• Review of Symbols
• Flux and Flux Density
• Gauss's Law and Kirchoff's Flux Law
• Relation to Ohm's Law
• DC Machines
  • Field Winding
  • Armature Winding
  • Separately Excited Machine
  • Shunt Connected Machine
  • Series Connected Machine
  • Permanent Magnet Machine
• Field Energy and Coenergy
• Force
• AC Machines
  • Mechanical Position
  • Electrical Position
  • Discrete Winding
  • Continuous Winding
  • Symmetry
  • Winding Arrangement Notation
  • Winding Types
  • Winding Function
    • Discrete Winding
    • Continuous Winding
  • Distributed MMF
  • Unbalanced Two Phase
  • Three Phase
• Flux Linkage and Magnetizing Inductance
• PM Terms
• Torque
• Reference Frame
• Useful Relations
• exam

Review of Symbols

• i - current in amperes
• H - magnetic field intensity (amperes/meter = Tesla)
• B - magnetic flux density (amperes/meter = Tesla)
• F - magneto-motive force (amperes)
• R - reluctance (1/henries)
• $λ$ - flux linkage in volt-seconds
• \(\phi\) - flux in webers

Flux and Flux Density

\[\phi = \int_S B \cdot \overrightarrow{dS}\]

Gauss's Law and Kirchoff's Flux Law

Since there are no magnetic monopoles, we have \[\oint B \cdot \overrightarrow{dS} = 0\] (Gauss's Law)

Written another way, using the relation from before: \(\sum \phi = 0\) (Kirchoff's Flux Law)

Relation to Ohm's Law

\(F = \phi \underbrace{R}_\text{reluctance}\)

DC Machines

Some DC machines have two separate electrical systems.

Field Winding

Armature Winding

The voltage across the armature winding can be given by

Separately Excited Machine

Shunt Connected Machine

In a shunt connected machine, \(v_t = v_a = v_f\), and \(i_t = i_a + i_f\).

Series Connected Machine

In a series connected machine, \(v_t = v_a + v_f\) and \(i_t = i_a = i_f\).

Permanent Magnet Machine

Field Energy and Coenergy

Force

force: \(f_e = - \frac{dW_f(\lambda, x)}{dx}\)

force: \(f_e = \frac{dW_c(\i, x)}{dx}\)

AC Machines

In an AC machine, the poles on the stator rotate (via switching) and the poles on the rotor follow them. Teeth in AC machines are designed to allow flux flow in one direction and current flow orthogonally.

For a given frequency, each additional pole reduces the machine speed by \(p/2\), i.e. \(\omega_{rm} = \frac{2}{p} w_e\)

Mechanical Position

\(\theta_{rm}\) - measures rotor position relative to stator - counterclockwise \(\phi_{rm}\) - an arbitrary position relative to rotor - counterclockwise \(\phi_{sm}\) - an arbitrary position relative to stator - counterclockwise

Electrical Position

\(\phi_s = \phi_{sm} \frac{p}{2}\) \(\phi_r = \phi_{rm} \frac{p}{2}\) \(\theta_r = \theta_{rm} \frac{p}{2}\)

Discrete Winding

\[N_{x,i}\] is the number of conductors of an `x` phase winding in the `i`th slot.

[slot and tooth locations] where \(y\) is either \(r\) for stator or \(r\) for rotor

slot locations: \(\phi_{ys,i} = 2 \pi (i - 1)/S_y + \phi_{ys,1}\) tooth locations: \(\phi_{yt,i} = 2 \pi(i - 1/2)/ S_y + \phi_{ys,1}\)

Conductors coming out of the page are positive, and going in are negative.

Summing the number of conductors in every slot should yield zero: \[\sum_{i=1}^{S_y} N_{x,i} = 0\]

Continuous Winding

\(n_x(\phi_{ym}\) is the conductor density

• \(x\) is the winding phase - 'a', 'b', 'c', …
• \(y\) is the location - 's' for stator

conductor density example

total conductors

Symmetry

Due to the construction of the machine, the following statements are always true.

Where moving one pole means \(\frac{2 \pi}{p}\) radians, \(\frac{S_y}{p}\) slots

Winding Arrangement Notation

\[N_{as} \right_{1-18} = N [ \begin{matrix}0 0 0 1 2 2 1 0 0 0 0 0 -1 -2 -2 -1 0 0 \end{matrix} ]\]

\(p = 4\) \(S_s = 36\)

Winding Types

The following windings have the same number of conductors in the slots, but they differ in terms of loss, amount of copper, and difficulty to wind.

