Electromechanical Energy Conversion and Coenergy Principles

2.1 Energy balance and signs

Physical model and mathematical relationship

Electrical input, mechanical output, field storage and losses obey conservation of energy. For a motor, electrical power in equals shaft power plus copper, core, mechanical, stray losses and rate of field-energy change. The core procedure is to select a reference polarity and sign convention, distinguish an ideal relation from an experimentally calibrated one, and account for quantities whose units differ. Electromagnetic quantities must be checked against the associated conservation law: electric current and flux linkage describe excitation; voltages and torques describe observable exchange; and losses reduce usable output. In rotating or time-varying equipment, steady-state expressions must not be extrapolated to startup without tracking stored energy.

Fully worked numerical example

An electrical input of 1.2 kW with 900 W output delivers efficiency 75%. A steady 300 W balance must be assigned to measured or estimated loss channels. Write the given quantities in SI units before substitution, then compute the predicted response and its physical meaning. If an ideal result conflicts with a device rating, the correct engineering action is to question the simplifying assumptions rather than force the physical device to satisfy the equation. For a measured system, use its nameplate, empirical characteristic curves and operating limits to refine the estimate.

Failure modes, units, and boundary conditions

Specify torque direction, speed sign and whether the device is operating as motor or generator before writing a power balance. A useful independent verification is dimensional analysis, followed by one limiting case: for example, zero excitation, zero frequency, or no mechanical load. In magnetic systems, material saturation invalidates proportional field laws; in electric machines, parasitic resistance and friction introduce heat; and in measurements, uncertainty prevents an exact empirical conclusion.

Engineering workflow

1. Draw and label the electrical or magnetic equivalent circuit and specify reference directions.
2. Identify which parameters are given, measured, fitted or assumed ideal.
3. Calculate one predicted output by hand with correct units and signs.
4. Change one parameter by 10% and explain whether the response is linear or nonlinear.
5. Compare conservation of energy and rate the result against thermal, electrical and mechanical operating limits.
6. Document what this model omits before using it for an experiment or design.

Concept checks

• Which assumption is most likely to fail near rated load, and what measurement would reveal it?
• What physical quantity is stored rather than dissipated, and how would you verify the energy balance?
• How does the predicted response change if the principal excitation is doubled, and why might a real device respond differently?

Comprehensive Theoretical Derivations & Analytical Modeling

In electrical machinery analysis, quantitative formulation begins from Maxwell's integral laws applied to ferromagnetic geometries. For 2.1 Energy balance and signs, we define the magnetic circuit relationship:

where the reluctance $\mathcal{R}$ is governed by path length $l$, cross-sectional area $A$, and effective permeability $\mu = \mu_0 \mu_r$:

When analyzing electromechanical energy conversion in Unit 2, the differential mechanical force developed along coordinate $x$ is derived rigorously from the magnetic coenergy $W_f'(i, x)$:

For rotational devices, the developed electromagnetic torque becomes:

Under steady-state machine operation, terminal voltage balances internal induced electromotive force $E_a$ and winding impedance drops:

where the sign reflects generator (+ for internal EMF) or motor (- for terminal supply) operating quadrants. Core loss segregation accounts for hysteresis loss $P_h = k_h f B_{\max}^n$ and eddy current loss $P_e = k_e f^2 B_{\max}^2$, providing complete physical fidelity.

2.2 Force from coenergy

Physical model and mathematical relationship

At fixed current, coenergy derivative gives force F = (partial Wprime / partial x)_i. For an ideal linear position-dependent inductor Wprime=0.5 L(x)i^2. The core procedure is to select a reference polarity and sign convention, distinguish an ideal relation from an experimentally calibrated one, and account for quantities whose units differ. Electromagnetic quantities must be checked against the associated conservation law: electric current and flux linkage describe excitation; voltages and torques describe observable exchange; and losses reduce usable output. In rotating or time-varying equipment, steady-state expressions must not be extrapolated to startup without tracking stored energy.

Fully worked numerical example

A solenoid model L(x)=0.05+0.020x H with x in metres, carrying 3 A, predicts F=0.590.020=0.09 N. The model is local and does not include magnetic saturation. Write the given quantities in SI units before substitution, then compute the predicted response and its physical meaning. If an ideal result conflicts with a device rating, the correct engineering action is to question the simplifying assumptions rather than force the physical device to satisfy the equation. For a measured system, use its nameplate, empirical characteristic curves and operating limits to refine the estimate.

