Electrical and Electronic System Models

Introduction

Electrical subsystems are central to mechatronic products. Sensors convert physical variables into electrical signals; power converters regulate energy; motors transform electrical energy into mechanical motion; and electronic controllers determine how the complete system behaves. A useful model must therefore represent both the electrical laws governing instantaneous variables and the dynamic storage of energy in inductors, capacitors, magnetic fields, and mechanical inertia.

This chapter develops models for circuits, power-electronic converters, electromagnetic components, motors, and drives. The emphasis is on models suitable for simulation and control design. The engineer must decide whether a model should be switching-level, averaged, linearized, or purely static. That decision depends on the purpose of analysis, the time scale of interest, and the required accuracy.

Learning Objectives

By the end of this chapter, the reader should be able to:

1. Formulate circuit models using Kirchhoff’s voltage and current laws.
2. Derive state-space models for RLC networks.
3. Explain equivalent circuits and their limitations.
4. Model diodes, switches, PWM converters, and DC-link dynamics.
5. Construct electromagnetic and electromechanical energy-conversion models.
6. Derive the basic equations of DC and induction motors.
7. Select an appropriate model fidelity for simulation and control.
8. Validate an electrical model using physical laws, units, and experimental data.

1. Modeling Foundations

An electrical model consists of variables, constitutive relationships, interconnections, parameters, initial conditions, and assumptions. The principal variables are voltage v(t) in volts and current i(t) in amperes. Passive sign convention assigns positive current entering the terminal marked positive for voltage. Under this convention, instantaneous power absorbed by an element is

where p is in watts. Positive power denotes absorption; negative power denotes delivery.

The fundamental interconnection laws are:

Kirchhoff’s current law (KCL): the algebraic sum of currents at a node is zero.

Kirchhoff’s voltage law (KVL): the algebraic sum of voltages around a closed loop is zero.

These laws follow from charge conservation and energy conservation in the lumped-circuit approximation. They are most appropriate when circuit dimensions are much smaller than the electromagnetic wavelength associated with the highest relevant frequency.

The constitutive equations of ideal passive elements are:

The ideal relations assume constant parameters, negligible parasitic effects, and compatible reference directions. In real components, resistance, leakage, saturation, core loss, and parasitic capacitance may be important.

2. Dynamic Circuit Models

2.1 First-order circuits

For a series RL circuit driven by a voltage source v_s(t), KVL gives

Hence,

The time constant is

for R > 0 and L > 0. If the source is suddenly changed to a constant value V, the current is

This expression assumes ideal switching at t = 0, constant parameters, and no magnetic saturation. The inductor current cannot change instantaneously because that would require an unbounded voltage.

For a series RC circuit, KVL and the capacitor relation give

The time constant is

The capacitor voltage cannot change instantaneously in a finite-current circuit. These continuity properties are valuable checks when constructing simulation models.

2.2 State-space formulation

A state is a minimum variable set that, together with the input and parameters, determines future behavior. Energy-storage variables are usually natural states: inductor currents and capacitor voltages. A general continuous-time state model is

where x is the state vector, u the input vector, and y the output vector. For a linear time-invariant model,

The matrices A, B, C, and D describe dynamics, input coupling, output selection, and direct feedthrough, respectively.

Consider a source feeding a series resistor and inductor, with capacitor voltage as a second state in a series RLC loop. Let x₁ = i and x₂ = v_C. KVL gives

and

Therefore,

The state-space matrices for output y = v_C are

The model is valid for fixed R, L, and C and for a lumped, linear circuit. If the inductor saturates, L becomes current-dependent and the model becomes nonlinear.

2.3 Frequency-domain and equivalent models

For sinusoidal steady-state analysis, differentiation may be replaced by multiplication by jω, where ω is angular frequency in radians per second. The impedances of ideal elements are

These are not time-domain dynamic models; they apply only to linear circuits under sinusoidal steady-state conditions. They are useful for frequency response, filtering, and impedance matching.

