions.

1. The Modelling Process

A disciplined modelling process normally consists of the following stages:

1. Define the purpose. State the question the model must answer, such as whether a converter satisfies a transient-voltage specification.
2. Define the system boundary. Identify inputs, outputs, internal states, disturbances, and interfaces.
3. State assumptions. Examples include linear components, lumped parameters, constant temperature, negligible parasitic capacitance, or balanced three-phase operation.
4. Select governing laws. Kirchhoff’s laws, constitutive relations, Newton’s laws, electromagnetic laws, conservation laws, and empirical relationships may be used.
5. Formulate equations. Equations should be dimensionally consistent and use clearly defined parameters.
6. Implement and solve. A suitable analytical or numerical method is selected.
7. Verify the implementation. Check units, limiting cases, conservation laws, and numerical convergence.
8. Validate the model. Compare predictions with independent measurements or trusted reference results.
9. Analyse uncertainty and sensitivity. Determine which parameters and assumptions influence conclusions.
10. Document the model. Record equations, parameter sources, software versions, solver settings, and data-processing methods.

Verification asks, “Was the model implemented correctly?” Validation asks, “Does the model represent the real system adequately for the intended use?” These are different activities and neither replaces the other.

Important distinctions

2. Circuit and Dynamic-System Models

2.1 Circuit modelling

For a lumped electrical network, Kirchhoff’s current law states that the algebraic sum of currents at a node is zero:

Kirchhoff’s voltage law states that the algebraic sum of voltages around a closed loop is zero:

For a resistor, inductor, and capacitor, the constitutive equations are:

Here, voltage is measured in volts, current in amperes, resistance in ohms, inductance in henries, and capacitance in farads. These ideal relationships assume lumped parameters and operation within the component’s valid frequency, temperature, and amplitude range.

For a series RLC circuit driven by an input voltage vₛ(t), Kirchhoff’s voltage law gives:

Since i(t) = C dv_C(t)/dt, an equivalent second-order equation is:

This equation is valid for ideal linear components with constant R, L, and C. Nonlinear magnetic saturation, capacitor voltage dependence, switching effects, and parasitic elements require an extended model.

2.2 Transfer functions

For a linear time-invariant system with zero initial conditions, the transfer function is defined as:

where U(s) and Y(s) are the Laplace transforms of input and output, and s is the complex frequency variable in radians per second. For the series RLC circuit with capacitor voltage as output:

A transfer function is useful for frequency response, poles, zeros, stability, and control design. It is less suitable when initial conditions, strong nonlinearities, switching, or time-varying parameters are central to the problem.

2.3 State-space representation

A general continuous-time state-space model is:

For a linear time-invariant system:

The vector x contains state variables, u contains inputs, y contains outputs, and A, B, C, and D are system matrices of compatible dimensions. The state-space form is especially valuable for multi-input, multi-output systems, control systems, power electronics, and systems with nonzero initial conditions.

2.4 Power and energy checks

A model should respect physical conservation laws. Instantaneous electrical power is:

Energy delivered over an interval [t₁, t₂] is:

For passive components, dissipated or stored energy should not become negative in a way inconsistent with the component model. Such checks often reveal incorrect signs, duplicated elements, or implementation errors.

3. Simulation Approaches and Tools

3.1 Analytical, numerical, and hybrid simulation

Analytical solutions provide exact expressions under stated assumptions. They are valuable for interpretation and benchmarking but may be unavailable for nonlinear or high-order systems. Numerical simulation calculates approximate solutions at discrete points. Hybrid approaches use analytical submodels together with numerical solvers.

Common simulation categories include:

• Time-domain simulation: predicts waveforms such as current, voltage, speed, or temperature.
• Frequency-domain analysis: evaluates sinusoidal steady-state response, impedance, gain, and phase.
• Steady-state simulation: determines an operating point or periodic solution.
• Small-signal analysis: linearises a nonlinear system near an operating point.
• Monte Carlo simulation: repeatedly samples uncertain parameters to estimate output distributions.
• Electromagnetic, thermal, or mechanical multiphysics simulation: couples domains where interaction is essential.

Tools may include SPICE-based circuit simulators, MATLAB/Simulink, Python scientific libraries, PSCAD, PLECS, finite-element software, or specialist power-system packages. Tool selection should follow the research question rather than software availability. A sophisticated tool cannot compensate for an inappropriate model or poor parameter data.

3.2 Numerical integration

Many electrical-engineering models can be written as an initial-value problem:

The explicit Euler method approximates the solution by:

where h is the time step. Euler’s method is simple but may be inaccurate or unstable for stiff systems. The classical fourth-order Runge–Kutta method uses four slope estimates and generally provides much better accuracy for smooth problems at comparable step sizes, although it requires more function evaluations.

