Lecture 14
Time Harmonic Fields
In this lecture you will learn:
• Complex mathematics for time-harmonic fields
• Maxwell’s equations for time-harmonic fields
• Complex Poynting vector
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Time-Harmonic Fields
E and H-fields for a plane wave are (from last lecture): r k = kx xˆ + ky yˆ + kzzˆ r r r r E()r ,t = nˆ Eo cos(ω t − k .r ) r r 2 2 2 2 ⇒ k .k = k = kx + ky + kz r r E r k .nˆ = 0 H()rr,t = ()kˆ × nˆ o cos()ω t − k .rr ηo ω = k c
• Fields for which the time variation is sinusoidal are called time-harmonic fields.
• Plane waves are just one example of time-harmonic fields
• In the rest of this course, 95% of the material will deal with time-harmonic fields
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
1 Time-Harmonic Signals in Circuits – Sinusoidal Steady State
Consider an RC circuit driven by a sinusoidal voltage source: R I(t ) + - + VR (t ) 1 V ()t = V cos (ω t ) ~ V (t ) o C C j ω C -
Remember phasors from ECE210…..
j ω t j ω t j ω t V ()t = Re[ Vo e ] VR ()t = Re[ VR (ω )e ] I()t = Re[ I (ω )e ]
j ω RC j ω C ⇒ V ()ω = V I()ω = V R o 1+ j ω RC o 1+ j ω RC
Time-average power dissipation in the resistor:
2 2 1 * Vo ⎡ (ω R C) ⎤ VR ()()t I t = Re[]VR ()ω I ()ω = ⎢ 2 ⎥ 2 2R ⎢⎣1+ ()ω RC ⎦⎥
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Time-Harmonic Fields and Complex Notation
Basic idea:
If the time-variation of fields is known a-priori to be sinusoidal (i.e. the fields are known to be time-harmonic) then, in order to simplify the math, one may not carry around the time dependence explicitly in calculations
Lets look at plane waves as an example to see how the complex notation can be used to factor out the sinusoidal time dependence
Some useful trigonometric Identities to recall before we start: j j e θ + e− θ jθ cos()θ = e = cos(θ ) + j sin(θ ) 2 e jθ − e− jθ sin()θ = e− jθ = cos(θ ) − j sin(θ ) 2 j Expression for the E-field of a plane wave in complex notation: r r r r E()r ,t = nˆ Eo cos(ω t − k .r )
r r r r r e j(ω t −k .r ) + e− j(ω t −k .r ) r r E()rr,t = nˆ E = Re⎡nˆ E e j()ω t −k .r ⎤ o 2 ⎢⎣ o ⎦⎥
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
2 Time-Harmonic Fields and Vector Phasors For the E-field of a plane wave we had… r r r r r E()rr,t = nˆ E cos(ω t − k .rr) ⇒ E()rr,t = Re⎡nˆ E e j(ω t −k .r )⎤ o ⎢⎣ o ⎦⎥ Do a little more manipulation … r r r E()rr,t = Re⎡nˆ E e j(ω t −k .r )⎤ ⎣⎢ o ⎦⎥ r r = Re⎡nˆ E e− j k .r e j ω t ⎤ ⎣⎢ o ⎦⎥
r r r r r r j ω t where: r r − j k .r E()r ,t = Re[ E ()r e ] E()r = nˆ Eo e
r The quantity E () rr , which is a time-independent complex vector, is a vector phasor for the plane wave
r In the book, the vector phasor has an additional under-line and written as: E ()rr
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Complex Notation For the E-field of a plane wave we had… r r r r r r r r j ω t E()r ,t = nˆ Eo cos(ω t − k .r ) ⇒ E()r ,t = Re[ E ()r e ]
Now generalize to all time-harmonic fields:
• All time-harmonic fields (not just plane waves) can be written in the form: r r E()rr,t = Re[ E ()rr e j ω t ] r where E () r r is a complex time-independent vector phasor r • Given a vector phasorE(rr) for a time-harmonic field, one can find the actual time-dependent field as follows: r r E()rr,t = Re[ E ()rr e j ω t ]
Example: Suppose I give you the following vector phasor for a plane wave: r E()rr = xˆ A e− j k z Then you can find the actual time-dependent E-field as follows: r r E()rr,t = Re[][]E ()rr e j ω t = Re xˆ A e− j k z e j ω t = xˆ Acos()ω t − k z
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
3 Maxwell’s Equations for Phasors - I
Let the time-harmonic E and H-fields be: r r r r E()rr,t = Re[ E ()rr e j ω t ] H()rr,t = Re[ H ()rr e j ω t ] Assume that the time-variations of charge density and current density are also sinusoidal: r r ρ()rr,t = Re[ ρ ()rr e j ω t ] J()rr,t = Re[ J ()rr e j ω t ]
Now we substitute these expressions in Maxwell’s equations one by one
