Harmonic Time Dependence

For fields with harmonic time dependence, it is convenient to compute the potentials using their complex phasor representation.

Scalar fields with harmonic time dependence are commonly written in terms of amplitudes and phases. For example, the scalar potential $ϕ$ can be written as:

Figure 1. EQUATION_DISPLAY

\begin{matrix} ϕ = \tilde{ϕ} \cos (ω t + θ_{ϕ}) \end{matrix}

(4244 4251)

where $ϕ$ follows a cosine law with amplitude $\tilde{ϕ}$ , angular frequency $ω$ , and phase $θ_{ϕ}$ .

The angular frequency $ω$ is related to the frequency $f$ through:

Figure 2. EQUATION_DISPLAY

ω = 2 π f

(4245)

Similarly, vector fields with harmonic time dependence have time-harmonic scalar components. For example, the magnetic field

H

can be written with local coordinate components using the matrix-vector notation:

Figure 3. EQUATION_DISPLAY

H = H_{i} i + H_{j} j + H_{k} k ≜ [\begin{matrix} {\begin{matrix} H \end{matrix}}_{i} i \\ \begin{matrix} H_{j} j \\ H_{k} k \end{matrix} \end{matrix}]

(4246)

where

i

j

, and

k

are local coordinate unit vectors and can be omitted, for convenience:

Figure 4. EQUATION_DISPLAY

H ≜ [\begin{matrix} {\begin{matrix} H \end{matrix}}_{i} \\ \begin{matrix} H_{j} \\ H_{k} \end{matrix} \end{matrix}]

(4247)

Each vector component has its own amplitude and phase:

Figure 5. EQUATION_DISPLAY

\begin{matrix} {\begin{matrix} H \end{matrix}}_{i} = {\tilde{H}}_{i} \cos (ω t + θ_{H_{i}}) \\ \begin{matrix} H_{j} = {\tilde{H}}_{j} \cos (ω t + θ_{H_{j}}) \\ H_{k} = {\tilde{H}}_{k} \cos (ω t + θ_{H_{k}}) \end{matrix} \end{matrix}

(4248)

Using the matrix-vector notation, $H$ can be written in terms of its amplitude and phase vectors:

Figure 6. EQUATION_DISPLAY

H = [\begin{matrix} \begin{matrix} {\tilde{H}}_{i} \\ {\tilde{H}}_{j} \end{matrix} \\ {\tilde{H}}_{k} \end{matrix}] ⊙ \cos (ω t + [\begin{matrix} θ_{H_{i}} \\ \begin{matrix} θ_{H_{j}} \\ θ_{H_{k}} \end{matrix} \end{matrix}]) = \tilde{H} ⊙ \cos (ω t + θ_{H})

(4249)

where $⊙$ denotes the Hadamard product, and the amplitude $\tilde{H}$ and phase $θ_{H}$ are:

Figure 7. EQUATION_DISPLAY

\tilde{H} = [\begin{matrix} \begin{matrix} {\tilde{H}}_{i} \\ {\tilde{H}}_{j} \end{matrix} \\ {\tilde{H}}_{k} \end{matrix}], θ_{H} = [\begin{matrix} \begin{matrix} θ_{H_{i}} \\ θ_{H_{j}} \end{matrix} \\ θ_{H_{k}} \end{matrix}]

(4250)

For fields with harmonic time dependence, it is convenient to compute the potentials using their complex phasor representations. Consider potentials with single harmonic time dependence:

Figure 8. EQUATION_DISPLAY

\begin{array}{l} A = \tilde{A} ⊙ \cos (ω_{0} t + θ_{A}) \\ ϕ = \tilde{ϕ} \cos (ω_{0} t + θ_{ϕ}) \end{array}

(4244 4251)

The potentials $A$ and $ϕ$ can be written in terms of their complex phasor representations $\hat{A}$ and $\hat{ϕ}$ :

Figure 9. EQUATION_DISPLAY

\begin{array}{l} A = ℜ [\hat{A}] = ℜ [\tilde{A} {⊙ e}^{i ω_{0} t} e^{i θ_{A}}] = ℜ [{\hat{A}}_{0} e^{i ω_{0} t}] \\ ϕ = ℜ [\hat{ϕ}] = ℜ [\tilde{ϕ} e^{i ω_{0} t} e^{i θ_{ϕ}}] = ℜ [{\hat{ϕ}}_{0} e^{i ω_{0} t}] \end{array}

(4252)

where $i$ is the imaginary unit, defined by $i^{2} = - 1$ , and ${\hat{A}}_{0} = \tilde{A} ⊙ e^{i θ_{A}}$ and ${\hat{ϕ}}_{0} = \tilde{ϕ} ⊙ e^{i θ_{ϕ}}$ are the complex amplitudes.