Implicit methods, such as backward Euler, calculate xₙ₊₁ through an equation involving the future state:

They require an algebraic or nonlinear solve at each step but are often more stable for stiff circuits and control systems.

The time step must resolve the fastest important dynamics. If a switching waveform has a period Tₛ, a simulation may require many time steps per switching period, depending on the desired accuracy and the solver. A convergence study—repeating the simulation with successively smaller step sizes—should demonstrate that reported conclusions do not depend materially on an arbitrary numerical setting.

3.3 Algebraic and nonlinear systems

Operating points often satisfy:

Newton–Raphson iteration updates an estimate according to:

where J_F is the Jacobian matrix of partial derivatives. The method may converge rapidly near a solution but can fail when the initial guess is poor, the Jacobian is singular, or the equations are badly scaled. Practical implementations often use damping, limits, scaling, or trust-region methods.

4. Stability, Accuracy, and Performance Metrics

A simulation result is not automatically reliable because it produces a smooth graph. Reliability requires numerical and physical assessment.

Stability

For a continuous linear system, internal asymptotic stability requires all eigenvalues of A to have negative real parts. For a discrete-time system xₙ₊₁ = A_d xₙ, asymptotic stability requires every eigenvalue of A_d to lie inside the unit circle. These conditions apply to linear time-invariant models; nonlinear systems require appropriate local or global analysis.

Accuracy and error

If x(t) is an exact solution and xₕ(t) is a numerical approximation, the absolute error is:

When the exact solution is unavailable, comparison with a refined solution, analytical limiting case, or trusted benchmark is appropriate. Relative error is:

provided x_ref is nonzero and an appropriate reference has been defined.

Engineering performance measures

Typical measures include peak value, settling time, rise time, overshoot, root-mean-square value, efficiency, total harmonic distortion, energy consumption, and steady-state error. For a periodic signal x(t) over period T, the RMS value is:

For sampled values x₁, …, x_N with equal spacing, the corresponding estimate is:

The integral form assumes a periodic waveform and complete period; the sampled form assumes representative, consistently sampled data.

5. Sensitivity, Uncertainty, and Design Comparison

Sensitivity analysis identifies parameters that strongly influence an output. For output y = f(p₁, p₂, …), a local dimensionless sensitivity of y to pᵢ may be defined as:

This expression is meaningful when y and pᵢ are nonzero and differentiability is reasonable. A large magnitude indicates that a small relative change in pᵢ may cause a comparatively large relative change in y.

For a simple RC time constant τ = RC, the relative differential is:

Thus, if resistance and capacitance errors are independent and small, a conservative worst-case estimate is the sum of their relative magnitudes. For statistically independent random uncertainties, a first-order root-sum-square estimate is often used:

The statistical expression assumes small uncertainties, approximate linearity, and independence. Correlation requires covariance terms.

Monte Carlo analysis samples uncertain inputs and produces an output distribution. It can estimate probability of violating a requirement, but its conclusions depend on the chosen distributions, sample size, and model validity. Sensitivity analysis does not prove causation in the physical system; it identifies dependence within the model.

Design alternatives should be compared using explicit criteria. A weighted score can be written as:

where qᵢ are normalised criteria and wᵢ are weights satisfying ∑wᵢ = 1. Weighting is appropriate only when criteria have been defined consistently and the decision-maker accepts the trade-offs. It should supplement, not conceal, raw technical data.

6. Worked Example: RC Low-Pass Response

Consider a first-order RC low-pass filter with R = 1.0 kΩ and C = 100 nF. The time constant is:

The cutoff angular frequency is:

and the cutoff frequency is:

The transfer function is:

For a sinusoidal input, the magnitude at frequency f is:

where ω = 2πf. At f = 1,000 Hz:

Therefore:

The output amplitude is approximately 84.6% of the input amplitude, corresponding to about −1.45 dB. The result demonstrates how a simple model can support a design decision. However, at sufficiently high frequency, resistor parasitic inductance, capacitor equivalent series resistance, layout capacitance, and measurement loading may invalidate the ideal model. A dissertation should state whether these effects are negligible for the frequency range being claimed.

7. Applied Case: Illustrative DC Motor Drive Model

Background

The following is an illustrative case, not a report of a particular commercial product. A student team must compare two control strategies for a permanent-magnet DC motor drive: proportional speed control and proportional–integral speed control.

Problem

The drive must reach a reference speed of 1,500 rpm with limited overshoot while rejecting a step increase in load torque. The team has motor resistance, inductance, torque constant, back-emf constant, inertia, and viscous-friction data from a manufacturer datasheet and bench tests.