Failure modes, units, and boundary conditions

An attractive actuator can be unstable against a spring when its force increases faster than spring restoring force. A useful independent verification is dimensional analysis, followed by one limiting case: for example, zero excitation, zero frequency, or no mechanical load. In magnetic systems, material saturation invalidates proportional field laws; in electric machines, parasitic resistance and friction introduce heat; and in measurements, uncertainty prevents an exact empirical conclusion.

Engineering workflow

1. Draw and label the electrical or magnetic equivalent circuit and specify reference directions.
2. Identify which parameters are given, measured, fitted or assumed ideal.
3. Calculate one predicted output by hand with correct units and signs.
4. Change one parameter by 10% and explain whether the response is linear or nonlinear.
5. Compare conservation of energy and rate the result against thermal, electrical and mechanical operating limits.
6. Document what this model omits before using it for an experiment or design.

Concept checks

• Which assumption is most likely to fail near rated load, and what measurement would reveal it?
• What physical quantity is stored rather than dissipated, and how would you verify the energy balance?
• How does the predicted response change if the principal excitation is doubled, and why might a real device respond differently?

Comprehensive Theoretical Derivations & Analytical Modeling

In electrical machinery analysis, quantitative formulation begins from Maxwell's integral laws applied to ferromagnetic geometries. For 2.2 Force from coenergy, we define the magnetic circuit relationship:

where the reluctance $\mathcal{R}$ is governed by path length $l$, cross-sectional area $A$, and effective permeability $\mu = \mu_0 \mu_r$:

When analyzing electromechanical energy conversion in Unit 2, the differential mechanical force developed along coordinate $x$ is derived rigorously from the magnetic coenergy $W_f'(i, x)$:

For rotational devices, the developed electromagnetic torque becomes:

Under steady-state machine operation, terminal voltage balances internal induced electromotive force $E_a$ and winding impedance drops:

where the sign reflects generator (+ for internal EMF) or motor (- for terminal supply) operating quadrants. Core loss segregation accounts for hysteresis loss $P_h = k_h f B_{\max}^n$ and eddy current loss $P_e = k_e f^2 B_{\max}^2$, providing complete physical fidelity.

2.3 Torque production

Physical model and mathematical relationship

At fixed current, T = (partial Wprime / partial theta)_i. A rotating machine transfers energy when magnetic flux linkages change with rotor position or conductor current interacts with field flux. The core procedure is to select a reference polarity and sign convention, distinguish an ideal relation from an experimentally calibrated one, and account for quantities whose units differ. Electromagnetic quantities must be checked against the associated conservation law: electric current and flux linkage describe excitation; voltages and torques describe observable exchange; and losses reduce usable output. In rotating or time-varying equipment, steady-state expressions must not be extrapolated to startup without tracking stored energy.

Fully worked numerical example

If Wprime(theta)=0.4+0.2 cos(2 theta) J at fixed excitation, torque is -0.4 sin(2 theta) N m; at theta=pi/4 it is -0.4 N m. Write the given quantities in SI units before substitution, then compute the predicted response and its physical meaning. If an ideal result conflicts with a device rating, the correct engineering action is to question the simplifying assumptions rather than force the physical device to satisfy the equation. For a measured system, use its nameplate, empirical characteristic curves and operating limits to refine the estimate.

Failure modes, units, and boundary conditions

Average torque must be computed over the appropriate electrical or mechanical cycle; peak torque is not necessarily sustainable. A useful independent verification is dimensional analysis, followed by one limiting case: for example, zero excitation, zero frequency, or no mechanical load. In magnetic systems, material saturation invalidates proportional field laws; in electric machines, parasitic resistance and friction introduce heat; and in measurements, uncertainty prevents an exact empirical conclusion.

Engineering workflow

1. Draw and label the electrical or magnetic equivalent circuit and specify reference directions.
2. Identify which parameters are given, measured, fitted or assumed ideal.
3. Calculate one predicted output by hand with correct units and signs.
4. Change one parameter by 10% and explain whether the response is linear or nonlinear.
5. Compare conservation of energy and rate the result against thermal, electrical and mechanical operating limits.
6. Document what this model omits before using it for an experiment or design.

Concept checks

• Which assumption is most likely to fail near rated load, and what measurement would reveal it?
• What physical quantity is stored rather than dissipated, and how would you verify the energy balance?
• How does the predicted response change if the principal excitation is doubled, and why might a real device respond differently?