Winding Function

Discrete Winding

For DC machines, MMF drop was defined as \(F = iN\), where \(i\) is the current and \(N\) is the number of windings.

For AC machines, we define the MMF drop in a similar way.

Continuous Winding

Suppose we would like to find the winding function at some point \(W(\phi + \delta \phi)\) given that we know \(W(\phi)\)

Distributed MMF

MMF is dependent on position as well as current.

\(\mcal{F}_s(\phi_m) = \sum_{x \in X_s} w_x(\phi_m) i_x\) missing stuff

#+begin_definiiton MMF drop

\[\mcal{F}_g = \int_{\text{rotor}}^\text{stator} H(\$ *missing stuff* #+end_definition #+begin_proof $F_g(0) - F_g(\phi_m) = i_{enc} = $\int_0^{ #+end_proof $J = \sum_{x} n_x i_x$, where $n_x$ is in units of conductors/radian. $F_g(0) - F_g(\phi_m) = \int_0^{\phi_m} \sum_{x} n_x i_x d \phi_m$ Since $i_x$ is not dependent on $\phi_m$ and since summation and integraiton are linear operators, $F_g(\phi_m) = F_g(0) - \sum_x i_x \int_0^{\phi_m} n_x$ \]\begin{aligned}F_g() &= -F_g(0)
&= F_g(0) - ∑_x i_x ∫₀^{2 π /p} n_x d ϕ_m \end{aligned}\[ so $F_g(0) = \sum_x i_x \int_0^{2 \pi / p} n_x d \phi_m$ plugging in $F_g(0)$, \]\begin{aligned}F_g(ϕ_m) &= ∑_x ∫₀^{2 π / p} n_x d ϕ_m
&= ∑_x i_x \left[
&= _Fs(ϕm) + _Fr(ϕm) \end{aligned}\[ where $X_s$ are the stator windings and $X_r$ are the rotor windings. Once we have the MMF drop, we can find flux density $\lambda$, winding inductance and have a better understanding of the machine. $F_g = \int_{\text{rotor}}^{\text{stator}} H \cdot dr \approx H_g g$. $B_g = \mu_0 \mu $B_g(\phi_m) = \frac{ $B_x = \frac{\mu_0 F_x(\phi)}{g(\phi)$ - useful for finding self and mutual inductance #+begin_examples 1. Suppose - $w_{as} = 100 \cos(4 \phi_{sm})$, $i_{as} = 10$ - $w_{as} = 50 \cos(4 \phi_{sm})$, $i_{bs} = 5$ - $g = 1e-3$ \]\begin{aligned} F_s &= w_as i_as + w_bs i_bs
&= 500 cos (4 ϕ_sm) + 500 sin (4 ϕ_sm)
&= 500 cos (4 ϕ_sm - π/4) \end{aligned}\[ $B = \frac{\mu_0 F_s}{g} = \underbrace{\frac{(4 \pi \cdot 10^{-7}) (500 \sqrt{2})}{10^{-3}}}_{\text{peak = 0.88 T}} \cos(4 \phi+{sm} - \pi/4)$ #+end_examples * Rotating MMF *missing* \]\begin{aligned} F_s = I_s
&= I_s cos ( ϕ_sm +

Notice that \(F, B, H\) are sinusoidal in space and time.

Setting \(\cos\) equal to \(0\), we get \(\frac{p}{2} \phi_{sm} - \omega_e t - \phi_i = 0\)

so \(\phi_{sm} = \frac{2}{p} ( \omega_e t + \phi_i)\)

and \(\frac{d \phi_{sm}}{dt} = \omega_{rm} = \frac{2}{p} \omega_e\)

Unbalanced Two Phase

Assume $i_as = $, \(i_{bs} = 0\).

then

\[\begin{aligned} F_s &= \frac{2 N_s}{p} \cos(\frac{p}{2} \phi_{sm}) \sqrt{2} I_s \cos (\omega_e t + \phi_i) \\ &= \frac{N_s \sqrt{2} I_s}{p} \left[ \underbrace{\cos(\frac{p}{2} \phi_{sm} - \omega_e t - \phi_i)}_\text{forward travelling} + \underbrace{\cos( \frac{p}{2} \phi_{sm} + \omega_e t + \phi_i)}_\text{backwards travelling} \right] \end{aligned}\]

forward travelling - travels at same speed as before : backward travelling - $ϕ_sm = - The backwards travelling waves introduce inefficiency to the machine.

