1. Bandwidth Bounds for Matching Coupled Loads
Ding Nie and Bertrand Hochwald
University of Notre Dame, USA
nding1@nd.edu
bhochwald@nd.edu
June 2, 2015
Ding Nie and Bertrand Hochwald (University of Notre Dame)Bandwidth Bounds for Coupled Loads June 2, 2015 1 / 12

2. Impedance Matching
1 2
8.27 nH
1.06 pF at 2.4 GHz
2
λ
0Z
LZ
GZ
Impedance matching maximizes
power transferred to the load
ZG = Z∗
L
Lossless networks match the
source and load at a design
frequency ω = ωd
Matching in a frequency band is
generally impossible 1 1.5 2 2.5 3 3.5 4 4.5 5
x 10
9
−300
−200
−100
0
100
200
300
400
Frequency (Hz)
Impedance(Ω)
Re{ZL
}
Im{ZL
}
Re{ZG
}
Im{ZG
}
Ding Nie and Bertrand Hochwald (University of Notre Dame)Bandwidth Bounds for Coupled Loads June 2, 2015 2 / 12

3. Impedance Matching
The reflection coefficient measures the quality of matching
Γ =
Z − Z0
Z + Z0
All incident power is absorbed without reflection when the source and
load are matched
Γ(2.4 GHz) = 0
8.27 nH
1.06 pF at 2.4 GHz
2
λ
0Z
( )jωΓ
1 2 3 4 5
x 10
9
−20
−15
−10
−5
0
Frequency (Hz)
Magnitude(dB)
|Γ(jω)|
Ding Nie and Bertrand Hochwald (University of Notre Dame)Bandwidth Bounds for Coupled Loads June 2, 2015 3 / 12

4. Bode-Fano Bounds
Bode-Fano bounds on a parallel RC load
Two-port
matching
network
C R
( )jωΓ
For a parallel RC load, the Bode-Fano bounds show the theoretical
limit on bandwidth
∞
0
log
1
|Γ(jω)|
dω ≤
π
RC
Finite bandwidth
Perfect matching at discrete frequency points
Perfect matching is harmful to the bandwidth
Broadband matching networks are needed
Pass band
ω
1
thr
| ( ) |jωΓ
Ding Nie and Bertrand Hochwald (University of Notre Dame)Bandwidth Bounds for Coupled Loads June 2, 2015 4 / 12

5. MIMO Systems
Multiple antennas in a compact
space
Significant electromagnetic
coupling between antennas
Much bigger problems in the
next generation massive MIMO
communication systems
Our contributions
We introduce a reflection measurement for coupled loads
We develop broadband matching bounds for coupled loads
Electromagnetic coupling between antennas can be beneficial for
bandwidth
Ding Nie and Bertrand Hochwald (University of Notre Dame)Bandwidth Bounds for Coupled Loads June 2, 2015 5 / 12

6. Impedance Matching for Coupled Loads
N uncoupled sources drive N coupled loads
through a 2N-port lossless matching network
No single reflection coefficient is defined for
coupled loads
b1(jω) = SLM(jω)a1(jω)
0Z
0Z
Lossless
2N-port
matching
network
S.
.
.
LMS
N-port
loads S
.
.
.
LS
LS
S
1a

1b

2a

2b

Power reflection ratio
The power reflection ratio r(ω) is defined as
r2
(ω) =
E b1(jω) 2
E a1(jω) 2
=
1
N
SLM(jω) 2
F
Elements of a1(jω) are assumed to be random and independent
Bandwidth is the frequency range where r(ω) ≤ τ
Ding Nie and Bertrand Hochwald (University of Notre Dame)Bandwidth Bounds for Coupled Loads June 2, 2015 6 / 12