Model formulation

A simplified armature model is:

The mechanical model is:

Here vₐ is armature voltage, iₐ armature current, ω angular speed, T_L load torque, Rₐ armature resistance, Lₐ armature inductance, Kₑ back-emf constant, Kₜ torque constant, J inertia, and B viscous-friction coefficient. In SI units, Kₑ and Kₜ have numerically compatible values under the usual motor conventions, but their units and definitions must still be documented.

The equations are implemented in a state-space or block-diagram model. Voltage saturation and current limiting are included because an unconstrained controller could demand physically impossible actuation. The model is simulated with a solver whose time step resolves the electrical time constant Lₐ/Rₐ and the mechanical response.

Analysis

The team performs four checks:

1. Unit and limiting-case checks: With zero load and zero friction, the model exhibits physically consistent acceleration; with zero voltage, current decays according to the electrical dynamics.
2. Steady-state check: At constant speed and current, derivatives become zero, allowing the algebraic voltage and torque relationships to be checked.
3. Validation: Simulated speed and current are compared with measured step-response data not used for parameter fitting.
4. Sensitivity: J, B, and Kₜ are varied over credible ranges to determine whether controller rankings change.

Outcome

The proportional controller reaches the target quickly but retains a steady-state speed error under load. The proportional–integral controller removes the steady-state error but produces excessive overshoot when integral action is too aggressive. After applying anti-windup and selecting gains through simulation followed by hardware testing, the PI controller meets the speed-error requirement while remaining within current and voltage limits.

Learning points

The case illustrates that simulation should support, rather than replace, engineering judgement. Inclusion of saturation changes the response substantially; parameter uncertainty may change the preferred gain; and validation data must be independent enough to test predictive capability. The final design decision is justified not by one attractive waveform but by requirements, constraints, sensitivity results, and experimental evidence.

8. Best Practices, Limitations, and Governance

A strong dissertation should:

• define every state, parameter, input, and output;
• state assumptions beside the equations they affect;
• use SI units consistently and show conversions;
• distinguish measured, estimated, assumed, and fitted parameters;
• preserve scripts, model files, raw data, and solver settings;
• report mesh size, time step, tolerances, and convergence evidence where relevant;
• separate calibration data from validation data;
• show uncertainty or credible parameter ranges;
• avoid selecting only simulations that support a preferred conclusion;
• compare simulation predictions with physical measurements whenever feasible.

Models have structural limitations. A model may omit temperature dependence, sensor noise, switching dead time, ageing, electromagnetic interference, manufacturing tolerances, or human interaction. Agreement with one experiment does not establish universal validity. Extrapolation beyond the validated operating range must be identified explicitly.

Ethical and governance issues are also important. Safety-critical claims require conservative verification and appropriate standards. Proprietary models and datasets must be used lawfully. If AI-assisted tools generate code, parameter estimates, or surrogate models, the researcher remains responsible for checking equations, units, numerical behaviour, data provenance, and reproducibility. AI-generated output should never be treated as independent validation.

Contemporary practice increasingly combines physics-based models with data-driven methods. Reduced-order models, digital twins, hardware-in-the-loop simulation, uncertainty quantification, and machine-learning surrogates can reduce computational cost. Nevertheless, a data-driven model may fail outside its training distribution, while a physics-based model may be biased by incorrect assumptions. Hybrid modelling is valuable when the physical structure is known but some parameters or sub-processes are difficult to describe analytically.

Chapter Summary

Modelling converts a physical engineering problem into a purposeful mathematical representation. Circuit laws, constitutive relationships, transfer functions, and state-space equations provide standard foundations for electrical-engineering models. Simulation uses analytical or numerical methods to predict time-domain, frequency-domain, steady-state, or uncertain behaviour.

Reliable engineering analysis requires more than graph generation. It requires correct implementation, dimensional consistency, suitable solver settings, convergence checks, stability assessment, validation against independent evidence, and explicit recognition of assumptions. Sensitivity and uncertainty analysis reveal whether a design conclusion is robust. Design alternatives should be compared using transparent technical criteria and realistic constraints.

In a dissertation, simulation is strongest when integrated with measurements, documented reproducibly, and interpreted critically. A model is an instrument for reasoning—not a substitute for physical understanding or engineering responsibility.

Key Terms

Model: A representation of a real system for a defined purpose.

Simulation: Computational execution of a mathematical or computational model.

Verification: Confirmation that a model and its implementation are mathematically and computationally correct.

Validation: Assessment of whether a model represents real behaviour adequately for its intended use.

Calibration: Estimation of model parameters using observed data.

State variable: A variable that, with inputs, describes the future behaviour of a dynamic system.

Transfer function: Ratio of output to input transforms under zero initial conditions.

State-space model: First-order vector representation of a dynamic system.

Lumped-parameter model: Model assuming distributed effects can be represented by discrete elements.

Numerical integration: Approximate calculation of a differential-equation solution.

Time step: Interval between successive numerical solution points.