Comprehensive Theoretical Derivations & Analytical Modeling

In electrical machinery analysis, quantitative formulation begins from Maxwell's integral laws applied to ferromagnetic geometries. For 2.3 Torque production, we define the magnetic circuit relationship:

where the reluctance $\mathcal{R}$ is governed by path length $l$, cross-sectional area $A$, and effective permeability $\mu = \mu_0 \mu_r$:

When analyzing electromechanical energy conversion in Unit 2, the differential mechanical force developed along coordinate $x$ is derived rigorously from the magnetic coenergy $W_f'(i, x)$:

For rotational devices, the developed electromagnetic torque becomes:

Under steady-state machine operation, terminal voltage balances internal induced electromotive force $E_a$ and winding impedance drops:

where the sign reflects generator (+ for internal EMF) or motor (- for terminal supply) operating quadrants. Core loss segregation accounts for hysteresis loss $P_h = k_h f B_{\max}^n$ and eddy current loss $P_e = k_e f^2 B_{\max}^2$, providing complete physical fidelity.

2.4 Speed, torque and operating quadrants

Physical model and mathematical relationship

Mechanical power P=T omega. Motor quadrant has torque aligned with rotation; braking has opposite signs. Back EMF constrains current and speed. The core procedure is to select a reference polarity and sign convention, distinguish an ideal relation from an experimentally calibrated one, and account for quantities whose units differ. Electromagnetic quantities must be checked against the associated conservation law: electric current and flux linkage describe excitation; voltages and torques describe observable exchange; and losses reduce usable output. In rotating or time-varying equipment, steady-state expressions must not be extrapolated to startup without tracking stored energy.

Fully worked numerical example

At 1500 rpm and torque 8 N m, omega=157.08 rad/s and shaft power is about 1.257 kW. If total input is 1.5 kW, efficiency is 83.8%. Write the given quantities in SI units before substitution, then compute the predicted response and its physical meaning. If an ideal result conflicts with a device rating, the correct engineering action is to question the simplifying assumptions rather than force the physical device to satisfy the equation. For a measured system, use its nameplate, empirical characteristic curves and operating limits to refine the estimate.

Failure modes, units, and boundary conditions

Include friction, inertia J d omega/dt and external load torque to predict acceleration and starting performance. A useful independent verification is dimensional analysis, followed by one limiting case: for example, zero excitation, zero frequency, or no mechanical load. In magnetic systems, material saturation invalidates proportional field laws; in electric machines, parasitic resistance and friction introduce heat; and in measurements, uncertainty prevents an exact empirical conclusion.

Engineering workflow

1. Draw and label the electrical or magnetic equivalent circuit and specify reference directions.
2. Identify which parameters are given, measured, fitted or assumed ideal.
3. Calculate one predicted output by hand with correct units and signs.
4. Change one parameter by 10% and explain whether the response is linear or nonlinear.
5. Compare conservation of energy and rate the result against thermal, electrical and mechanical operating limits.
6. Document what this model omits before using it for an experiment or design.

Concept checks

• Which assumption is most likely to fail near rated load, and what measurement would reveal it?
• What physical quantity is stored rather than dissipated, and how would you verify the energy balance?
• How does the predicted response change if the principal excitation is doubled, and why might a real device respond differently?

Comprehensive Theoretical Derivations & Analytical Modeling

In electrical machinery analysis, quantitative formulation begins from Maxwell's integral laws applied to ferromagnetic geometries. For 2.4 Speed, torque and operating quadrants, we define the magnetic circuit relationship:

where the reluctance $\mathcal{R}$ is governed by path length $l$, cross-sectional area $A$, and effective permeability $\mu = \mu_0 \mu_r$:

When analyzing electromechanical energy conversion in Unit 2, the differential mechanical force developed along coordinate $x$ is derived rigorously from the magnetic coenergy $W_f'(i, x)$:

For rotational devices, the developed electromagnetic torque becomes:

Under steady-state machine operation, terminal voltage balances internal induced electromotive force $E_a$ and winding impedance drops:

where the sign reflects generator (+ for internal EMF) or motor (- for terminal supply) operating quadrants. Core loss segregation accounts for hysteresis loss $P_h = k_h f B_{\max}^n$ and eddy current loss $P_e = k_e f^2 B_{\max}^2$, providing complete physical fidelity.

Unit 2: Electromechanical energy conversion

Learning outcomes: derive relevant equations, solve at least one rated-condition example, explain limiting assumptions, and connect model parameters to a measurable device characteristic.

2.1 Energy balance and signs

Physical model and mathematical relationship

Electrical input, mechanical output, field storage and losses obey conservation of energy. For a motor, electrical power in equals shaft power plus copper, core, mechanical, stray losses and rate of field-energy change. The core procedure is to select a reference polarity and sign convention, distinguish an ideal relation from an experimentally calibrated one, and account for quantities whose units differ. Electromagnetic quantities must be checked against the associated conservation law: electric current and flux linkage describe excitation; voltages and torques describe observable exchange; and losses reduce usable output. In rotating or time-varying equipment, steady-state expressions must not be extrapolated to startup without tracking stored energy.