Gauss’ Law: r r r ∇ .εo E()r ,t = ρ(r ,t ) r r j ω t r j ω t ⇒ Re[]∇ .εo E()r e = Re[]ρ()r e The only way the above can be true for all time is if: r r r ∇ .εo E()r = ρ ()r (1) Gauss’ Law for the Magnetic Field:
r r r r (2) ∇ . µo H()r ,t = 0 ⇒ ∇ . µo H(r ) = 0
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Maxwell’s Equations for Phasors - II The time-harmonic E and H-fields are: r r r r E()rr,t = Re[ E ()rr e j ω t ] H()rr,t = Re[ H ()rr e j ω t ] r r r r j ω t ρ()rr,t = Re[ ρ ()rr e j ω t ] J()r ,t = Re[ J ()r e ] Faraday’s Law: r r ∂ µ H(rr,t ) ∇ × E()rr,t = − o ∂t r r j ω t r r j ω t ⇒ Re[∇ × E()r e ]= Re[− j ω µo H()r e ] The only way the above can be true for all time is if: r r r r ∇ × E()r = − j ω µo H(r ) (3) Ampere’s Law: r r r ∂ ε E(rr,t ) ∇ × H()rr,t = J ()rr,t + o ∂t r r j ω t r v j ω t r r j ω t ⇒ Re[∇ × H()r e ]= Re[J()r e + j ω εo E()r e ] The only way the above can be true for all time is if: r r r v r r ∇ × H()r = J()r + j ω εo E(r ) (4)
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
4 Maxwell’s Equations for Phasors - III Let the time-harmonic E and H-fields be given as: r r r r E()rr,t = Re[ E ()rr e j ω t ] H()rr,t = Re[ H ()rr e j ω t ] r r r r j ω t ρ()rr,t = Re[ ρ ()rr e j ω t ] J()r ,t = Re[ J ()r e ]
Maxwell’s equations for the vector phasors of time-harmonic fields are then:
Gauss’ Law: r r r ∇ .εo E()r = ρ(r ) Gauss’ Law for the Magnetic Field: r r ∇ . µo H()r = 0
Faraday’s Law: r r r r ∇ × E()r = − j ω µo H(r ) Ampere’s Law: r r r v r r ∇ × H()r = J(r ) + j ω εo E(r )
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
Calculations in the Complex Notation Suppose for a plane wave we know the E-field to be: r r r r r j ω t r r − j k .rr E()r ,t = Re[ E ()r e ] where E()r = nˆ Eo e How does one find the vector phasor for the H-field? r r H()rr,t = Re[ H ()rr e j ω t ] Use Faraday’s law for time-harmonic fields: r r r r ∇ × E()r = − j ω µo H(r ) r j r ⇒ H()rr = ∇ × E()rr ω µo r r r j ⎛ − j k .rr ⎞ ⇒ H()r = ∇ × ⎜ nˆ Eo e ⎟ ω µo ⎝ ⎠ r r r j v − j k .rr ⇒ H()r = ()− j k × nˆ Eo e ω µo r r r r k ˆ − j k .r r ⇒ H()r = ()k × nˆ Eo e k = k kˆ ω µo r E r r ⇒ H()rr = ()kˆ × nˆ o e− j k .r ηo
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
5 Complex Poynting Vector Suppose for a plane wave we know the E-field and H-field phasors to be: r r r − j k .rr E()r = nˆ Eo e r E r r H()rr = ()kˆ × nˆ o e− j k .r ηo How does one find the time-average power per unit area carried by the wave? Define a complex Poynting vector as: r r r S()rr = E ()rr × H *(rr)
Claim: The time-average power per unit area is one-half of the real part of the complex Poynting vector Check: r 1 r S()rr,t = Re[]S()rr 2 1 r r = Re[]E()rr × H * ()rr 2 1 ⎡ E2 ⎤ E2 = Re⎢nˆ × ()kˆ × nˆ o ⎥ = kˆ o which is indeed the 2 ⎣⎢ ηo ⎦⎥ 2ηo right answer
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
More Calculations in the Complex Notation - I Example:
Consider a plane wave with E-field of amplitude Eo and pointing in a direction 45-degrees w.r.t. the x-axis (as shown) and traveling in the +z-direction
Write expression for the E-field phasor: y
r r ⎛ xˆ + yˆ ⎞ − j k z E x E()r = ⎜ ⎟ Eo e ⎝ 2 ⎠ H Write expression for the H-field phasor: r j r H()rr = ∇ × E()rr ω µo z r r j ⎛⎛ xˆ + yˆ ⎞ − j k z ⎞ ⇒ H()r = ∇ × ⎜⎜ ⎟ Eo e ⎟ ω µo ⎝⎝ 2 ⎠ ⎠ r r j ⎛ ⎛ xˆ + yˆ ⎞⎞ − j k z r ⇒ H()r = ⎜ − j k zˆ × ⎜ ⎟⎟ Eo e k = k zˆ ω µo ⎝ ⎝ 2 ⎠⎠ r r k ⎛ yˆ − xˆ ⎞ − j k z ⇒ H()r = ⎜ ⎟ Eo e ω µo ⎝ 2 ⎠ r r ⎛ yˆ − xˆ ⎞ Eo − j k z ⇒ H()r = ⎜ ⎟ e ⎝ 2 ⎠ ηo
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
6 More Calculations in the Complex Notation - II y The E-field phasor is: E x r r ⎛ xˆ + yˆ ⎞ − j k z E()r = ⎜ ⎟ Eo e H ⎝ 2 ⎠
The H-field phasor is:
r r ⎛ yˆ − xˆ ⎞ Eo − j k z H()r = ⎜ ⎟ e z ⎝ 2 ⎠ ηo Find the complex Poynting vector: 2 r r r r r * r ⎛ xˆ + yˆ ⎞ ⎛ yˆ − xˆ ⎞ Eo S()r = E ()r × H ()r = ⎜ ⎟ × ⎜ ⎟ ⎝ 2 ⎠ ⎝ 2 ⎠ ηo E2 = zˆ o ηo Find the time-average power per unit area:
r 1 r E2 S()rr,t = Re[]S()rr = zˆ o 2 2ηo
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
ECE 303 – Fall 2007 – Farhan Rana – Cornell University
7