Exercise:
A resistor circuit has the shape of a tetrahedron with a battery on edge a and resistors on the other edges b to f see figure. center includegraphics#image_path:resistor-tetrahedron-# center abcliste abc Choose directions for the currents I_a I_b dots I_f through the six edges of the tetrahedron and derive the s for the four nodes P Q R and S Kirchhoff's node rule. Express I_d I_e and I_f through I_a I_b and I_c. abc Derive the s for the four loops PQR PRS PSQ and QRS Kirchhoff's loop rule. abc For the case of five identical resistors R_b R_c s R_f R find formal expressions for the currents I_a to I_f. abc Determine the currents for a system with Delta V VO R_b R_d RbO and R_c R_e R_f RcO. abcliste

Solution:
abcliste abc The s for the nodes are I_e + I_f I_d quad textnode P labelnode:P I_d + I_e I_a quad textnode Q labelnode:Q I_a I_b + I_c quad textnode R labelnode:R I_b I_c + I_f quad textnode S labelnode:S After some manipulations of the s we find I_d fracI_a + I_b - I_c labeleq:Id I_e fracI_a - I_b + I_c labeleq:Ie I_f I_b - I_c labeleq:If This means that we can express the loop s in b by only three currents I_a I_b and I_c reducing the number of unknown quantities to three. abc With the current directions from a we find the following loop s: Delta V R_e I_e + R_d I_d quad textloop PQR labelloop:PQR -R_e I_e + R_b I_b + R_f I_f quad textloop PRS labelloop:PRS -R_f I_f + R_c I_c - R_d I_d quad textloop PSQ labelloop:PSQ Delta V R_b I_b + R_c I_c quad textloop QRS labelloop:QRS By substituting the expressions for I_d I_e and I_f this can be simplified to Delta V R_d + R_e I_a + R_d - R_e I_b + R_e - R_d I_c labelsimp:PQR -R_e I_a + R_b + R_e + R_f I_b - R_e + R_f I_c labelsimp:PRS -R_d I_a - R_d + R_f I_b + R_c + R_d + R_f I_c labelsimp:PSQ Delta V R_b I_b + R_c I_c labelsimp:QRS abc Substituting R for each of the resistances R_b to R_f yields Delta V R I_a labelsym:PQR -R I_a + R I_b - R I_c labelsym:PRS -R I_a - R I_b + R I_c labelsym:PSQ Delta V R I_b + R I_c labelsym:QRS Solving this system of linear s yields I_a fracDelta VR I_b I_c fracDelta V R and with s refeq:Id to refeq:If I_d I_e fracDelta V R I_f abc With R RO we can write the system of linear s as Delta V R I_a + R I_b - R I_c labelasym:PQR -R I_a + R I_b - R I_c labelasym:PRS - R I_a - R I_b + R I_c labelasym:PSQ Delta V R I_b + R I_c labelasym:QRS By solving this system we find for the currents I_a frac I_ IaP I_b frac I_ IbP I_c frac I_ IcP I_d frac I_ IdP I_e frac I_ IeP I_f -frac I_ IfP with I_ Delta V/R IP. vspacemm The negative sign of I_f tells us that the ased direction is wrong. abcliste