Fully worked numerical example

An electrical input of 1.2 kW with 900 W output delivers efficiency 75%. A steady 300 W balance must be assigned to measured or estimated loss channels. Write the given quantities in SI units before substitution, then compute the predicted response and its physical meaning. If an ideal result conflicts with a device rating, the correct engineering action is to question the simplifying assumptions rather than force the physical device to satisfy the equation. For a measured system, use its nameplate, empirical characteristic curves and operating limits to refine the estimate.

Failure modes, units, and boundary conditions

Specify torque direction, speed sign and whether the device is operating as motor or generator before writing a power balance. A useful independent verification is dimensional analysis, followed by one limiting case: for example, zero excitation, zero frequency, or no mechanical load. In magnetic systems, material saturation invalidates proportional field laws; in electric machines, parasitic resistance and friction introduce heat; and in measurements, uncertainty prevents an exact empirical conclusion.

Engineering workflow

1. Draw and label the electrical or magnetic equivalent circuit and specify reference directions.
2. Identify which parameters are given, measured, fitted or assumed ideal.
3. Calculate one predicted output by hand with correct units and signs.
4. Change one parameter by 10% and explain whether the response is linear or nonlinear.
5. Compare conservation of energy and rate the result against thermal, electrical and mechanical operating limits.
6. Document what this model omits before using it for an experiment or design.

Concept checks

• Which assumption is most likely to fail near rated load, and what measurement would reveal it?
• What physical quantity is stored rather than dissipated, and how would you verify the energy balance?
• How does the predicted response change if the principal excitation is doubled, and why might a real device respond differently?

2.2 Force from coenergy

Physical model and mathematical relationship

At fixed current, coenergy derivative gives force F = (partial Wprime / partial x)_i. For an ideal linear position-dependent inductor Wprime=0.5 L(x)i^2. The core procedure is to select a reference polarity and sign convention, distinguish an ideal relation from an experimentally calibrated one, and account for quantities whose units differ. Electromagnetic quantities must be checked against the associated conservation law: electric current and flux linkage describe excitation; voltages and torques describe observable exchange; and losses reduce usable output. In rotating or time-varying equipment, steady-state expressions must not be extrapolated to startup without tracking stored energy.

Fully worked numerical example

A solenoid model L(x)=0.05+0.020x H with x in metres, carrying 3 A, predicts F=0.590.020=0.09 N. The model is local and does not include magnetic saturation. Write the given quantities in SI units before substitution, then compute the predicted response and its physical meaning. If an ideal result conflicts with a device rating, the correct engineering action is to question the simplifying assumptions rather than force the physical device to satisfy the equation. For a measured system, use its nameplate, empirical characteristic curves and operating limits to refine the estimate.

Failure modes, units, and boundary conditions

An attractive actuator can be unstable against a spring when its force increases faster than spring restoring force. A useful independent verification is dimensional analysis, followed by one limiting case: for example, zero excitation, zero frequency, or no mechanical load. In magnetic systems, material saturation invalidates proportional field laws; in electric machines, parasitic resistance and friction introduce heat; and in measurements, uncertainty prevents an exact empirical conclusion.

Engineering workflow

1. Draw and label the electrical or magnetic equivalent circuit and specify reference directions.
2. Identify which parameters are given, measured, fitted or assumed ideal.
3. Calculate one predicted output by hand with correct units and signs.
4. Change one parameter by 10% and explain whether the response is linear or nonlinear.
5. Compare conservation of energy and rate the result against thermal, electrical and mechanical operating limits.
6. Document what this model omits before using it for an experiment or design.

Concept checks

• Which assumption is most likely to fail near rated load, and what measurement would reveal it?
• What physical quantity is stored rather than dissipated, and how would you verify the energy balance?
• How does the predicted response change if the principal excitation is doubled, and why might a real device respond differently?

2.3 Torque production

Physical model and mathematical relationship

At fixed current, T = (partial Wprime / partial theta)_i. A rotating machine transfers energy when magnetic flux linkages change with rotor position or conductor current interacts with field flux. The core procedure is to select a reference polarity and sign convention, distinguish an ideal relation from an experimentally calibrated one, and account for quantities whose units differ. Electromagnetic quantities must be checked against the associated conservation law: electric current and flux linkage describe excitation; voltages and torques describe observable exchange; and losses reduce usable output. In rotating or time-varying equipment, steady-state expressions must not be extrapolated to startup without tracking stored energy.

