04 Matching Networks
4th unit in course 3, RF Basics and Components
Dipl.-Ing. Dr. Michael Gebhart, MSc
RFID Qualification Network, University of Applied Sciences, Campus 2
WS 2013/14, September 30th
Content
Introduction: How to match the chip output to the antenna impedance?
-
Loop antenna
Quality factor
Low impedance or load matching: Power and Q
Modulation envelope and Q-requirement (single resonance)
Network transformation: differential to single-ended
The general matching solution: π and T-topology networks
Impedance adjustment with L-topology
- Antenna impedance
- Q-factor adjustment for operating frequency
- Determination of serial and parallel capacitance for impedance adjustment
The resonant coupling system
- How near-field coupling affects the air interface
page 2
How to match the chip to the antenna?
NFC device in Reader Mode (TX), a network adjusts (transforms) the antenna
impedance to a desired value for the chip driver output. This allows optimum
power transfer and to meet other contactless property requirements.
NFC Chip
Matching Network
NFC Antenna
?
chip output impedance
e.g. 5 Ω
conjugate reactance
antenna impedance @ 13.56 MHz
e.g. 0.6 + j 110 Ω
conjugate reactance
Narrow-band “matching” for the 13.56 MHz carrier frequency in reader mode
page 3
Loop antenna for Contactless Communication
The loop antenna is a distributed component with
inductance (L) as main element and capacitance (C) and
resistance (R) as parasitic network elements.
End of coil turns
Start of coil turns
C0
L0
R0
CA LA
L1
Ln
C1
Cn
R1
Rn
For simulation it must be
represented by an equivalent
circuit network of lumped
elements.
elements Over a wide frequency
range this can be a loose coupled
reactive ladder network of
resonance circuits - it has several
resonances in frequency domain.
At 13.56 MHz carrier frequency we use the fundamental
(lowest) resonance.
resonance So we can simplify the equivalent circuit
RA e.g. to a parallel resonance circuit (since losses are mainly
determined by chip current consumption in Proximity Systems).
Note: This is a narrow-band approximation only!
page 4
The Quality factor
Originally, the quality factor reports the quality at which the component (coil,
capacitor) represents the pure network element (inductance, capacitance).
RS
LS
Resistance
Q(ω ) =
ω LS
Q (ω ) = ω C P RP
RS
CP
RP
RP MAX
RPRP
22
∆f
f RES
For a 2nd order resonant LCR circuit, Q can be determined in
frequency domain.
domain Q is related to the bandwidth and can be
measured from the Resistance trace.
RRPP
Note: This is only valid, if the broad
2 2
ω RES band equivalent circuit
Qω =
2 ∆ f representation really is a parallel
resonant circuit!
Frequency
( )
For a 2nd order resonant LCR circuit, Q can also be
determined in time domain.
domain Q is related to the
envelope according to
e(t ) = Ae
−t
Q (@ ω RES ) =
τ
−t ⋅
= Ae
ω RES
2Q
= Ae −(ζωRES )t
ω RES τ
2
page 5
Driver concepts – Power and Q requirement
If we consider the quality factor, it depends how the load / antenna is matched
to the source / driver:
Low output impedance
QANT ~ QSYS, e.g. ~ 12.5
RSOURCE ~ 0
RANTENNA = 50 Ω
PSYS ~ PANT ~ 1/QANT, e.g. ~ 400 mW
Load Matching
QANT ~ 2QSYS, e.g. ~ 25
PSYS ~ PSOURCE + PANT ~ 2PANT ~ 2/2QANT
e.g. ~ 2 * 200 mW
RSOURCE = 50 Ω
RANTENNA = 50 Ω
We need a certain operational system Q to achieve time constants for modulation (e.g. ~ 12.5).
In Load Matching, QSYS is half the value of the open antenna – QANT can be doubled.
The power consumed in the antenna is related to 1/Q.
For Load Matching, the required total power is the same, as for low output impedance.
But for low output impedance no power is dissipated in the amplifier, all in the antenna network.
page 6
Power and Q requirement
The unloaded H-field strength emitted by an antenna with resonance at carrier
frequency can be approximated by...
Power Consumption over H -field strength
where X = ω L
Z = R + jX
X
Q=
R
→ R=
ωL
Q
I 2ω L
⇒P=
Q
⇒ H ~ I ANT
0,5
R
P ⋅ QANT
≈
⋅N
ω ⋅ LANT
Driver Power in Watt to 50 Ohm load
P = U ⋅ I = I 2R = U
2
Q = 35
Q = 30
Q = 25
Q = 20
Q = 15
0,5
0,4
0,4
0,3
0,3
0,2
0,2
0,1
1,4
1,5
1,6
1,7
1,8
1,9
2,0
2,1
H -field strength in A/m (rms)
page 7
Modulation envelope and Q requirement
To emit high H-field from low driver power, Q should be high,
To meet modulation timing specifications, Q should be low
Q is only clearly defined for a single-resonance circuit.
For this we get for ISO/EC14443 Type B specifications a maximum system Q…
Start of t f
End of t r
End of tf
Start of t r
hr
tf
tr
hf
1
0
b
t
To note: Modulation index and overshoots may be even more stringent!
page 8
Modulation envelope and Q requirement
For Type A specifications...
110 %
100 %
90 %
Start of t1
End of t3
60 %
End of t4
Start of t2
0
5%
t
End of t1 and t2
Start of t3 and t4
t4
t2
t1
t3
page 9
Network transformation: single-ended, differential
In practice, the TX output is usually differential (to be able to have double output
voltage swing from a single supply voltage).
For simplicty reasons, RF System behaviour can be considered for a singleended network. Component values for the differential network can then be
calculated by the following considerations:
Source and load (antenna) impedance is split up for single-ended consideration
Serial components:
C
Source
Parallel components:
2C ½L
L
RPA
VPA +
R
Network Load
½ RPA
+
½ VPA
+
Source
2C ½L
½R
½ RPA
½ VPA
Network Load
C
½R
+
RPA
+
+
VPA
Source
Network Load
Source
RDIFF = 1 RSINGLE
2
RDIFF = 1 RSINGLE
2
LDIFF = 1 LSINGLE
2
LDIFF = 1 LSINGLE
2
CDIFF = 2CSINGLE
CDIFF = 2CSINGLE
L
R
Network Load
page 10
Network transformation: single-ended, differential
So for the typical matching network, values for a single-ended equivalent circuit
are following quantities of the differential network:
Source
C
2C ½L
½R
ZIN
2 RPA
+
+ VPA
0
½ RPA
RPA
R
VPA +
Network Load
0V
GND
2C ½L
L
Source
Network Load
&
½ VPA
C
½R
+
RPA
+
+
VPA
Source
Network Load
Source
L
R
Network Load
ZANT
2 L0
½ C0
½ C1
2 RQ
k, M
LANT
½ C2
CANT
RANT
LTP
CTP RTP
VCARD_AC
½ RPA
+
½ VPA
+
VCARD_AC
VCARD_DC
Limiter &
Rectifier
page 11
Impedance Matching: π or T topology
Any complex load can be matched to a source impedance using an LC element
network in π - or T topology (general solution). As the antenna impedance varies
over frequency, this matching is for the carrier frequency only.
Z1
RD
ZC
Z2
Z3
RA
RD
ZA
ZB
RA

