Tensor Formulation of Electric Circuit Theory: Geometric Origin of Circuit Laws

Tensor'Formulation'of'Electric'Circuit'Theory:'Geometric' Origin'of'Circuit'Laws' Se#Kyun#Kwon# Department#of#Physics,#Pohang#University#of#Science#and#Technology,# Pohang#37673,#Republic#of#Korea# (Correspondence:#[email protected])# # Abstract' Classical#electric#circuit#theory#represents#voltages,#currents,#and#impedances#using# complex#numbers,#a#convention#adopted#historically#for#algebraic#convenience#rather# than#physical#necessity.# Here#we#develop#a#complete#real-tensor'formulation#of#circuit#theory#in#which# voltages#and#currents#are#genuine#vectors#in#a#two-dimensional#real#space,#while# impedances#are#second-order#tensors#composed#of#an#isotropic#scaling#operator#and#the# antisymmetric#generator#of#the#rotation#group# SO(2) .# Within#this#framework,#electrical#circuit#operation#is#redefined#as#a'local–global' geometric'synchronization:#Kirchhoff’s#current#law#emerges#as#a#local#divergence-free# condition,#while#Kirchhoff’s#voltage#law#arises#as#a#global#holonomy#constraint#on# closed#loops.# The#traditional#complex#impedance# 𝑍=𝑅+𝑗𝑋 #is#replaced#by#the#tensor# 𝐙=𝑅𝕀+𝑋𝕁, 𝑍01=𝑅𝛿01+𝑋𝐽01, ( 𝛼,𝛽=1,2 )# where# 𝕀 # is#the#identity#and# 𝕁 #is#the#canonical#90°#rotation#tensor.# We#define#a#mapping# Φ:9ℂ→ℝ=×=, Φ(𝑥+𝑗𝑦)=𝑥𝕀+𝑦𝕁 # which#establishes#an# algebra# isomorphism# between# complex# numbers# and# the# twodimensional#real#subalgebra#spanned#by# 𝕀 #and# 𝕁 .# This#demonstrates#that# the# complex# formalism# is#merely# the# algebraic# projection# of# a# richer#real-geometric#structure.# The#fundamental'circuit'law#is#expressed#as#the#coordinate-invariant#tensor#equation,# # 𝐕=𝐙 ( 𝐈 ) , 𝑉0= D 𝑍01𝐼1, # constituting#a#genuine#geometric#physical#law.# Phase#shift,#active#and#reactive#power,#resonance,#and#impedance#matching#emerge# naturally#as#geometric#phenomena#in# ℝ= .# Power#flow#is#encoded#in#the#power'tensor# 𝐓=𝐕⨂𝐈9, 𝑇01 =𝑉0𝐼1, # whose#symmetric#part#describes#dissipative#transfer#of#energy#and#antisymmetric#part# encodes#reversible#oscillatory#exchange.# This#tensor#formulation#reveals#that#complex#AC#analysis#is#not#intrinsically#complexvalued#physics#but#a#compressed#representation#of#real#two-dimensional#geometry,# offering#a#physically#transparent#and#systematically#extensible#foundation#for#circuit# theory.# # 1.'Introduction' Since#the#pioneering#work#of#Heaviside#and#Steinmetz#in#the#late#nineteenth#century,# complex#numbers#have#served#as#the#dominant#language#of#AC#circuit#analysis.# The#representation# 𝑍=𝑅+𝑗𝑋 # has#proven#remarkably#effective#for#computation,#yet#it#obscures#the#underlying# geometric#and#physical#structure#of#sinusoidal#systems.# In#standard#formulations,#the#imaginary#unit# 𝑗 #is#treated#as#an#abstract#algebraic#symbol# rather#than#a#physical#operation,#and#the#relations#among#resistance,#reactance,#phase# shift,#energy#flow,#and#power#decomposition#remain#embedded#in#algebraic#shorthand# rather#than#expressed#as#geometric#entities.