Two questions about an AC electric circuit: Have I interpreted the dots correctly and where does the negative sign come from?
So I am looking at two mutually coupled circuits that are not at resonance. In particular I am looking at a pair of electric circuits discussed on pgs: 259 - 261 of the book "Electricity and Magnetism", 4th edition by <PERSON>. I focus on pg 260 and a sketch of the two circuits is shown below (Fig 10.14a in <PERSON>). Apologies for the lob sided drawing.
Sketch of two mutually coupled circuits.
My first question pertains to my understanding of how to assign the potential differences across the coils when applying <PERSON>'s voltage law to each mesh. Before showing what I have done, I will show Fig 9.13 and Fig 10.1 from <PERSON> which use and discuss the dot notation and quote the explanation of how to assign the potential difference across each coil given by <PERSON> in the paragraph preceding Fig 10.1.
<PERSON>'s explanation of what the dots mean for coupled coils. Fig9.13
The meaning of the dots in relation to mutual induction.Fig 10.1
In the paragraph preceding Fig 10.1 <PERSON> explains how to assign the potential difference across each coil using the dot notation and the direction of the current in each coil as follows:
" For a mutual inductor, details of the directions of the windings are avoided by using the dot notation of Fig. 628
9.13. A current flowing in at the dotted end of one coil produces a voltage $V = M\, dI/dt$ across the other which is higher at its dotted end. The $sign$ attached to this voltage will depend on the assumed direction of current in the second coil. In any given circuit with directions for all currents decided, taking one dot to the other end of its coil will reverse the sign of $M$."
So with this in mind I think the potential differences across each coil are as shown in
Assigning the potential difference across each coil.
where plus and minus signs are in reference to the potential difference for the mutual inductance effects.
This leads me to the following for <PERSON>'s equations for the physical voltage and currents in each mesh:
$$v_1 - i_1R_{\rm p} - L_{\rm p}\frac{di_{1}}{dt} + M\frac{di_2}{dt} = 0\ \,\ M\frac{di_1}{dt} - i_2R_{\rm s} - L_{\rm s}\frac{di_2}{dt} - v_2 = 0\,. $$ In terms of phasors, with ${\rm {\bf I}}1 = I_1e^{j\omega t}$ and ${\rm {\bf I}}_2 = I_2e^{j\omega t}$ (so $i_1 = {\rm Re}({\rm {\bf I}}_1)\, {\rm etc.},)$, where $j=\sqrt{-1}\,,$ $\omega$ is the driving frequency of the AC source of emf, ${\rm {\bf Z}}{1} = R_{\rm p} + j\omega L_{\rm p}$, and ${\rm {\bf Z}}{\rm s} = R{\rm s} + j\omega L_{\rm s}$ the above equations can be written as $$ {\rm {\bf V}}1 - {\rm {\bf I}}_1{\rm {\bf Z}}_1 + M\frac{d{\rm {\bf I}}_2}{dt} = 0\ \,\ M\frac{d{\rm {\bf I}}_1}{dt} - {\rm {\bf I}}_2{\rm {\bf Z}}{\rm s} - {\rm {\bf V}}_2 = 0\,. $$
Letting ${\rm {\bf V}}2 = {\rm {\bf I}}_2{\rm {\bf Z}}{\rm L}$ and defining ${\rm {\bf Z}}{2} = {\rm {\bf Z}}{\rm s} + {\rm {\bf Z}}_{\rm L}\,$ this can be written as
$$ {\rm {\bf V}}1 - {\rm {\bf I}}{1}{\rm {\bf Z}}{1} + j \omega M{\rm {\bf I}}_2 = 0\ \,\ j\omega M{\rm {\bf I}}{1} - {\rm {\bf I}}{2}{\rm {\bf Z}}{\rm 2} = 0\,. 628