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In my textbook, it is given that \$\mathcal {E} = -L\frac{\Delta I}{\Delta t}\$. The graph of resistor voltage (\$V_R\$) and inductor voltage (\$V_L\$) in a series RL AC circuit is given similar to below:

V_R is presented as a sine wave, while V_L is presented as a cosine wave (90 degrees out of phase)

Where the red wave is \$V_R\$, and the blue wave is \$V_L\$.

It justifies this by implying that since \$V_L\$ is a multiple of the rate of change of current (\$\mathcal {E} = -L\frac{\Delta I}{\Delta t}\$) and current is proportional to \$V_R\$ (\$V_R=IR\$), then the graph of \$V_L\$ is simply the rate of change of \$V_R\$ multiplied by some constant.

The problem I see is, wouldn't the negative sign in the induced emf equation mean that the \$V_L\$ should have a negative amplitude (i.e. be flipped)?.

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The negative sign in the equation \$E = -L\frac{di}{dt}\$ implies back emf i.e. the voltage induced in the coil due to current flowing. Given that you might want to calculate the current that flows in an inductor for an applied voltage, the formula is \$V = L\frac{di}{dt}\$.

Note that I used the voltage symbol "E" for back emf and "V" for applied voltage but please get used to them being called "V" for both occasions and recognizing that the negative sign implies back (or induced) voltage (or emf).

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  • \$\begingroup\$ What do you mean by the applied voltage in this case? How is it different from back emf? \$\endgroup\$ – Ziscmac Oct 18 at 20:04
  • \$\begingroup\$ It’s the opposite polarity ie acting against the applied voltage. \$\endgroup\$ – Andy aka Oct 19 at 7:52
  • \$\begingroup\$ I'm not quite sure what you mean by applied voltage. Is that the supply voltage? \$\endgroup\$ – Ziscmac Oct 20 at 8:33
  • \$\begingroup\$ Applied voltage is the supplied excitation voltage. I’m making a distinction about not calling it supply voltage because, the term supply voltage usually can be confused with other things. Applied voltage is the proper term to use or, excitation voltage. \$\endgroup\$ – Andy aka Oct 20 at 11:00
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Load power is positive V=LdI/dt while generated power is negative including back EMF while coasting a motor with generated output.

Dont confuse these conventions for L phase current polarity.

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