Video summary

تئوری جامع ماشین‌های الکتریکی، جلسه 1

Main summary

Key takeaways

Educational

Main ideas, concepts, and lessons (Session 1)

1) Course plan: what will be covered

The instructor outlines the structure of the semester’s modeling theory for electric machines:

  • Brief review of coupled magnetic circuits
  • Modeling coupled magnetic circuits
  • Principles of electromechanical conversion
  • After modeling electromechanical components:
    • Calculation of inductance of large windings
    • Reference frame theory and ABC ↔ QD0 transformations
  • Analysis and modeling of:
    • DC machines
    • Induction machines
    • If time remains: synchronous machines

Scheduling note: Because time is tight, DC machine modeling may be moved later. If needed, the focus shifts toward induction machine modeling.

2) Reference sources (books mentioned)

  • “Electric Machinery and Systems” by Cross and colleagues
    • Especially:
      • Chapter 3: conversion ABC to 0DQ and vice versa
      • Chapter 2: principles of energy conversion and analysis of induction motors/DC machines
  • “Electric Machines” (the instructor’s own book)
    • Detailed treatment of:
      • magnetic circuits
      • electromechanical converters
      • numerical exercises (with solutions)

3) Magnetic circuits foundations

Ampere’s law and closed magnetic paths

A magnetic circuit is described as a closed path for magnetic flux energized by current.

  • The governing relation is Ampere’s law over the closed path.
  • If along the path H = 1 (uniform segments in the simplest case) and leakage is ignored:
    • the integral form simplifies conceptually into expressions proportional to the enclosed current:
      • effectively leading to forms like Σ(A·H·dl) → N·I (as described by the instructor).

Leakage flux (why it exists)

  • Flux tends to follow the easiest path (analogous “least resistance”).
  • Besides the main flux path, some flux travels through secondary paths.
  • This portion is defined as leakage flux.

Fringing / scattering flux in the air gap

In an air gap, the flux spreads outward due to available parallel paths. This outward scattering is called:

  • fringing flux
  • or scattering flux

The instructor emphasizes that leakage and fringing magnitudes are not easily computed manually—instead, they are typically obtained using numerical simulation.

4) Flux linkage: definition and role

Circuit voltage equation and Faraday’s law

The basic coil relationship revisits the idea that terminal voltage includes:

  • a resistive drop (armature/winding resistance term)
  • plus an induced emf term from time-varying flux linkage

A key point: emf is linked to the derivative of flux linkage with respect to time.

Flux linkage as sum over turns/loops

Treating a coil as multiple loops/turns:

  • each loop links some flux
  • the total flux linkage is the sum over all linked loop fluxes

This leads to:

  • λ (lambda) as the total linkage flux
  • emf proportional to dλ/dt

Special case: no leakage

If leakage is absent:

  • each turn links the same flux
  • flux linkage reduces to:
    • λ = N·φ
  • producing the familiar induced emf relation using N·φ

5) Equivalent leakage flux and equivalent inductance

Because real machines have leakage, the instructor introduces a concept to handle leakage in modeling.

Decomposing flux contributions

For each loop/turn, the flux has:

  • a part through the main path
  • plus parts representing leakage paths

Hypothetical leakage flux linkage (Φ_l or equivalent leakage)

The instructor defines a hypothetical flux quantity representing leakage linkage as if it were uniformly linking all turns.

The goal is to convert leakage effects into:

  • an effective leakage flux linkage

so inductance can separate into:

  • a magnetizing/main-path part
  • a leakage inductance part

Inductance split

The inductance is decomposed into:

  • Magnetizing inductance (often denoted (L_M))
  • Leakage inductance (often denoted (L_l))

6) Inductance vs. operating point: linear vs nonlinear magnetic circuits

Core linearity vs nonlinearity

  • If the magnetic circuit is linear (magnetic sense):
    • λ vs i behaves like a straight-line relationship
  • If the magnetic circuit is nonlinear (core saturation):
    • the λ–i curve becomes nonlinear, but inductance is still defined via the λ–i relationship

Magnetic “line” depends on current

Inductance can vary with current because:

  • the slope of the λ(i) curve changes with (i)
  • therefore, (L) is not constant unless the circuit is linear

Scaling from B–H to λ–i (no air gap case)

For a uniform core (and no air gap), the instructor describes mapping:

  • use the iron core B–H characteristic
  • convert it to an equivalent λ–i characteristic via geometric scaling:
    • λ depends on linked flux (from (B) and area)
    • i depends on (H) and magnetic path length
  • axis scaling yields λ–i

Air gap case: requires iterative extraction of points

With an air gap, the relationship changes. Conceptually:

  1. choose currents (i)
  2. for each (i), obtain corresponding (H) from the B–H curve
  3. compute λ using core and air-gap contributions
  4. repeat across multiple operating points to reconstruct λ(i)

7) Nonlinear inductance as a function of electrical and mechanical variables

Independent variables on the electrical side

On the electrical side, either:

  • λ or i can be chosen as independent variables,

since (under a no-hysteresis assumption) λ and i are one-to-one.

Mechanical side parameter

The mechanical angle/position (e.g., θ or x) determines air-gap geometry. The instructor also uses (i_k) as a mechanical-side indicator (position).