The Thevenin equivalent of a linear two-terminal network is a voltage source V_th in series with resistance or impedance Z_th. The Norton equivalent is a current source I_N in parallel with the same impedance. For a resistive network,

The equivalent is valid at the selected terminals and for the assumed operating conditions. A nonlinear load or a network containing active switching devices may require a local linearization rather than a single global equivalent.

3. Semiconductor and Power-Electronic Models

Power electronics changes the topology of a circuit. Consequently, a converter is often a hybrid system: continuous states evolve according to differential equations, while discrete switch states change according to gate signals and device conditions.

3.1 Diodes and controlled switches

An ideal diode conducts when forward biased and blocks when reverse biased. A practical piecewise-linear model may use

when conducting, where V_F is the forward-voltage approximation and r_D is the incremental resistance. When blocking, i_D ≈ 0. Reverse recovery, leakage, junction capacitance, and thermal dependence may be added when switching transients matter.

An ideal transistor switch has two states: ON, approximately a short circuit, and OFF, approximately an open circuit. Real switches have conduction resistance, voltage limits, switching losses, dead time, and finite transition duration. A simulation model should not ignore these features when predicting efficiency, electromagnetic interference, or device stress.

3.2 Buck converter

For an ideal buck converter operating in continuous-conduction mode, the switch connects the source voltage V_in to the inductor for a fraction D of each switching period. Under steady-state volt-second balance,

so

Here D is duty ratio, V_o output voltage, and V_in input voltage. The relationship assumes ideal components, continuous inductor current, constant output voltage over one switching period, and sufficiently high switching frequency.

The two switching submodels are:

ON state:

OFF state:

Averaging over one switching period gives

For an output capacitor and resistive load R_load,

The averaged model is efficient for control design but cannot reproduce individual switching ripple, diode recovery, or high-frequency electromagnetic interference.

3.3 DC-link capacitor

A DC-link capacitor connected to a converter satisfies

where i_in is current entering the capacitor from the source-side converter and i_load is current delivered to the load-side converter. Its energy is

A useful design check follows from energy balance. If a current imbalance ∆i persists for a short interval ∆t, then approximately

provided the voltage change is small and C_dc is constant.

4. Electromagnetic Components and Energy Conversion

4.1 Magnetic circuits

For a magnetic path with mean length l_m, cross-sectional area A_c, and permeability μ, reluctance is

Magnetomotive force is 𝓕 = Ni, where N is turns and i is current. The approximate magnetic flux is

This magnetic-circuit analogy assumes a reasonably uniform flux path and neglects leakage and fringing. Ferromagnetic materials are nonlinear because permeability varies with flux density and may exhibit hysteresis.

For a coil with constant inductance, flux linkage is λ = Li, and induced voltage is

In general, if inductance depends on position or current, the more fundamental relation is v = dλ(i,x)/dt.

4.2 Transformers

An ideal transformer with turns ratio n = N₁/N₂ satisfies

and, neglecting losses,

The apparent power is approximately conserved. Real transformer models include winding resistance, leakage inductance, magnetizing inductance, and core-loss resistance. These additions are necessary for voltage regulation, inrush-current, and transient studies.

4.3 Electromechanical coupling

In an electromechanical device, electrical and mechanical domains are coupled through force or torque. A general energy-based relation uses co-energy W′. Electromagnetic torque is

where θ is mechanical position and the derivative is taken at constant current. This relation is particularly useful for reluctance and permanent-magnet machines, although the exact co-energy function depends on the machine model.

Mechanical dynamics are represented by

where J is rotor inertia in kg·m², ω_m angular speed in rad/s, T_e electromagnetic torque, T_L load torque, and B viscous-friction coefficient in N·m·s/rad. Position satisfies dθ_m/dt = ω_m. Coulomb friction, gear backlash, and elastic couplings may be added when relevant.

5. Motor and Drive Models

5.1 Separately excited DC motor

The armature circuit is modeled by

The back electromotive force is

and electromagnetic torque is

In SI units, K_e and K_t have numerically equal values when consistent mechanical and electrical units are used. Combining the equations gives the state model with states i_a and ω_m:

The model assumes constant field flux. It is widely used for servo drives because it clearly separates electrical current dynamics from mechanical speed dynamics.