In single phase wired houses, many devices use a single phase input then create a secondary shifted phase with a capacitor. This increases the forward travelling wave and reduces the backwards travelling wave.

Three Phase

\(n_{as}(\phi_{sm}) = N_{s1} \sin(P \phi_{sm} / 2) - N_{s3} \sin(3 p \phi_{sm} /2 )\) \(n_{bs}(\phi_{sm}) = N_{s1} \sin(P \phi_{sm} / 2 - 2 \pi / 3) - N_{s3} \sin(3 p \phi_{sm} /2 )\) \(n_{cs}(\phi_{sm}) = N_{s1} \sin(P \phi_{sm} / 2 + 2 \pi / 3) - N_{s3} \sin(3 p \phi_{sm} /2 )\)

missing winding function

missing currents

\(\mathbb{F}_s = \frac{3 \sqrt{2} N_{s1} I_s}{P} \cos( P \phi_{sm} /2 - \omega_e t - \phi_i)\)

Flux Linkage and Magnetizing Inductance

\(\lambda_x = \underbrace{\lambda_{xl}_{\text{leakage}} + \underbrace{\lambda_{xm}}_{\text{magnetizing}}\)

missing latex right | \(L_{xy} = \frac{\lambda_x \text{due to }i_y}{i_y} = \frac{\lambda_{xl}}{i_y} + \underbrace{\frac{\lambda_{xm}}{i_y}}_{L_{xym}\)

missing eqn vv $L_xym = \frac{λ_xym{i_y} = μ

PM Terms

\[\begin{aligned} \lambda_{asm} &= \int_0^{2 \pi} w_{as} B_{pm} r l d\frac{2}{p} d \phi_s \\ &= \int_0^{2 \pi} w_{as} B_{pm} r l d \phi_s \\ &= \int_0^{2 \pi} \frac{2N_s}{p} \cos( \theta_r + \phi_r ) B_{pm} r d \phi_r \\ &= \frac{2N_s}{p r l \int_0^{2 \pi} B_{pm} \cos( \theta_r + \phi_r ) d \phi_r \\ &= \underbrace{\frac{8 N_s r l B_{pm}}{p}}_{\lambda_m'} \sin ( \theta_r ) \end{aligned}\]

Torque

Reference Frame

Useful Relations

reluctance : \(R = \frac{l}{A \mu}\ \text{H}^{-1}\), where \(l\) is length, \(A\) is area, and \(\mu\) is permittivity note that \(\mu\) is not always constant and can depend on conductor current

flux : \[\phi = \frac{Ni}{R} = \int_A \overrightarrow{B} \cdot d \overrightarrow{S}\], where \(N\) is the number of loops, \(i\) is the current, \(R\) is reluctance, \(B\) is the flux density, and \(A\) is the cross sectional area

flux linkage : \(\lambda = N \phi = Li\), where \(N\) is the number of loops

MMF drop : \[F = Ni = \phi R = \oint \overrightarrow{H} \cdot d \overrightarrow{l}\]

\(V = \frac{d \lambda}{dt}\)

co energy : \[W_c = \left[ \int_{0}^{i_{1,f}} \lambda_1 di_1 \right]_{\substack{i_2 = 0 \\ ...}} + \dots + \left[ \int_{0}^{i_{n,f}} \lambda_n di_n \right]_{\substack{\dots \\ i_{n-1} = i_{n-1,f}}} = \frac{1}{2} i^T L i\]

conservative system : \[\left[ \begin{matrix} \lambda_1 \\ \vdots \\ \lambda_n \end{matrix} \right] = \underbrace{\left[ \begin{matrix} L_{11} & \dots & L_{1n} \\ \vdots & \ddots & \vdots \\ L_{n1} & \dots & L_{nn} \end{matrix} \right]}_{\text{inductance matrix}} \left[ \begin{matrix} i_1 \\ \vdots \\ i_n \end{matrix} \right]\]

if the inductance matrix is diagonally symmetric, then the system is conservative because \[\frac{d \lambda_j}{d i_k} = \frac{d \lambda_k}{d i_j}\]

force : \[f_e = \frac{1}{2} \frac{dL}{dx} i^2\] torque : \[T_e = \frac{1}{2} \frac{d \lambda}{d \theta_r} i^2\]

nick north : \[\frac{(\overrightarrow{H} \times \overrightarrow{C}) \times \hat{z}}{\left|(\overrightarrow{H} \times \overrightarrow{C}) \times \hat{z}\right|}\]

exam

• 27 stepper motor notes to end
• dc machines
• no lecture set 5