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Block 09/10 — Transformers and Magnetic Coupling
Learning objectives
After this 90-minute block, you can
- explain how two coils can exchange energy by a common magnetic flux \(\Phi\).
- use the ideal transformer equations
\[ \begin{align*} \frac{\underline{U}_1}{\underline{U}_2}=\frac{N_1}{N_2}=n, \qquad \frac{\underline{I}_1}{\underline{I}_2}=-\frac{1}{n} \end{align*} \] with a clear sign convention.
- explain mutual inductance \(M\) using flux linkage and magnetic reluctance \(R_{\rm m}\).
- distinguish main flux, leakage flux, copper losses, and iron losses in a real transformer.
- refer secondary-side quantities to the primary side using \( \underline{U}'_2=n\underline{U}_2\), \( \underline{I}'_2=\frac{1}{n}\underline{I}_2\), \(R'_2=n^2R_2\), and \(X'_{2\sigma}=n^2X_{2\sigma}\).
- interpret the no-load test and short-circuit test using the reduced equivalent circuit.
- calculate short-circuit voltage \(u_{\rm k}\), continuous short-circuit current \(I_{\rm 1k}\), and an estimated initial peak short-circuit current.
- connect transformer parameters to engineering applications in mechatronics and robotics, such as isolated power supplies, motor current measurement, welding transformers, and safety transformers.
Preparation at Home
Well, again
- read through the present chapter and write down anything you did not understand.
- Repeat the EEE1 ideas of magnetic flux and induction, magnetic circuits, and inductance and magnetic energy.
For checking your understanding please do the quick checks in the exercise section.
90-minute plan
- Warm-up (10 min):
- Where do transformers occur in robots and automation systems?
- Recall: Faraday induction from EEE1 — a changing magnetic flux induces a voltage.
- Recall: in AC analysis we use RMS phasors \(\underline{U}\), \(\underline{I}\), and impedances \(j\omega L\).
- Core concepts and derivations (55 min):
- Ideal transformer: common flux, voltage ratio, current ratio, power balance.
- Mutual inductance: how flux from one coil links another coil.
- Magnetic coupling with reluctance \(R_{\rm m}\).
- Real transformer: winding resistances, leakage inductances, iron-loss resistance.
- Reduced equivalent circuit: refer secondary quantities to the primary side.
- No-load and short-circuit operation: what can be measured, what can be neglected.
- Practice (20 min):
- Quick ratio calculations for step-up and step-down transformers.
- Unit checks for \(j\omega L\), \(j\omega N\Phi\), and \(u_{\rm k}\).
- Short-circuit current calculation for a transformer used in an actuator supply.
- Wrap-up (5 min):
- Summary box: ideal transformer, mutual inductance, real transformer, reduced circuit, short-circuit parameters.
- Common pitfalls checklist.
Conceptual overview
- A transformer is not a DC component. It needs a changing magnetic flux. In normal operation this is usually a sinusoidal flux created by AC voltage.
- The transformer does not “create power”. Ideally, it trades voltage for current:
\[ \begin{align*} \text{higher voltage} \quad \Longleftrightarrow \quad \text{lower current} \end{align*} \]
- The link between the two windings is the magnetic field in the iron core. This continues directly from EEE1:
- induction explains why a changing flux induces voltage.
- magnetic circuits explains why the iron core guides the flux.
- inductance explains how flux linkage and current are connected.
- Mutual inductance \(M\) measures how strongly one coil “notices” the changing current in another coil.
- A real transformer is almost ideal, but not quite:
- \(R_1, R_2\): copper losses in the windings.
- \(L_{1\sigma}, L_{2\sigma}\): leakage flux that does not couple both windings.
- \(R_{\rm Fe}\): iron losses in the core.
- \(L_{\rm H}\): main magnetizing inductance needed to create the main flux.
- In engineering, transformer data such as \(u_{\rm k}\) are not abstract: they determine voltage drop, fault current, thermal stress, and protection design.
Core content
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Common pitfalls
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Exercises
Worked examples
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Embedded resources
Explanation (video): …