Stability: Tendency of a system or numerical method to remain bounded or converge appropriately.

Convergence: Approach of numerical results toward a limiting solution as discretisation is refined.

Sensitivity analysis: Study of output changes caused by parameter changes.

Uncertainty: Incomplete knowledge of model inputs, parameters, or structure.

Monte Carlo simulation: Repeated simulation using sampled uncertain inputs.

Operating point: Equilibrium condition about which system behaviour may be analysed.

Surrogate model: A computationally cheaper approximation of a more expensive model.

Digital twin: A model connected to data to represent and monitor a physical asset or process.

Review Questions

Short Answer Questions

1. Distinguish model verification from model validation.
2. State Kirchhoff’s current law and explain its use in circuit modelling.
3. What information is represented by the state vector in a state-space model?
4. Why is a convergence study important in numerical simulation?
5. Give two limitations of an ideal component model.

Long Answer Questions

1. Explain a complete modelling workflow for an electrical-engineering dissertation, from problem definition to validation.
2. Derive the differential equation and transfer function of a series RLC circuit with capacitor voltage as output, stating assumptions.
3. Compare explicit Euler, Runge–Kutta, and implicit numerical methods in terms of accuracy, stability, and computational cost.
4. Explain how sensitivity and uncertainty analysis can affect a design decision.
5. Discuss reproducibility, ethical responsibility, and governance in simulation-based engineering research.

Analytical / Application Questions

1. An RC filter uses R = 2.2 kΩ and C = 47 nF. Calculate its time constant and cutoff frequency, and state the assumptions behind the calculation.
2. A simulated converter waveform changes significantly when the time step is halved. Diagnose possible causes and propose a verification plan.
3. Two motor-control designs have different speed error, overshoot, energy use, and implementation cost. Construct a transparent comparison method and explain how you would test whether the preferred design is robust to parameter uncertainty.

Multiple Choice Questions with Answers

Question 1

Which statement best distinguishes verification from validation?

A. Verification checks whether the implementation is correct; validation checks whether the model represents the real system adequately.

B. Verification uses measurements; validation uses only equations.

C. Verification estimates parameters; validation selects solver tolerances.

D. Verification applies only to hardware; validation applies only to software.

Correct Answer: A. Verification checks whether the implementation is correct; validation checks whether the model represents the real system adequately.

Question 2

For an ideal resistor, which relationship is correct?

A. v(t) = L di(t)/dt

B. i(t) = C dv(t)/dt

C. v(t) = R i(t)

D. p(t) = ∫v(t)i(t)dt

Correct Answer: C. v(t) = R i(t)

Question 3

For a continuous-time linear system ẋ = Ax, asymptotic stability requires:

A. All eigenvalues of A have positive real parts.

B. All eigenvalues of A have negative real parts.

C. All eigenvalues have magnitude exactly one.

D. The trace of A must be zero.

Correct Answer: B. All eigenvalues of A have negative real parts.

Question 4

A circuit has R = 1 kΩ and C = 100 nF. What is its time constant?

A. 10 μs

B. 100 μs

C. 1 ms

D. 10 ms

Correct Answer: B. 100 μs

Question 5

Which numerical method generally requires solving an equation involving the future state at each step?

A. Forward Euler

B. Backward Euler

C. Direct plotting

D. Algebraic substitution only

Correct Answer: B. Backward Euler

Question 6

The main purpose of sensitivity analysis is to:

A. Guarantee that a model is valid under all conditions.

B. Identify which parameters strongly influence an output.

C. Remove all measurement noise from data.

D. Replace physical testing in every application.

Correct Answer: B. Identify which parameters strongly influence an output.

Question 7

Which expression gives instantaneous electrical power?

A. p(t) = v(t)i(t)

B. p(t) = v(t)/i(t)

C. p(t) = L di(t)/dt

D. p(t) = RC

Correct Answer: A. p(t) = v(t)i(t)

Question 8

For the RC low-pass filter in the worked example, the approximate cutoff frequency is:

A. 15.9 Hz

B. 159.2 Hz

C. 1,591.5 Hz

D. 15,915 Hz

Correct Answer: C. 1,591.5 Hz

Question 9

Why should validation data preferably be independent of calibration data?

A. To make the model more complex.

B. To test predictive performance rather than merely fit quality.

C. To eliminate the need for uncertainty analysis.

D. To ensure the simulation has no numerical error.

Correct Answer: B. To test predictive performance rather than merely fit quality.

Question 10

Which practice most improves reproducibility of a simulation study?

A. Reporting only the final graph.

B. Changing solver settings until the desired result appears.

C. Archiving equations, code, data, parameters, and solver settings.

D. Using the most expensive software available.

Correct Answer: C. Archiving equations, code, data, parameters, and solver settings.