# Complex#numbers#were#adopted#for#their#algebraic#convenience—not#because#AC# circuits#are#intrinsically#complex-valued.# In#this#work,#we#show#that#the#complex#representation#is#not#fundamental#to#AC#circuit# theory.# Instead,#it#emerges#from#a#real,#two-dimensional#tensor'geometry#in#which#voltages# and#currents#are#vectors#in# ℝ= #and#the#imaginary#unit#is#the#canonical#90°#rotation# operator.# Impedance#is#not#a#scalar#but#a#genuine#second-order#tensor#whose#symmetric#and# antisymmetric#parts#describe#dissipation#and#rotation.# We#construct#an#explicit#algebra#isomorphism#between#the#field#of#complex#numbers# and#a#two-dimensional#real#matrix#subalgebra,#showing#that#the#traditional#phasor# formalism#is#simply#a#compressed#notation#for#this#tensor#algebra.# Within#this#geometric#framework,#the#fundamental'circuit'law#assumes#the# coordinate-invariant#tensor#form# 𝐕=𝐙 ( 𝐈 ) , 𝑉0= D 𝑍01𝐼1. # which#holds#in#any#orthonormal#basis#of# ℝ= .# This#formulation#restores#the#geometric#content#of#AC#circuit#theory#and#provides# transparent#interpretation#of#phase,#resonance,#reactive#energy,#and#power#flow.# Furthermore,#the#tensor#formulation#offers#a#natural#platform#for#extending#circuit# theory#to#nonlinear#elements,#three-phase#machines,#distributed#systems,#and#realgeometric#physics.# # 2.'Real-Geometric'Representation'of'Phasors' A#complex#voltage#phasor# 𝑉=𝑉J+𝑗𝑉K # is#identified#with#a#real#two-dimensional#vector# 𝐕=(𝑉J, 𝑉K)M. # Similarly,#a#current#phasor#is# 𝐈=(𝐼J, 𝐼K)M. # We#introduce#two#fundamental#rank-2#tensors#on# ℝ= .# The#Euclidean#metric#tensor#in# ℝ= #is# 𝕀=𝛿01= N 1 0 0 1 P , # and#the#canonical#generator#of#90°#rotation#is# 𝕁=𝐽01= N 0 −1 1 0 P , 𝕁==−𝕀, 𝑒S𝕁 = N cos𝜃 −sin𝜃 sin𝜃cos𝜃 P . # Multiplication#by#the#imaginary#unit#corresponds#to#the#action#of# 𝕁 ,# 𝑗𝐼 9 ⟷ 9 𝕁 ( 𝐈 ) , D 𝐽01𝐼1. # Thus,#the#imaginary#unit#is#not#an#algebraic#symbol#but#a#concrete#linear#transformation# of#the#vector#space.# The#complex#plane#is#simply#the#real#vector#space# ℝ= #equipped#with#the#rotation# tensor# 𝕁 .# # 3.'Impedance'as'a'Second-Order'Tensor' The#classical#scalar#impedance# 𝑍=𝑅+𝑗𝑋 # is#naturally#lifted#to#the#impedance#tensor# 𝐙=𝑅𝕀+𝑋𝕁, 𝑍01=𝑅𝛿01+𝑋𝐽01. # Explicitly,# 𝐙=𝑍01= N 𝑅 −𝑋 𝑋 𝑅 P . # This#representation#has#a#clear#geometric#interpretation.# The#term# 𝑅𝕀 # is#a#symmetric#isotropic#scaling#tensor#representing#dissipation,#whereas# the#term# 𝑋𝕁 'is#an#antisymmetric#rotation#tensor#generating#a#vector#orthogonal#to#the# input.# Because# 𝕁 #generates#the#Lie#algebra# 𝔰𝔬 ( 2 ),#the#reactance#is#fundamentally#a#rotation# rate#in#the#voltage-current#plane.# The#fundamental#circuit#law#becomes#the#tensor#equation# 𝐕=𝐙 ( 𝐈 ) =𝑅𝐈+𝑋𝕁 ( 𝐈 ) , 𝑉0= D 𝑍01𝐼1=𝑅 𝐼0+𝑋 D 𝐽01𝐼1. # Voltage#is#therefore#the#vector#sum#of#a#component#parallel#to#the#current# 𝑅𝐈 ,#which# transfers#dissipative#power,#and#a#component#orthogonal#to#the#current# 𝑋𝕁 ( 𝐈 ),#which# participates#only#in#reactive#energy#exchange.