Worked Example: DC Motor Acceleration

An illustrative motor has R_a = 2 Ω, L_a = 0.5 H, K_e = K_t = 0.1, J = 0.02 kg·m², and B = 0.001 N·m·s/rad. A step voltage of 12 V is applied, with zero load torque and initial rest.

At the initial instant, ω_m = 0 and i_a = 0, so

The initial torque rate is therefore

The initial angular acceleration is zero because the current initially is zero. After the current rises, torque accelerates the rotor. This illustrates why a motor model must include both electrical and mechanical states; a static voltage-to-speed gain cannot predict starting current or acceleration.

5.2 Permanent-magnet synchronous motor

For a surface-mounted permanent-magnet synchronous motor in the rotating d–q reference frame, a common model is

Electromagnetic torque is

where R_s is stator resistance, L_d and L_q are d–q inductances, ψ_f is permanent-magnet flux linkage, p is the number of pole pairs, and ω_e = pω_m is electrical angular speed. The first torque term is magnet torque; the second is reluctance torque. This model assumes balanced three-phase quantities and a defined Park transformation. Parameter conventions and sign conventions must be stated because different software packages use different transformations.

5.3 Induction motor equivalent circuit

The per-phase steady-state equivalent circuit contains stator resistance and leakage reactance, a magnetizing branch, and rotor quantities referred to the stator. Rotor slip is

where ω_s is synchronous mechanical angular speed expressed consistently with ω_m. The rotor resistance term referred to the stator is commonly represented as R′_r/s. As slip approaches zero, rotor current and torque decrease; at standstill, s = 1.

A steady-state equivalent circuit is suitable for torque–speed prediction and efficiency estimation. It does not directly reproduce electromagnetic transients unless additional differential equations are used.

6. Switching, Averaging, and Linearization

Three model levels are particularly important:

For a nonlinear model ẋ = f(x,u), linearization about equilibrium (x₀,u₀) uses perturbations ∆x and ∆u:

where

The approximation is accurate for sufficiently small perturbations and away from discontinuities such as switch transitions, saturation boundaries, and diode conduction changes. A converter may require separate linear models for different operating modes.

7. Simulation Practice and Validation

A reliable simulation workflow begins with a physical specification: voltage and current ranges, switching frequency, mechanical load, thermal limits, and required outputs. The engineer then selects states, writes equations, defines initial conditions, and chooses parameter values from datasheets or experiments.

Important validation checks include:

• Dimensional consistency: every term in an equation must have compatible units.
• Continuity: inductor current and capacitor voltage should not jump without a physically justified impulse.
• Energy balance: input energy should equal stored energy, delivered energy, and losses within numerical tolerance.
• Limiting cases: setting resistance to zero, load torque to zero, or duty ratio to zero should produce physically sensible behavior.
• Steady-state checks: derivatives should approach zero in a constant operating condition.
• Parameter sensitivity: uncertain resistance, inductance, friction, and delay should be varied to determine robustness.

Numerical solvers must be selected with regard to stiffness and discontinuity. Fast switching and slow mechanical motion can create widely separated time scales. Variable-step solvers may be efficient for continuous models, whereas fixed-step solvers are often preferable for real-time control implementation. Solver tolerances should not be treated as substitutes for correct equations.

8. Applied Case: Servo-Driven Conveyor Axis

Background

Consider an illustrative conveyor axis driven by a permanent-magnet motor through a reduction gearbox. A current-controlled inverter supplies the motor, while an encoder measures position. The axis must move packages accurately and reject disturbances caused by changing loads.

Problem

A purely static motor model predicts speed from commanded voltage but fails to predict current peaks, acceleration time, DC-link ripple, and position error during load changes. Excessive current may also trigger inverter protection.

