#06 Circuit Theory-2 #06 Mutual Inductance and the Ideal Transformer

Below are all the lines written in the notebook together with the full narration transcript.

1. 1. Mutual inductance

   Corrected solution notes. The audio is the original recording and may contain errors or incomplete phrases. Where they differ, use the corrected written solution.

   Changing current in one winding creates changing flux that can link the other. Mutual inductance M measures this coupling for the stated linear model.

   Narration transcript

   When two inductors are close enough, they should not be treated as isolated pieces anymore. A changing current in one winding can create a changing magnetic flux that links the other winding. That cross-coupling is mutual inductance. It is the bridge between ordinary inductors and the ideal transformer.

2. 2. Flux linkage

   For coupled linear windings the flux linkages contain self and mutual contributions.

   Induced voltage depends on the time derivative of linked flux, not merely the sign of instantaneous current.

   Narration transcript

   If current in winding one changes, the magnetic flux around it changes as well. Part of that changing flux can link winding two, and winding two then sees an induced voltage. The better the shared flux linkage, the larger the mutual inductance M. So mutual inductance is really a measure of how strongly one winding can impress itself on the other.

3. 3. Correct dot convention

   With both voltage references positive at the dotted ends and both current references entering the dots, the mutual terms have positive coefficient M.

   v1=L1 di1/dt+M di2/dt; v2=M di1/dt+L2 di2/dt.

   A positive current entering a dot does not alone establish instantaneous induced polarity; its derivative may be positive, negative or zero.

   Narration transcript

   The dots tell us the sign of that induced voltage. If current enters the dotted terminal of one winding, the induced voltage makes the dotted terminal of the other winding positive. If the reference current leaves the dotted end, the sign flips. The dots do not tell magnitude. They tell polarity consistency.

4. 4. Ideal voltage ratio

   Under the same dotted voltage references, V1/V2=N1/N2 for an ideal transformer.

   This ideal model neglects losses and leakage and imposes ideal flux coupling.

   Narration transcript

   An ideal transformer is the cleanest version of this coupling idea. All useful flux links both windings, and we ignore loss. The voltage ratio follows the turns ratio: V one over V two equals N one over N two. More turns on the secondary means a larger secondary voltage. Fewer turns means a smaller one.

5. 5. Current direction matters

   If both current references enter the dotted terminals, N1I1+N2I2=0.

   If secondary current is instead defined leaving its dot into the load, I1/I2=N2/N1. Magnitudes scale inversely to turns; ideal input and delivered power agree.

   Narration transcript

   Current scales in the opposite direction. I one over I two equals N two over N one. So a step-up voltage transformer is also a step-down current transformer. In the ideal case, input power equals output power, which is why the voltage and current ratios invert each other.

6. 6. Impedance reflection

   Zin=(N1/N2)² Zload for an ideal transformer and consistently connected secondary load.

   This follows by combining the voltage and load-current ratios; the load's physical impedance is not changed internally.

   Narration transcript

   This is why transformers are so useful in circuit analysis. A load connected to the secondary can be reflected to the primary as an equivalent impedance. Z in equals the turns ratio squared times Z load, or N one over N two all squared times Z load. The transformer changes how the source experiences the load.

7. 7. Review

   Dots fix reference signs, while derivatives determine instantaneous mutual voltage.

   State current directions before using transformer ratios; voltage, current and reflected impedance must use one consistent convention.

   Narration transcript

   Mutual inductance explains how one winding affects another. Dot convention fixes the sign, and the ideal transformer turns that coupling into clean voltage, current, and impedance ratios. In the next lesson, we move to instantaneous and average power.

Source video: Circuit Theory-2 #06 Mutual Inductance and the Ideal Transformer (2:50)