The time derivative of $A$ is then:

Figure 10. EQUATION_DISPLAY

\frac{\partial A}{\partial t} = ℜ [i ω_{o} \hat{A}]

(4253)

By replacing the potentials and the time derivatives with their complex representations, Eqn. (4241) and Eqn. (4242) become:

Figure 11. EQUATION_DISPLAY

ℜ [\nabla\times {(\hat{μ})}^{- 1} \nabla\times \hat{A} + i ω_{0} \hat{σ} \hat{A}] = ℜ [- \hat{σ} \nabla \hat{ϕ} + {\hat{J}}_{e x}]

(4254)

Figure 12. EQUATION_DISPLAY

\begin{matrix} ℜ [- \nabla \cdot (\hat{σ} \nabla \hat{ϕ})] = ℜ [\nabla \cdot (i ω_{0} \hat{σ} \hat{A}) + {\hat{S}}_{ϕ}] \end{matrix}

(4255)

where ${\hat{J}}_{e x}$ is a user-defined source and ${\hat{S}}_{ϕ} = \nabla\cdot {\hat{J}}_{e x}$ is a user-defined source that accounts for unresolved physics (and is typically zero).

The complex amplitudes ${\hat{A}}_{0}$ and ${\hat{ϕ}}_{0}$ can be written in terms of their real and imaginary parts as:

Figure 13. EQUATION_DISPLAY

\begin{array}{l} {\hat{A}}_{0} = A_{0}^{'} + i A_{0}^{''} \\ {\hat{ϕ}}_{0} = ϕ_{0}^{'} + i ϕ_{0}^{''} \end{array}

(4256)

or in polar form:

Figure 14. EQUATION_DISPLAY

\begin{array}{l} {\hat{A}}_{0} = \tilde{A} ⊙ e^{i θ_{A}} = \tilde{A} ⊙ (\cos θ_{A} + i \sin θ_{A}) \\ {\hat{ϕ}}_{0} = \tilde{ϕ} e^{i θ_{ϕ}} = \tilde{ϕ} (\cos θ_{ϕ} + i \sin θ_{ϕ}) \end{array}

(4257)

with $A_{0}^{'} = ℜ [{\hat{A}}_{0}]$ , $A_{0}^{''} = ℑ [{\hat{A}}_{0}]$ , $ϕ_{0}^{'} = ℜ [{\hat{ϕ}}_{0}]$ , $ϕ_{0}^{''} = ℑ [{\hat{ϕ}}_{0}]$ .

Similarly, the complex potentials themselves can be written in terms of real and imaginary parts as:

Figure 15. EQUATION_DISPLAY

\begin{matrix} \hat{A} = A^{'} + i A^{''} = A + i A^{''} \\ \hat{ϕ} = ϕ^{'} + i ϕ^{''} = ϕ + i ϕ^{''} \end{matrix}

(4258)

and in polar form

Figure 16. EQUATION_DISPLAY

\begin{matrix} \hat{A} = {\hat{A}}_{0} e^{i ω_{0} t} = (A_{0}^{'} + i A_{0}^{''}) (\cos ω_{0} t + i \sin ω_{0} t) = (A_{0}^{'} \cos ω_{0} t - A_{0}^{''} \sin ω_{0} t) + i (A_{0}^{''} \cos ω_{0} t + A_{0}^{'} \sin ω_{0} t) \\ \hat{ϕ} = {\hat{ϕ}}_{0} e^{i ω_{0} t} = (ϕ_{0}^{'} + i ϕ_{0}^{''}) (\cos ω_{0} t + i \sin ω_{0} t) = (ϕ_{0}^{'} \cos ω_{0} t - ϕ_{0}^{''} \sin ω_{0} t) + i (ϕ_{0}^{''} \cos ω_{0} t + ϕ_{0}^{'} \sin ω_{0} t) \end{matrix}

(4259)

The electric current density source is a time-harmonic vector which can be written as the real part of a complex quantity

Figure 17. EQUATION_DISPLAY

J_{e x} = {\tilde{J}}_{e x} ⊙ \cos (ω_{0} t + θ_{J_{e x}}) = ℜ [{\hat{J}}_{e x}]