Analysis

The model is divided into four interacting parts:

1. A DC-link capacitor model describes bus-voltage variation.
2. An averaged inverter model converts duty commands into d–q voltages.
3. The motor d–q equations predict currents and electromagnetic torque.
4. The mechanical equation predicts speed and position through the gearbox.

The load torque is reflected to the motor shaft. For an ideal gearbox with speed ratio N = ω_m/ω_L, reflected load inertia is approximately

and reflected torque is

where motor speed is ω_m, load speed is ω_L, load inertia is J_L, and load torque is T_L. These relations assume negligible gearbox losses and a rigid transmission.

Approach and Solution

The design uses a detailed switching model initially to check semiconductor current and voltage stress. An averaged motor-drive model is then used to tune the current and position controllers. Finally, a linearized model around the rated operating point is used to examine bandwidth and disturbance rejection. Friction and belt compliance are included after experimental tests reveal position oscillation near a structural resonance.

Outcome

The multi-fidelity approach reduces simulation time while preserving the details needed for each engineering decision. The switching model identifies ripple and peak current; the averaged model supports controller tuning; and the mechanical extension explains oscillation that was absent from the rigid-body model.

Learning Points

A model should be no more detailed than necessary, but no less detailed than the phenomenon being investigated. Electrical, power-electronic, and mechanical states must be connected through explicit energy and torque relationships. Validation against measured current, speed, and bus voltage is essential before using the model for safety-critical decisions.

9. Limitations, Safety, and Contemporary Practice

Electrical models can conceal hazards if ideal switches, zero delays, or unlimited current are assumed. Design verification must consider overvoltage, short circuits, isolation, grounding, thermal rise, electromagnetic compatibility, and safe discharge of capacitors. Standards and institutional procedures should govern insulation coordination, protective devices, machine safety, and functional safety.

Model governance is also important. Engineers should record parameter sources, sign conventions, transformations, assumptions, solver settings, and validation evidence. Reproducible models are easier to audit and less likely to cause unsafe design decisions.

Modern digital twins increasingly combine first-principles models with measured data. Machine-learning estimators can identify friction, temperature-dependent resistance, battery state, or unmodeled load torque. However, data-driven components should be bounded and validated against physical constraints. AI may assist parameter estimation and fault detection, but it does not remove the need for conservation laws, safety analysis, and human engineering approval.

Chapter Summary

Electrical and electronic models are built from Kirchhoff’s laws, element constitutive equations, energy-storage relationships, and electromechanical coupling. Resistors dissipate energy, whereas inductors and capacitors store it and create dynamic states. State-space models provide a systematic representation for simulation and control.

Power converters are hybrid systems whose topology changes with switching state. Detailed switching models capture ripple and device stress; averaged models describe slower energy flow; and linearized models support local stability and controller design. Motors require both electrical and mechanical equations, with back electromotive force and torque providing the principal coupling. Equivalent circuits are useful but valid only under their stated assumptions.

Reliable engineering practice requires dimensional checks, energy-balance checks, sensible limiting cases, parameter validation, and awareness of numerical and safety limitations. The appropriate model is determined by the engineering question, not by maximum mathematical complexity.

Key Terms

Kirchhoff’s current law: Conservation of current at an electrical node.

Kirchhoff’s voltage law: Conservation of voltage around a closed loop.

Passive sign convention: Reference convention in which positive current enters the positive-voltage terminal.

State variable: A variable that, with inputs and parameters, determines future system behavior.

Time constant: Characteristic exponential-response time of a first-order system.

Impedance: Frequency-domain ratio of sinusoidal voltage to current.

Thevenin equivalent: A two-terminal voltage-source representation of a linear network.

Norton equivalent: A two-terminal current-source representation of a linear network.

Duty ratio: Fraction of a switching period for which a switch is commanded on.

Continuous conduction: Converter operation in which inductor current does not reach zero.

Averaged model: Model obtained by averaging switching-state equations over a switching period.

DC link: Intermediate electrical energy-storage stage, commonly containing a capacitor.

Magnetic reluctance: Opposition offered by a magnetic path to flux production.

