Measuring length with polar coordinates.

I have polar coordinate functions that describe segments of a coil using Z, R and phi as the variables. I want to measure how long the coil is using these functions. Here is an example:

[Mathieu NOE]

hello

convert your polar coordinates in cartesian x,y,z

then you know that the curve length increment ds is given by :

ds = sqrt(dx² + dy² + dz²)

compute dx,dy,dz from x,y,z using diff function and then do the summation

s = sum(ds)

Answer 1

[Milan Bansal]

Hi Benjamin,

As per my understanding, you want to calculate the length of the coil using polar coordinates (cylindrical coordinates). The coordinates of the coil "Z", "R" and "Phi" are described as function of parameter "alpha" which varies from 0 to "alpha_c".

To measure the length of a coil represented in cylindrical coordinates as functions of a parameter "alpha", specify the functions Z_func(alpha) , R_func(alpha), and phi_func(alpha) that describe the coil. The length of the coil can be calculated by integrating the arc length differential along the path parameterized by "alpha".

Please refer to the following steps to calculate the length of the coil using polar coordinates:

% 1. Define alpha as symbolic expression 
syms alpha; 

% 2. Define the symbolic functions for Z, R and phi 
% Replace these with your actual functions 
Z = Z_func(alpha); % Example function for Z 
R = R_func(alpha); % Example function for R 
phi = sin(alpha); % Example function for phi 

% 3. Compute the derivatives of Z, R, and phi with respect to alpha using "diff" function. 
dZ_dalpha = diff(Z, alpha); 
dR_dalpha = diff(R, alpha); 
dphi_dalpha = diff(phi, alpha); 

% 4. Define the differential arc integral. 
integrand = sqrt(dR_dalpha^2 + (R * dphi_dalpha)^2 + dZ_dalpha^2); 


% 5. Calculate the symbolic integration. "vpaintegral" function calculates approximate definite integrals. 
alpha1 = 0; % Replace with your actual starting value of alpha 
alpha2 = 2*pi; % Replace with your actual ending value of alpha 

% 6. Compute arc length 
arc_length_sym = vpaintegral(integrand, alpha, alpha1, alpha2); 

% Convert the symbolic arc length to a numeric value 
arc_length = double(arc_length_sym); 

Please refer to the following documentation links to learn more about diff function and “vpaintegral” function.

• https://www.mathworks.com/help/matlab/ref/diff.html
• https://www.mathworks.com/help/symbolic/sym.vpaintegral.html

Hope this helps!

[Benjamin]

Fantastic works well thank you so much. I checked this with my own less effecient code, and got very close answers.

[Benjamin]

@Milan Bansal if the range instead of an angle is a length, how can i configure this code to calculate the length over a variable length range. Ie the variable lin =(0;10) mm

[Benjamin]

i tried this: @Milan Bansal

%segment 3

% 1. Define alpha as symbolic expression

syms l;

% 2. Define the symbolic functions for Z, R and phi

% Replace these with your actual functions

Z = Z0+l*sin(alphaCR)*cos(alphaS); % Example function for Z

R = R0+(Z-Z0)*tan(alphaS); % Example function for R

phi = phi0+(l/Rt)*cos(alphaCR); % Example function for phi

% 3. Compute the derivatives of Z, R, and phi with respect to alpha using "diff" function.

dZ_dl = diff(Z, l);

dR_dl = diff(R, l);

dphi_dl = diff(phi, l);

% 4. Define the differential arc integral.

integrand = sqrt(dR_dl^2 + (R * dphi_dl)^2 + dZ_dl^2);

% 5. Calculate the symbolic integration. "vpaintegral" function calculates approximate definite integrals.

l1 = 0; % Replace with your actual starting value of alpha

l2 = lin; % Replace with your actual ending value of alpha

% 6. Compute arc length

arc_length_sym = vpaintegral(integrand, l, l1, l2);

% Convert the symbolic arc length to a numeric value

arc_length3 = double(arc_length_sym)

Z0=Z0+lin*sin(alphaCR)*cos(alphaS);

R0=R0+(Z-Z0)*tan(alphaS);

phi0=phi0+(lin/Rt)*cos(alphaCR);

but I get the error: Error using symengine

Unable to convert expression containing symbolic variables into double array. Apply 'subs' function first to substitute values for

variables.

Error in sym/double (line 872)

Xstr = mupadmex('symobj::double', S.s, 0);

Related documentation

[Dyuman Joshi]

There are undefined variables in your code, so we can not run it and reproduce the error you got.

%segment 3
% 1. Define alpha as symbolic expression 
syms l; 
% 2. Define the symbolic functions for Z, R and phi 
% Replace these with your actual functions 
Z = Z0+l*sin(alphaCR)*cos(alphaS); % Example function for Z 

Unrecognized function or variable 'Z0'.