# # 4.'Algebraic'Isomorphism'between'Complex'Numbers'and'Tensor'Algebra' We#now#formalize#the#statement#that#the#complex#algebra# ℂ #is#isomorphic#to#a#real# matrix#subalgebra#generated#by# 𝕀 #and# 𝕁 .# Define#the#mapping# Φ:9ℂ→ℝ=×=,999999999Φ(𝑥+𝑗𝑦)=𝑥𝕀+𝑦𝕁, # where# 𝕀= N 1 0 0 1 P , 𝕁= N 0 −1 1 0 P . # 4.1'Linearity' For#complex#numbers# 𝑧_=𝑥_+𝑗𝑦_ #and# 𝑧==𝑥=+𝑗𝑦= ,# Φ(𝑧_+𝑧=)=(𝑥_+𝑥=)𝕀+(𝑦_+𝑦=)𝕁=Φ(𝑧_)+Φ(𝑧=). # Thus,# Φ #is#additive.# Homogeneity#with#respect#to#real#scalars#follows#immediately,# Φ ( 𝑐𝑧 ) =𝑐Φ ( 𝑧 ) , 𝑐∈ℝ. # # 4.2'Multiplicativity' The#product#in# ℂ #is# 𝑧_𝑧==(𝑥_+𝑗𝑦_)(𝑥=+𝑗𝑦=)=(𝑥_𝑥=−𝑦_𝑦=)+𝑗(𝑥_𝑦=+𝑦_𝑥=). # On#the#matrix#side,# Φ(𝑧_)Φ(𝑧=)= ( 𝑥_𝕀+𝑦_𝕁 )( 𝑥=𝕀+𝑦=𝕁 ) 9=(𝑥_𝑥=−𝑦_𝑦=)𝕀+(𝑥_𝑦=+𝑦_𝑥=)𝕁, # because# 𝕁==−𝕀 #and# 𝕀 #commutes#with# 𝕁 .# Therefore# Φ(𝑧_𝑧=)=Φ(𝑧_)Φ(𝑧=), # and# Φ #is#an#algebra'homomorphism.# # 4.3'Isomorphism'onto'a'Subalgebra' The#image#of# Φ #is#exactly#the#two-dimensional#real#subspace# 𝒜 ={𝑥𝕀+𝑦𝕁∣𝑥,𝑦∈ℝ}⊂ℝ𝟐×𝟐. # 𝒜 #is#closed#under#matrix#addition#and#multiplication,#and# Φ #is#clearly#injective.# Hence# Φ #is#an#algebra'isomorphism# ℂ≅𝒜. # This#proves#that#classical#complex#AC#theory#is#a#special#case#of#real#tensor#algebra# on# ℝ𝟐 .# Complex#numbers#are#simply#a#convenient#notation#for#matrices#of#the#form# 𝑥𝕀+𝑦𝕁 .# # 5.'Coordinate'Invariance'and'Physical'Fundamentality' In#the#tensor#framework,#voltages#and#currents#are#vectors,#and#impedance#is#a#secondorder#tensor.# Under#a#orthonormal#transformation#of#basis#represented#by#a#rotation#matrix# 𝑅01∈ SO(2) ,# 𝑉 i 0= D 𝑅01𝑉1, 𝐼 j 0= D 𝑅01𝐼1, # and#the#impedance#tensor#transforms#as# 𝑍 j 01= D 𝑅0k𝑅1l𝑍kl # The#circuit#law,# 𝐕=𝐙 ( 𝐈 ) , 𝑉0= D 𝑍01𝐼1, # retains#its#form#in#the#new#basis#as# 𝐕 m =𝐙 i( 𝐈 j) , 𝑉 i 0= D 𝑍 j 01𝐼 j 1. # Thus,#the#tensor#equation#is#coordinate-invariant#establishing#that#the#fundamental# circuit#law#is#a#physical'law#independent#of#the#particular#axis#orientation#in# ℝ𝟐 .# By#contrast,#the#complex#equation# 𝑉=𝑍𝐼 # implicitly#assumes#a#specific#identification#of# the#real#and#imaginary#axes#with#the#chosen#coordinate#axes#in# ℝ𝟐 .# Changing#the#basis#corresponds#to#a#nontrivial#transformation#of#the#complex# representation.# The#tensor#framework#is#therefore#more'fundamental;#it#encodes#the#geometry#and# physics#in#a#basis-independent#manner,#while#the#complex#notation#corresponds#to#a# particular#coordinate#choice.# # 6.'Geometry'of'Circuit'Phenomena' 6.1'Phase'shift' The#phase#angle# 𝜙 #between#voltage#and#current#is#simply#the#geometric#angle#between# the#vectors# 𝐕 #and# 𝐈 'in# ℝ𝟐 .# For#a#single#impedance,# 𝐙=𝑅𝕀+𝑋𝕁= N 𝑅 −𝑋 𝑋 𝑅 P , # we#may#factor# 𝐙 #into#its#magnitude#and#rotation#components,# 𝐙= o 𝑅=+𝑋= p cos𝜙 −sin𝜙 sin𝜙cos𝜙 q = o 𝑅=+𝑋=𝑒r𝕁. # Here,# 𝜙=tanu_ ( 𝑋 𝑅 ⁄) ,cos𝜙= 𝑅 √ 𝑅=+𝑋=,sin𝜙= 𝑋 √ 𝑅=+𝑋=9. # This#decomposition#shows#that#the#phase#angle#is#determined#by#the#ratio#of#the# antisymmetric#and#symmetric#components#of#the#impedance#tensor.# Thus,#the#phase#shift#in#AC#circuits#is#not#an#abstract#complex-number#operation,#but# the#physical#rotation#generated#by#the#antisymmetric#tensor# 𝑋𝕁 # relative#to#the# symmetric#part# 𝑅𝕀 .