Flux linkage: Total magnetic flux linked by a winding, usually represented by λ.

Back electromotive force: Voltage induced by motion in an electromechanical device.

Slip: Relative speed difference between synchronous field and rotor in an induction motor.

Linearization: Local approximation of a nonlinear model about an operating point.

Model fidelity: Degree to which a model represents physical detail.

Energy balance: Accounting of supplied, stored, converted, and dissipated energy.

Review Questions

Short Answer Questions

1. State Kirchhoff’s current law and explain its physical basis.
2. Why are inductor current and capacitor voltage normally continuous at a switching instant?
3. Define duty ratio and state the ideal continuous-conduction buck-converter voltage relation.
4. What is the difference between a detailed switching model and an averaged model?
5. Identify the electrical and mechanical states in the basic DC-motor model.

Long Answer Questions

1. Derive the state-space equations of a series RLC circuit and state the assumptions used.
2. Explain Thevenin and Norton equivalents, including their applicability and limitations.
3. Derive the averaged inductor-current and capacitor-voltage equations of an ideal buck converter.
4. Explain how back electromotive force and electromagnetic torque couple the electrical and mechanical parts of a DC motor.
5. Compare switching, averaged nonlinear, and small-signal linearized models for drive-system design.

Analytical / Application Questions

1. A buck converter has V_in = 48 V and duty ratio D = 0.25. Under ideal continuous-conduction assumptions, calculate the average output voltage and explain why the result may differ in a practical converter.
2. A DC motor has R_a = 1 Ω, L_a = 0.2 H, K_e = 0.08, and is initially at rest with 10 V applied. Calculate the initial armature-current derivative and explain the subsequent role of back emf.
3. Design a validation plan for a motor-drive model intended to predict current ripple, acceleration time, and steady-state speed. Identify measurements, parameter uncertainties, and acceptance criteria.

Multiple Choice Questions with Answers

Question 1

Which variable is naturally selected as a state for an ideal inductor?

A. Inductor voltage

B. Inductor current

C. Resistor power

D. Switch duty ratio

Correct Answer: B. Inductor current

Question 2

For an ideal buck converter in continuous-conduction steady state, the average output voltage is

A. V_in/D

B. (1−D)V_in

C. D V_in

D. V_in + D

Correct Answer: C. D V_in

Question 3

What does Kirchhoff’s current law express?

A. Conservation of charge at a node

B. Conservation of angular momentum

C. Conservation of magnetic reluctance

D. Conservation of resistance

Correct Answer: A. Conservation of charge at a node

Question 4

The time constant of a series RL circuit is

A. R/L

B. L/R

C. RL

D. 1/(RL)

Correct Answer: B. L/R

Question 5

In a DC motor, back emf is primarily proportional to

A. Rotor speed

B. Load inertia only

C. Armature resistance only

D. Duty ratio only

Correct Answer: A. Rotor speed

Question 6

Which model is most appropriate for studying individual semiconductor switching transients?

A. Static gain model

B. Detailed switching model

C. Steady-state Thevenin model

D. Rigid mechanical model

Correct Answer: B. Detailed switching model

Question 7

The energy stored in an ideal capacitor is

A. C v

B. 1/2 C v²

C. 1/2 L i²

D. R i²

Correct Answer: B. 1/2 C v²

Question 8

For an ideal transformer, if the turns ratio N₁/N₂ increases, the voltage ratio V₁/V₂

A. Decreases to zero

B. Equals N₁/N₂

C. Equals N₂/N₁ only for current

D. Becomes independent of turns

Correct Answer: B. Equals N₁/N₂

Question 9

Linearization of a nonlinear model is generally valid

A. Only near a selected operating point

B. At every switching transition

C. Only when all parameters are zero

D. Without any assumptions

Correct Answer: A. Only near a selected operating point

Question 10

For a gearbox with speed ratio N = ω_m/ω_L and negligible losses, reflected load inertia is

A. J_L N²

B. J_L/N

C. J_L/N²

D. N/J_L

Correct Answer: C. J_L/N²