Nonlinear inductance depends on both sides

“Nonlinear inductance” is defined as:

  • a function of electrical state and mechanical position, e.g.:
    • (L = L(\lambda, i_k))
  • or equivalently using (i) instead of λ

Two inductance definitions: equivalent vs incremental/differential

  • Equivalent (static) inductance
    • (L = \lambda / i) at a given operating point
  • Differential (incremental) inductance
    • slope of the λ–i curve:
    • (L_{\text{diff}} = \frac{d\lambda}{di})

This distinction is used later for dynamic equations.

8) Modeling and simulation (what the instructor means)

Definition of modeling

Modeling = representing a real phenomenon (physical or mathematical) to analyze behavior safely and cheaply.

Examples:

  • Transformers
    • simulate an internal short circuit and predict current
  • Earthquake engineering
    • simulate earthquake loading to identify weak points
  • Dams/buildings
    • create scaled models and validate withstand capability before construction

Definition of simulation

Simulation = using a model to examine system behavior, often by numerically solving equations.

9) MATLAB environment workflow: preparing BH and λ–i curves

The instructor demonstrates a MATLAB-like workflow:

  • parameters are entered as matrices
  • operations like:
    • matrix multiplication/division require matching dimensions
    • inversion and plotting
  • for BH curve input:
    • need matching B and H parameter vectors
  • for λ–i:
    • λ is computed from:
      • geometry: turns (N), area (A), path length (l)
      • magnetic assumptions
    • then inductance can be obtained by numerical differentiation:
      • (d\lambda/di)

Practical issue: If BH input points are too sparse or disconnected, the derivative becomes discontinuous.

Fix: Add more points to smooth/complete the curve (increase continuity).

Also noted:

  • BH data may be entered in tables (e.g., B in one column and H in another), and λ(i) can be computed per air-gap case/position.

10) Simulink demonstration: discrete-time sampling and sine wave fidelity

Sampling/time-step effect

Simulink can distort signals if sampling is too coarse:

  • if the time step is large relative to the waveform frequency:
    • a sine wave is drawn with few points → it appears piecewise
  • fix:
    • reduce maximum step size / increase time resolution so each cycle has enough points

Frequency units issue

Simulink expects sinusoid frequency in radians per second (ω), not Hz directly. For 50 Hz:

  • ( \omega = 100\pi )

Logging/memory limitation

Scope/logging may show only the last portion of long simulations due to buffer limits. Lowering the sample interval increases sample count → fewer cycles shown.

11) System-level model example in Simulink (RL-type / integrator and phase shift)

The instructor shows an RL-type model block:

  • states relationships using state/derivative structure
  • demonstrates phase behavior:
    • output and integral output differ in phase (90°) when integrating derivative-like signals
  • uses an integrator to recover the original waveform from its derivative
  • adds DC bias to show expected waveform shift

The instructor stops before a deeper electromechanical example, indicating continuation in the next session.

Methodology / instructions extracted

A) Building magnetic-circuit relations (conceptual steps)

  • Define a closed magnetic path for flux in the coupled magnetic circuit
  • Apply Ampere’s law over the closed path
  • If segments are uniform and leakage/fringing are ignored:
    • simplify the integral into a sum proportional to enclosed ampere-turns (N·i)
  • When leakage/fringing exists:
    • treat leakage as flux in secondary paths
    • treat fringing as flux dispersion around/in the air gap
  • Use numerical simulation methods if leakage/fringing must be quantified

B) Deriving flux linkage and induced voltage

  • Represent the coil as multiple loops/turns
  • For each loop:
    • identify linked flux (main + leakage contributions)
  • Compute total flux linkage by summing linked fluxes over turns
  • Apply Faraday’s law:
    • induced emf ∝ d(λ)/dt

C) Constructing inductance from λ–i characteristics

  • Build/obtain the λ(i) curve:
    • no air gap: scale from B–H to λ–i using geometry
    • with air gap: compute λ at multiple operating points using BH + air-gap contribution
  • For each operating point:
    • Equivalent inductance: (L_{eq} = \lambda/i)
    • Differential inductance: (L_{diff} = d\lambda/di) from the slope

D) Nonlinear inductance modeling with mechanical position

  • Choose electrical independent variable as either:
    • ( \lambda ) or ( i )
  • Choose mechanical independent variable as:
    • position/angle (e.g., (x) or (i_k))
  • Express inductance as a two-variable function:
    • (L = L(\lambda, x)) or (L = L(i, x))

E) Simulink fidelity: ensure adequate sampling

  • When generating sine inputs:
    • ensure the time step is small enough for many samples per cycle
  • If the waveform looks polygonal:
    • reduce maximum step size
  • Use correct frequency units:
    • sine frequency in radians/sec (ω)
  • Watch logging limits:
    • buffers may truncate earlier data

F) MATLAB/curve preparation for smooth differentiation

  • Enter BH data as arrays with matching sizes
  • Compute λ values at each point
  • Ensure the curve is smooth/continuous before differentiating
  • If derivative becomes discontinuous:
    • add more BH points (densify, especially near saturation)

Speakers / sources featured

  • Speaker: The course instructor (unnamed in the subtitles)

  • Sources cited:

    • “Electric Machinery and Systems” by Cross and colleagues (chapter references: conversion ABC↔0DQ; energy conversion and motor/DC analysis)

    • “Electric Machines” by the instructor (author referenced as having written it)

Original video