7. Bandwidth Bound For Coupled Loads
Bandwidth bound for coupled loads
Let SL(s) be the N × N S-matrix model for the coupled loads, then
∞
0
log
1
r(ω)
dω ≤
−π
2N
i
pL,i +
i
zL,i ,
where pL,i , zL,i are the poles and zeros of SL(s)
The bound depends only on the loads
Can be used to
- examine the effects of coupling between loads on bandwidth
- judge whether a multiport matching network with a prescribed
bandwidth is realizable
- design multiport matching networks with near-optimal bandwidth
Ding Nie and Bertrand Hochwald (University of Notre Dame)Bandwidth Bounds for Coupled Loads June 2, 2015 7 / 12

8. Capacitively-Coupled Loads
2
1
3
4
N
...
1
cZ
s
0Z
2
1
3
4
N
...
0C
1C
2C
(a) (b)
The loads have a circulant structure
The bound is ∞
0
log
1
r(ω)
dω ≤
π
Z0Ceq
Ceq is the capacitance seen from any port relative to ground
Bandwidth may go to zero if Ceq → ∞ as N → ∞
Ding Nie and Bertrand Hochwald (University of Notre Dame)Bandwidth Bounds for Coupled Loads June 2, 2015 8 / 12

9. Two Coupled Parallel Dipoles
Two coupled parallel dipoles
with λ/2 length and λ/10
separation at 2.4 GHz
The bound is
10π×109
2π×109
log
1
r(ω)
dω ≤ 1.84×1010
The decoupling network achieves
10π×109
2π×109
log
1
r(ω)
dω = 5.36×109
Narrowband matching networks
achieve integrals significantly
smaller than the bound
Magnitude(dB)
Frequency (Hz)
2.2 2.3 2.4 2.5 2.6
x 10
9
−15
−10
−5
012.5
mm
62.5
mm
2.5
mm 1
2
3
4
7.76 nH
5.92 nH
5.42 nH
6.45 nH
0.15 pF
0.59 pF
0.57 pF
0.63 pF
r(ω)
Step of applying the bound
1 Obtain SL(jω) through
simulation or measurement
2 Find a passive rational
model SL(s) for SL(jω)
3 Calculate the poles and
zeros pL,i , zL,i from SL(s)
4 Calculate the bound using
pL,i , zL,i
Ding Nie and Bertrand Hochwald (University of Notre Dame)Bandwidth Bounds for Coupled Loads June 2, 2015 9 / 12

10. Comparing the Isolated and Coupled Dipoles
0 0.5 1 1.5
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
x 10
10
Spacing (wavelength)
Bound
Even mode
Odd mode
Bound vs. dipole separation
Small spacing: a single dipole
Large spacing: two isolated dipoles
Bound maximized at about a
quarter wavelength spacing
Conclusion
The bounds are easy to apply, have various of applications
Coupling can be beneficial for bandwidth
Designs of multiport broadband matching networks for coupled
antennas are needed
Ding Nie and Bertrand Hochwald (University of Notre Dame)Bandwidth Bounds for Coupled Loads June 2, 2015 10 / 12

11. Summary
Bandwidth bounds for coupled loads are developed, applicable to
MIMO communication systems
Electromagnetic coupling between the antennas can be beneficial for
the bandwidth
Design of multiport broadband matching networks for coupled
antennas are needed to achieve the bounds
Ding Nie and Bertrand Hochwald (University of Notre Dame)Bandwidth Bounds for Coupled Loads June 2, 2015 11 / 12

12. References
H. W. Bode, Network Analysis and Feedback Amplifier Design, New
York, NY, USA: Van Nostrand 1945.
R. M. Fano, “Theoretical limitations on the broadband matching of
arbitrary impedances,” Journal of the Franklin Institute, 249(1) pp.
57–83, and 249(2), pp. 139–154, 1950.
D. Nie, B. Hochwald, “Broadband matching bounds for coupled
loads,” IEEE Transaction on Circuits and Systems I: Regular Papers,
vol. 62, no. 4, pp. 995–1004, April 2015.
Ding Nie and Bertrand Hochwald (University of Notre Dame)Bandwidth Bounds for Coupled Loads June 2, 2015 12 / 12