# The#familiar#phase#lag#or#phase#lead#arises#from#the#geometric#action#of#the#rotation# generator# 𝕁 #acting#on#the#current#vector.# # 6.2'Resonance'as'vanishing'antisymmetric'part' For#a#series# 𝑅𝐿𝐶 # circuit,#the#frequency-dependent#reactance#is# 𝑋 ( 𝜔 ) =𝜔𝐿−1 𝜔𝐶9, # and#the#impedance#tensor#is# 𝐙 ( 𝜔 ) =𝑅𝕀+𝑋 ( 𝜔 ) 𝕁, 𝑍01(𝜔)=𝑅𝛿01+𝑋(𝜔) 𝐽01. # Resonance#occurs#when# 𝑋(𝜔{)=0, # so#that# 𝐙 ( 𝜔{ ) =𝑅𝕀, 𝑍01(𝜔{)=𝑅𝛿01. # Geometrically,#the#rotation#component#vanishes,#and#the#voltage#becomes#collinear#with# the#current.# Resonance#is#therefore#the#condition#that#the#antisymmetric'part'of'the'impedance' tensor'vanishes,#leaving#a#purely#symmetric#scaling#operator.# # 6.3'Power'tensor'and'energy'interpretation' Define#the#power'tensor' 𝐓=𝐕⨂𝐈9, 𝑇01 =𝑉0𝐼1, # Decompose#it#into#symmetric#and#antisymmetric#parts:# 𝐓|=1 2 ( 𝐓+𝐓M ) , 𝑇 ( 01 ) =1 2 } 𝑇01 +𝑇10 ~ , # 𝐓•=1 2 ( 𝐓−𝐓M ) , 𝑇 [ 01 ] =1 2 } 𝑇01 −𝑇10 ~ . # # The#active'(real)'power#is# 𝑃=⟨𝐕,𝐈⟩= D 𝑉0𝐼0=𝑉J𝐼J+𝑉K𝐼K. # This#can#be#expressed#as#the#trace#of#the#symmetric#part#of#the#power#tensor,# 𝑃=tr ( 𝐓 ) = D 𝑇 ( 00 ) . # The#reactive'power#is# 𝑄=⟨𝐕,𝕁 ( 𝐈 ) ⟩=− ( 𝐕×𝐈 ) ∙𝐳 ‰ = D 𝐽01𝑉0𝐼1=− } 𝑉J𝐼K−𝑉K𝐼J ~ . # Using#the#Levi-Civita#symbol# 𝜀01 # #(with# 𝜀_= =1 ),#we#have# 𝑄=− D 𝜀01𝑇01. # Since# 𝜀01 #is#antisymmetric,#this#contraction#selects#the#antisymmetric#part#of# 𝐓 ,# 𝑄=− D 𝜀01𝑇 [ 01 ] . # Thus,# 𝑃 # arises#from#the#symmetric'part#of# 𝐓 #and#represents#net'energy'transfer'and' dissipation.# On#the#contrary,# 𝑄 # arises#from#the#antisymmetric'part#of# 𝐓 #and#represents# oscillatory'energy'exchange#between#electric#and#magnetic#fields,#or#between#storage# elements,#with#no#net-work#over#a#cycle.# The#geometric#orthogonality# ⟨𝕁 ( 𝐈 ) ,𝐈⟩=0 # explains#why#reactive#power#does#not#contribute#to#net#energy#transfer.# # 6.4'Circuit'Operation'as'Local–Global'Geometric'Synchronization' The#operation#of#electrical#circuits#can#be#interpreted#as#a#synchronization#of#two# fundamental#geometric#principles:#local'conservation'laws#and#global'topological' constraints.# These#principles#are#embodied#in#Kirchhoff’s#current#law#(KCL)#and#Kirchhoff’s#voltage# law#(KVL),#respectively,#which#govern#the#behavior#of#currents#and#voltages#in#electrical# circuits.# In#this#section,#we#present#circuit#operation#as#a#structural'synchronization#between# local#divergence-free#conditions#and#global#holonomy#conditions.# Local&conservation—KCL&as&a&divergence-free&condition& Kirchhoff’s#current#law#enforces#the#local#conservation#of#charge#at#each#node#of#a# circuit,#implying#that#the#net#current#entering#and#exiting#any#node#must#vanish.# This#condition#is#expressed#in#continuum#form#as#the#divergence-free#constraint,# 𝛁⋅𝐉=0. # When#integrated#over#a#small#control#volume#surrounding#a#node,#this#equation#yields# D 𝐈Ž=0, # where# 𝐈Ž # denote#the#currents#flowing#through#the#branches#connected#to#the#node.# Geometrically,#KCL#can#be#viewed#as#a#pointwise'constraint#on#the#current#field,# ensuring#the#local#balance#of#charge.# Global&consistency—KVL&as&a&holonomy&condition&