Fully worked numerical example

If Wprime(theta)=0.4+0.2 cos(2 theta) J at fixed excitation, torque is -0.4 sin(2 theta) N m; at theta=pi/4 it is -0.4 N m. Write the given quantities in SI units before substitution, then compute the predicted response and its physical meaning. If an ideal result conflicts with a device rating, the correct engineering action is to question the simplifying assumptions rather than force the physical device to satisfy the equation. For a measured system, use its nameplate, empirical characteristic curves and operating limits to refine the estimate.

Failure modes, units, and boundary conditions

Average torque must be computed over the appropriate electrical or mechanical cycle; peak torque is not necessarily sustainable. A useful independent verification is dimensional analysis, followed by one limiting case: for example, zero excitation, zero frequency, or no mechanical load. In magnetic systems, material saturation invalidates proportional field laws; in electric machines, parasitic resistance and friction introduce heat; and in measurements, uncertainty prevents an exact empirical conclusion.

Engineering workflow

1. Draw and label the electrical or magnetic equivalent circuit and specify reference directions.
2. Identify which parameters are given, measured, fitted or assumed ideal.
3. Calculate one predicted output by hand with correct units and signs.
4. Change one parameter by 10% and explain whether the response is linear or nonlinear.
5. Compare conservation of energy and rate the result against thermal, electrical and mechanical operating limits.
6. Document what this model omits before using it for an experiment or design.

Concept checks

• Which assumption is most likely to fail near rated load, and what measurement would reveal it?
• What physical quantity is stored rather than dissipated, and how would you verify the energy balance?
• How does the predicted response change if the principal excitation is doubled, and why might a real device respond differently?

2.4 Speed, torque and operating quadrants

Physical model and mathematical relationship

Mechanical power P=T omega. Motor quadrant has torque aligned with rotation; braking has opposite signs. Back EMF constrains current and speed. The core procedure is to select a reference polarity and sign convention, distinguish an ideal relation from an experimentally calibrated one, and account for quantities whose units differ. Electromagnetic quantities must be checked against the associated conservation law: electric current and flux linkage describe excitation; voltages and torques describe observable exchange; and losses reduce usable output. In rotating or time-varying equipment, steady-state expressions must not be extrapolated to startup without tracking stored energy.

Fully worked numerical example

At 1500 rpm and torque 8 N m, omega=157.08 rad/s and shaft power is about 1.257 kW. If total input is 1.5 kW, efficiency is 83.8%. Write the given quantities in SI units before substitution, then compute the predicted response and its physical meaning. If an ideal result conflicts with a device rating, the correct engineering action is to question the simplifying assumptions rather than force the physical device to satisfy the equation. For a measured system, use its nameplate, empirical characteristic curves and operating limits to refine the estimate.

Failure modes, units, and boundary conditions

Include friction, inertia J d omega/dt and external load torque to predict acceleration and starting performance. A useful independent verification is dimensional analysis, followed by one limiting case: for example, zero excitation, zero frequency, or no mechanical load. In magnetic systems, material saturation invalidates proportional field laws; in electric machines, parasitic resistance and friction introduce heat; and in measurements, uncertainty prevents an exact empirical conclusion.

Engineering workflow

1. Draw and label the electrical or magnetic equivalent circuit and specify reference directions.
2. Identify which parameters are given, measured, fitted or assumed ideal.
3. Calculate one predicted output by hand with correct units and signs.
4. Change one parameter by 10% and explain whether the response is linear or nonlinear.
5. Compare conservation of energy and rate the result against thermal, electrical and mechanical operating limits.
6. Document what this model omits before using it for an experiment or design.

Concept checks

• Which assumption is most likely to fail near rated load, and what measurement would reveal it?
• What physical quantity is stored rather than dissipated, and how would you verify the energy balance?
• How does the predicted response change if the principal excitation is doubled, and why might a real device respond differently?

Unit 2 assessment

1. Calculation: Recompute the first worked example in this unit after increasing its central excitation (current, voltage, flux, speed or load) by 10%. Declare whether linear scaling is justified and show units.

2. Model critique: Select a different lesson within this unit and identify two mechanisms absent from its ideal equation. Explain how each mechanism would alter a specific measurement.

3. Design decision: Develop a one-paragraph specification for a laboratory device from this unit, including a rated quantity, a safety boundary, and a verification test.

4. Interpretation: Describe one conservation law that constrains the unit's numerical answers, and show its corresponding terms.

Marking guide: For question 1, the numeric answer depends on the worked example and whether its equations are linear; give credit for correctly substituting and rechecking assumptions. For questions 2–4, require explicit physical mechanisms, units, realistic protection and an objective acceptance criterion.