SMG-8156 Time-harmonic electromagnetic
systems
Saku Suuriniemi
saku.suuriniemi@tut.fi
Tampere University of Technology
Institute of Electromagnetics
2007
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Lecture set 3:
Complex Fourier series
Time-harmonic fields, superposition
Fourier transform
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1
Complex Fourier series
• A common choice in RF engineering: complex exponential
functions as basis functions and complex numbers as
coefficients:
∞
X
cn ejnωt .
f (t) =
n=−∞
• The sum of the terms of frequencies nω and −nω is
cn ejnωt + c−n e−jnωt .
• Rewritten:
Re {cn } cos(nωt) − Im {cn } sin(nωt)
+j(Im {cn } cos(nωt) + Re {cn } sin(nωt))
+Re {c−n } cos(nωt) + Im {c−n } sin(nωt)
+j(Im {c−n } cos(nωt) − Re {c−n } sin(nωt))
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Question 1:
• When is the function real-valued?
• When is the signal purely a combination of sines?
• When is the signal purely a combination of cosines?
Re {cn } cos(nωt) − Im {cn } sin(nωt)
+j(Im {cn } cos(nωt) + Re {cn } sin(nωt))
+Re {c−n } cos(nωt) + Im {c−n } sin(nωt)
+j(Im {c−n } cos(nωt) − Re {c−n } sin(nωt))
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• When the function is real-valued, its value is
2 Re {cn } cos(nωt) − 2 Im {cn } sin(nωt),
and the power at frequency nω is (2 Re {cn })2 + (2 Im {cn })2
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2
= 4(Re {cn } + Im {cn } ) = 4|cn |2 .
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• Recipe for coefficient computation: the functions
orthonormal over [−π, π].
√1 ejkt
2π
1. Map f (x) from x ∈ [a, b] to t ∈ [−π, π] using
b+a
x(t) = b−a
2π t + 2 , i.e. f (t) = f (x(t)).
Rπ
1
1
jkt
2. Compute ck = hf (t), √2π e i = √2π −π f (t)e−jkt dt.
3. Map series back using t(x) =
∞
X
π
b−a (2x
− (b + a)):
1 jkt(x)
.
ck √ e
f (x) =
2π
k=−∞
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are
Example: f (x) = x2 at x ∈ [0, 1].
1
t + 21 )2 at t ∈ [−π, π].
• Maps to f (t) = ( 2π
Rπ 1
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• Coefficients: ck = √2π −π ( 2π t + 21 )2 e−jkt dt.
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They become ck = √12π k1 ( kπ
+ j)(−1)k .
P∞
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• Series f (x) = 2π k=−∞ k1 ( kπ
+ j)(−1)k ejk(2πx−π)
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• Approximations for N Fourier coefficients can be computed
from N discrete data points.
• If the points are equidistant and there are 2n of them, there is a
very efficient way to do this, the FFT (fast Fourier transform).
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Time-harmonic fields
• So far only time-harmonic scalar quantities have been
discussed.
• Easy to formally extend the time-harmonicity to vectors: vector
can be expressed with components that are scalar quantities.
– Let the components of a vector A(t) be time harmonic (all
with angular frequency ω).
– Use the complex exponential function and (implicitly) take
the real part:
A(t) = iAx0 ejωt + jAy0 ejωt + kAz0 ejωt .
Here all of Ax0 , Ay0 , Az0 are complex numbers.
– Then the vector has representation
Â(t) = Â0 ejωt ,
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where Â0 is a complex-valued vector.
– The actual quantity is the real part
Re{Â0 } cos(ωt) − Im{Â0 } sin(ωt).
• What can vectors with time harmonic components be like,
geometrically speaking?
1. If the real and imaginary parts are collinear, i.e.
a Re{Â0 } = b Im{Â0 } holds for some a, b ∈ R, the vector
oscillates in one dimension.
2. If the real and imaginary parts are perpendicular and equal
in magnitude, the vector rotates along a circle in the plane
they span.
3. Else the vector rotates along an ellipse in the plane spanned
by the real and imaginary parts.
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• Let us generalize the time harmonicity to systems that can be
characterized by vector fields.
• Vector field is a mapping from a domain to a vector space. Let
us map the domain to time-harmonic vector values:
F̂(r, t) = F̂0 (r)ejωt .
• Field F̂0 (r) is a mapping F̂0 : Ω → Cn : Clearly a vector field,
because Cn is a vector space.
• Real part:
Re{F̂0 (r)} cos(ωt) − Im{F̂0 (r)} sin(ωt).
• Do fields like this ever occur?
• Yes: Time-harmonic sources in a system with linear media.
Very commonplace in electrical engineering. Even more useful
than that, we’ll see in a while . . .
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Question 2: Give examples of practical electromagnetic devices
whose operation can be modelled with time harmonic fields.
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Superposition of fields
• Let us have two sources of frequencies ω1 and ω2 in a linear
system (i.e. linear media).
• In steady state, we can find out the responses to sources
individually, and then add or superpose the solutions.
• The solution will be
F̂1 (r)ejω1 t + F̂2 (r)ejω2 t .
• Furthermore, the following works:
1. Decompose any periodic source quantity to a sum of
time-harmonic sources,
2. analyze the responses individually (other sources absent),
3. superpose the solutions.
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J(z, t) = f1 (z) sin(ωt)J(z, t) = f2 (z) sin(2ωt)
Figure 1: Two antennas with current densities oscillating with different frequencies
J(z, t) = f1 (z) sin(ωt) + f2 (z) sin(2ωt)
Figure 2: One antenna with a current density that is a sum of current
densities oscillating with different frequencies
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Fourier transform
• Any periodic function (or field) could be approximated to any
precision by a sum of harmonic functions.
• In the limit, the sum was extended to Fourier series.
• What if the function is not time periodic, but aperiodic?
• This corresponds to the period length approaching infinity:
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• The period of function
∞
X
f (t) =
ck ejkω0 t ,
k=−∞
is
2π
ω0 .
As the period approaches infinity,
1. the base angluar frequency ω0 must approach zero, and
2. the difference of consecutive frequencies approaches zero.
• Let us see how a Fourier series behaves when the period gets
longer and the function approaches aperiodicity.
1
−nπ
−π/2 π/2
• The results for n = 1, 2, 3 are
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• Instead of summing a series of weighted functions we integrate
over a continuum of functions, i.e. construct a Fourier integral
Z ∞
1 jωt
f (t) =
F (ω) √ e dω.
2π
−∞
– For Fourier series, Fourier coefficients ck needed.
– For Fourier integral, function F (ω) needed. It is called the
Fourier transform of function f (t).
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