(4260)

where

Figure 18. EQUATION_DISPLAY

{\hat{J}}_{e x} = {\tilde{J}}_{e x} ⊙ e^{i θ_{J_{e x}}} e^{i ω_{0} t} = {\hat{J}}_{{e x}_{0}} e^{i ω_{0} t} = (J_{{e x}_{0}}^{'} + i J_{{e x}_{0}}^{''}) e^{i ω_{0} t}

(4261)

which can be further extended as

Figure 19. EQUATION_DISPLAY

{\hat{J}}_{e x} = (J_{{e x}_{0}}^{'} \cos ω_{0} t - J_{{e x}_{0}}^{''} \sin ω_{0} t) + i (J_{{e x}_{0}}^{''} \cos ω_{0} t + J_{{e x}_{0}}^{'} \sin ω_{0} t)

(4262)

The electrical conductivity is considered as a time and frequency independent (symmetric tensor) quantity which, to account tor the displacement currents, can be written as:

Figure 20. EQUATION_DISPLAY

\begin{matrix} \hat{σ} = σ^{'} + i σ^{"} = [\begin{matrix} σ_{i i}^{'} & σ_{j i}^{'} & σ_{i k}^{'} \\ σ_{i j}^{'} & σ_{j j}^{'} & σ_{j k}^{'} \\ σ_{i k}^{'} & σ_{j k}^{'} & σ_{k k}^{'} \end{matrix}] + i [\begin{matrix} σ_{i i}^{''} & σ_{i j}^{''} & σ_{i k}^{''} \\ σ_{i j}^{''} & σ_{j j}^{''} & σ_{j k}^{''} \\ σ_{i k}^{''} & σ_{j k}^{''} & σ_{k k}^{''} \end{matrix}] = [\begin{matrix} {\tilde{σ}}_{i i} e^{i θ_{σ_{i i}}} & {\tilde{σ}}_{i j} e^{i θ_{σ_{i j}}} & {\tilde{σ}}_{i k} e^{i θ_{σ_{i k}}} \\ {\tilde{σ}}_{i j} e^{i θ_{σ_{i j}}} & {\tilde{σ}}_{j j} e^{i θ_{σ_{j j}}} & {\tilde{σ}}_{j k} e^{i θ_{σ_{j k}}} \\ {\tilde{σ}}_{i k} e^{i θ_{σ_{i k}}} & {\tilde{σ}}_{j k} e^{i θ_{σ_{j k}}} & {\tilde{σ}}_{k k} e^{i θ_{σ_{k k}}} \end{matrix}] \end{matrix}

(4263)

At low frequencies, the magnetic permeability can be assumed to be a real quantity. Eqn. (4254) then assumes the simple form:

Figure 21. EQUATION_DISPLAY

ℜ [\nabla\times μ^{- 1} \times (A^{'} + i A^{''}) + i ω_{0} (σ^{'} + σ^{''}) (A^{'} + i A^{''})] = ℜ [- (σ^{'} + i σ^{''}) \nabla \hat{ϕ} + (J_{e x}^{'} + i J_{e x}^{''})]

(4264)

Looking at the real parts of the expanded expressions for the time-harmonic complex representations it is seen that, in order to satisfy the differential equations, both real and imaginary parts of these equations with complex magnitudes must be satisfied.

Figure 22. EQUATION_DISPLAY

\nabla\times μ^{- 1} \times (A_{0}^{'} + i A_{0}^{''}) + i ω_{0} (σ^{'} + σ^{''}) (A_{0}^{'} + i A_{0}^{''}) = - (σ^{'} + i σ^{''}) \nabla {\hat{ϕ}}_{0} + (J_{e x_{0}}^{'} + i J_{{e x}_{0}}^{''})

(4265)

By using vector identities and imposing the Coulomb gauge, $\nabla\cdot A = 0$ , the real and imaginary parts of Eqn. (4264) assume the simple form:

Figure 23. EQUATION_DISPLAY

\begin{array}{l} \begin{matrix} - \nabla\cdot (μ^{- 1} \nabla A_{0}^{'}) - ω_{0} σ^{"} A_{0}^{'} - ω_{0} σ^{'} A_{0}^{''} + σ^{'} \nabla ϕ_{0}^{'} - σ^{"} \nabla ϕ_{0}^{''} - J_{e x}^{'} = 0 \end{matrix} \\ \begin{matrix} - \nabla\cdot (μ^{- 1} \nabla A_{0}^{''}) + ω_{0} σ^{'} A_{0}^{'} - ω_{0} σ^{"} A_{0}^{''} + σ^{'} \nabla ϕ_{0}^{''} + σ^{"} \nabla ϕ_{0}^{'} - J_{e x}^{"} \end{matrix} = 0 \end{array}

(4266)

Similarly, the real and imaginary parts of Eqn. (4255) assume the simple form:

Figure 24. EQUATION_DISPLAY

\begin{array}{l} \begin{matrix} - \nabla\cdot (σ^{'} \nabla ϕ_{0}^{'}) + \nabla\cdot (σ^{"} \nabla ϕ_{0}^{''}) + ω_{0} \nabla\cdot (σ^{'} A_{0}^{''}) + ω_{0} \nabla\cdot (σ^{"} A_{0}^{'}) - S_{ϕ}^{'} = 0 \end{matrix} \\ \begin{matrix} - \nabla\cdot (σ^{'} \nabla ϕ_{0}^{''}) - \nabla\cdot (σ^{"} \nabla ϕ_{0}^{'}) - ω_{0} \nabla\cdot (σ^{'} A_{0}^{'}) + ω_{0} \nabla\cdot (σ^{"} A_{0}^{''}) - S_{ϕ}^{"} = 0 \end{matrix} \end{array}

(4267)

The complex magnetic flux density $\hat{B}$ and magnetic field $\hat{H}$ are calculated from:

Figure 25. EQUATION_DISPLAY

\begin{array}{l} \hat{B} = \nabla\times A_{0}^{'} + i \nabla\times A_{0}^{''} \\ \hat{H} = μ^{- 1} (\nabla\times A_{0}^{'} + i \nabla\times A_{0}^{''}) \end{array}

(4268)

The complex electric field $\hat{E}$ and the total electric current density $\hat{J}$ are:

Figure 26. EQUATION_DISPLAY

\begin{array}{l} \hat{E} = - \nabla ϕ_{0}^{'} + ω_{0} A_{0}^{''} - i (\nabla ϕ_{0}^{''} + ω_{0} A_{0}^{'}) \\ \hat{J} = J^{'} + i J^{"} = (σ^{'} + {i σ}^{"}) \hat{E} \end{array}

(4269)

The magnitude of the electric current density, $| \hat{J} |$ , is also referred to as the peak value. The root mean square current density, $J_{R M S}$ , which represents the time-averaged current density for sinusoidal systems, is related to $| \hat{J} |$ through $| \hat{J} | = \sqrt{2} J_{R M S}$ .

Harmonic Transverse Magnetic Potential

Similarly, for magnetic fields lying on the plane defined by a 2D domain, Eqn. (4243) can be written as:

Figure 27. EQUATION_DISPLAY

\begin{array}{l} \begin{matrix} - \nabla\cdot (μ^{- 1} \nabla A_{n}^{'}) - ω_{0} σ^{"} A_{n}^{'} - ω_{0} σ^{'} A_{n}^{"} - J_{n, e x}^{'} = 0 \end{matrix} \\ \begin{matrix} - \nabla\cdot (μ^{- 1} \nabla A_{n}^{"}) + ω_{0} σ^{'} A_{n}^{'} - ω_{0} σ^{"} A_{n}^{'} - J_{n, e x}^{"} \end{matrix} = 0 \end{array}

(4270)

where the total electric current density is normal to the 2D domain:

Figure 28. EQUATION_DISPLAY

{\hat{J}}_{n} = J_{n}^{'} + i J_{n}^{"}

(4271)

Solution

Simcenter STAR-CCM+ provides several harmonic balance models that compute the complex potentials using either the finite volume method or the finite element method. For information on the integral equations that are solved for each model, see:


Harmonic Balance FV Electrodynamic Potential Harmonic Balance FV Magnetic Vector Potential	These models solve the coupled system Eqn. (4266)-Eqn. (4267) for the complex potentials using the finite volume method.
Harmonic Balance FE Magnetic Vector Potential	This model solves Eqn. (4254) for the complex magnetic vector potential using the finite element method. In Eqn. (4254), the term containing the electric potential is ignored.
Harmonic Balance FV Transverse Magnetic Potential	This model solves Eqn. (4270) for the complex magnetic potential using the finite volume method.