R = R0+(Z-Z0)*tan(alphaS); % Example function for R
phi = phi0+(l/Rt)*cos(alphaCR); % Example function for phi 
% 3. Compute the derivatives of Z, R, and phi with respect to alpha using "diff" function. 
dZ_dl = diff(Z, l); 
dR_dl = diff(R, l); 
dphi_dl = diff(phi, l); 
% 4. Define the differential arc integral. 
integrand = sqrt(dR_dl^2 + (R * dphi_dl)^2 + dZ_dl^2); 
% 5. Calculate the symbolic integration. "vpaintegral" function calculates approximate definite integrals. 
l1 = 0; % Replace with your actual starting value of alpha 
l2 = lin; % Replace with your actual ending value of alpha 
% 6. Compute arc length 
arc_length_sym = vpaintegral(integrand, l, l1, l2)
% Convert the symbolic arc length to a numeric value 
arc_length3 = double(arc_length_sym)
Z0=Z0+lin*sin(alphaCR)*cos(alphaS);
R0=R0+(Z-Z0)*tan(alphaS);
phi0=phi0+(lin/Rt)*cos(alphaCR);

[Benjamin]

%Value Inputs: All inputs are in radians and millimetres

Qs=96; %no. of stator slots

Lfe=400; %core length

hw=1; %height of wedge

ho=1; %height of slot opening

D1=735; %core id

hd=1;%middle strip depth

hwi=1;%wedge packer depth

hi=1;%aux wedge packer depth

hc=22.106;%coil stack height

bc=11.328;%voil stack width

y1s=10;%no. slots per coil span

bs=(pi*D1*y1s)/Qs;% total coil span

Rp=40;%coil exit bend pin radius

Rl=25;%radius coil loop

llin=10;%length between stator and coil curvature

Rc=Rp+0.5*bc;%Coil slot exit bend radius

alphaedegrees=10;%angle of bevelling of coil loop

alphae= deg2rad(alphaedegrees);%radians version

bi=4.5; %legspace

taus=(pi*D1)/Qs;%slot pitch

alphac=asin((bc+bi)/taus);%angle of coil involutes

alphaprimec=(pi/2)-alphac;%central angle of coil curvature

alphaS=(2*pi)/Qs;%extension of winding overhang

lin=(bs-4*Rc*(1-sin(alphac))-2*Rl*sin(alphae))/(2*cos(alphaS));%length of involute

alphaCR=atan(tan(alphac)*cos(alphaS));%real angle of the coil involute

%segment 1

Z=llin;

Rt=D1/2 + ho+hw+hwi+hi+hc/2;

R=Rt;

phi=0;

Z0=Z;

R0=R;

phi0=phi;

arclength1=llin

%segment 2

% 1. Define alpha as symbolic expression

syms alpha;

% 2. Define the symbolic functions for Z, R and phi

% Replace these with your actual functions

Z = Z0+Rc*sin(alpha)*cos(alphaS); % Example function for Z

R = R0+(Z-Z0)*tan(alphaS); % Example function for R

phi = phi0+(Rc/Rt)*(1-cos(alpha)); % Example function for phi

% 3. Compute the derivatives of Z, R, and phi with respect to alpha using "diff" function.

dZ_dalpha = diff(Z, alpha);

dR_dalpha = diff(R, alpha);

dphi_dalpha = diff(phi, alpha);

% 4. Define the differential arc integral.

integrand = sqrt(dR_dalpha^2 + (R * dphi_dalpha)^2 + dZ_dalpha^2);

% 5. Calculate the symbolic integration. "vpaintegral" function calculates approximate definite integrals.

alpha1 = 0; % Replace with your actual starting value of alpha

alpha2 = alphaprimec; % Replace with your actual ending value of alpha

% 6. Compute arc length

arc_length_sym = vpaintegral(integrand, alpha, alpha1, alpha2);

% Convert the symbolic arc length to a numeric value

arc_length2 = double(arc_length_sym)

Z0=Z0+Rc*sin(alphaprimec)*cos(alphaS);

R0=R0+(Z-Z0)*tan(alphaS);

phi0=phi0+(Rc/Rt)*(1-cos(alphaprimec));

%segment 3

% 1. Define alpha as symbolic expression

syms l;

% 2. Define the symbolic functions for Z, R and phi

% Replace these with your actual functions

Z = Z0+l*sin(alphaCR)*cos(alphaS); % Example function for Z

R = R0+(Z-Z0)*tan(alphaS); % Example function for R

phi = phi0+(l/Rt)*cos(alphaCR); % Example function for phi

% 3. Compute the derivatives of Z, R, and phi with respect to alpha using "diff" function.

dZ_dl = diff(Z, l);

dR_dl = diff(R, l);

dphi_dl = diff(phi, l);

% 4. Define the differential arc integral.

integrand = sqrt(dR_dl^2 + (R * dphi_dl)^2 + dZ_dl^2);

% 5. Calculate the symbolic integration. "vpaintegral" function calculates approximate definite integrals.

l1 = 0; % Replace with your actual starting value of alpha

l2 = lin; % Replace with your actual ending value of alpha

% 6. Compute arc length

arc_length_sym = vpaintegral(integrand, l, l1, l2);

% Convert the symbolic arc length to a numeric value

arc_length3 = double(arc_length_sym)

Z0=Z0+lin*sin(alphaCR)*cos(alphaS);

R0=R0+(Z-Z0)*tan(alphaS);

phi0=phi0+(lin/Rt)*cos(alphaCR);