$vec{h}$ is defined in such a way that it has only 2 components, which in my eyes form only a 1-sphere. The number of turns should always be trivial, based on homotopy arguments (I don`t really understand that). The counterargument is that if we draw the flowchart of $vec{h}$, we see a vertex structure just like the vertex structure in the XY model of statistical mechanics, which is one of the main points of topology in physics. The winding that gives the desired output voltage by mutual induction is commonly referred to as the “secondary winding”. This is the “second coil” in the diagram above. The principle of operation of a transformer is very simple. Mutual induction between two or more windings (also called coils) allows electrical energy to be transferred between circuits. This principle is explained in more detail below. The ideal transformer has no losses.

There is no magnetic leakage, ohmic resistance in its windings and no loss of iron in the core. The winding number is closely related to the continuous Heisenberg ferromagnetic equations of dimension (2 + 1) and their integrable extensions: the Ishimori equation, etc. The solutions of the last equations are classified according to the number of revolutions or topological charge (topological invariant and/or topological quantum number). Finally, winding tension and straight band tension controls are directly related to production efficiency. When winding the rollers, stress levels must be controlled to exclude air between the layers of material. Air pockets can roll the roller towards the telescope, with layers sliding in and out of the coil. Too much tension between layers can reduce air trapping, but cause too much stretching and/or deformation of the material, or the elasticity of the material can cause the material to try to return to its original shape, deforming the finished roller. In any case, without proper tension control, the roller may be too soft due to a lack of voltage regulation, or the roller core may be damaged or the roller telescopic due to high voltage. E2 = terminal voltage (theoretical or calculated) at the secondary winding. The primary and secondary windings are insulated from each other with insulating materials. We reveal the geometric meaning of the number of turns and use it to characterize topological phases in non-hermit one-dimensional chiral systems. While chiral symmetry ensures that the number of turns of Hermitian systems is integer, for non-hermit systems it may require half-integers.

We give a geometric interpretation of half-integers by showing that the number ν of a non-hermit system is equal to half the sum of two winding numbers ν1 and ν2, each assigned to two extraordinary points. The reversal numbers ν1 and ν2 represent the times of the real Hamilton part in the momentum space surrounding the extraordinary points, and can only assume integers. We further note that the difference of ν1 and ν2 is related to the second number of windings or vortices of energy. By applying our scheme to a non-Hermitian Su-Schrieffer-Heeger model and an extended version of it, we show that topologically different phases can be well characterized by rotating numbers. In addition, we show that the existence of edge states in left and right zero mode is closely related to the number of turns ν1 and ν2. Since γ {displaystyle gamma } is a closed curve, the total change of ln ( r ) {displaystyle ln(r)} is zero, and therefore the integral of d z z {textstyle {frac {dz}{z}}} is equal to i {displaystyle i} multiplied by the total change of θ {displaystyle theta }. Therefore, the number of turns of the closed path is γ {displaystyle gamma } around the origin by the expression [3] The form dθ (defined on the complement of the origin) is closed but not exact, and produces the first Rham cohomology group of the dotted plane. In particular, if ω is a differentiable closed form defined on the complement of the origin, then the integral of ω along the closed loops gives a multiple of the number of revolutions. Case $d = $1: We now have a card $T^1to S^2$. Note that $T^1 = S^1$, so such a map must be topologically trivial: if you put a closed chain on the surface of a sphere, you can always continually reduce it to a point.

So no topological phases in 1D. Unless we impose additional restrictions on $H_{boldsymbol k}$. For example, if we require our Hamilton to be invariant under complex conjugation, then the component $sigma_y$ must be zero, i.e. $boldsymbol n_{boldsymbol k} = left(n^x_{boldsymbol k},0,n^z_{boldsymbol k} right)$. Now we actually have the card $T^1 at S^1$. But it is the mapping from circle to circle, which can clearly have a non-trivial turn number! As above, this is now calculated by $$propto int_{T^1} mathrm d k; boldsymbol n_{boldsymbol k} cdot left( nabla times boldsymbol n_{boldsymbol k} right) $$ So we see that as long as we apply time invariance, we can have topological band isolators in $d= 1$. These are usually called topological (one-dimensional) insulators. This is different from integer quantum states, which do not require symmetry protection (although this is not entirely clear from the above analysis $d=$2, as I suspected translational symmetry and charge conservation – it turns out these are not essential). The purpose of the transformer core is to provide a low-reluctance path through which the maximum amount of flow generated by the primary winding is traversed and connected to the secondary winding. When another winding is brought closer to this winding, some of this alternating current connects to the second winding. Since this flow is constantly changing in its amplitude and direction, there must be a changing flow connection in the second winding or coil.

1. A transformer has 600 turns of the primary winding and 20 turns of the secondary winding. Determine the secondary voltage when the secondary circuit is open and the primary voltage is 140 V. Case $d=$2: Then we have a board $T^2to S^2$. This map has an easy-to-imagine topological invariant: imagine taking one torus and two spheres. One sphere is located outside the torus (contains it completely), the other sphere is located in the torus tube. Now imagine inflating the smallest ball until it hits the biggest ball. The torus between the two spheres is flattened. This now visually defines a map from the torus (pressed) to the sphere.

In addition, you can see that this map is topologically non-trivial: by dragging and stretching the torus locally, you cannot reduce it to a point. We say that this torus meanders once around the sphere, and such a number can be measured by calculating the number of revolutions: $$propto iint_{T^2} mathrm d k_1 mathrm d k_2 ; boldsymbol n_{boldsymbol k} cdot left( frac{partial boldsymbol n_{boldsymbol k}}{partial k_1} times frac{partial boldsymbol n_{boldsymbol k}}{partial k_2} right) $$ (Note that this formula is intuitive: the cross product measures the surface on the sphere covered with a small piece of torus $mathrm d k_1 mathrm d k_2$. So this formula essentially measures the (signed) area covered by the torus on the sphere, so if it is equal to the area of the sphere, we know that the torus winds once.) The above example of a curve wrapping around a point has a simple topological interpretation.