R cos(ϕ ) − RD RA
RD RA  RD


(
)
=−j
cos
ϕ
−
1
Z1 = − j D

sin (ϕ )
sin (ϕ )  RA

 RA

RD RA sin (ϕ )
ZA = j
= j RD RA sin (ϕ )
cos(ϕ ) − 1
RA cos(ϕ ) − RD R A
 RD


R cos(ϕ ) − RD RA
RD RA  RA


(
)
Z2 = − j A
=−j
cos
ϕ
−
1

sin (ϕ )
sin (ϕ )  RD

 RD

RD RA sin (ϕ )
ZB = j
= j RD RA sin (ϕ )
cos(ϕ ) − 1
RD cos(ϕ ) − RD R A
 RA

Z3 = − j
RD RA
sin (ϕ )
ϕ....Phase deviation
output to input
−1
−1
Z C = j RD RA sin (ϕ )
Literature: F. E. Terman, „Network Theory,
Filters, and Equalizers“
page 12
Towards a more specific impedance adjustment
Any complex load can be matched to any source impedance using π - or T
topology.
However, there is at least one inductor L, which introduces losses…
We may not need to adjust any load:
- antennas are an inductive load (1st self-resonance > fCAR)
- phase relation output to input for the carrierfrequency is irrelevant
So there is a more specific solution, consisting only of capacitors
- HF capacitors of C0G or NP0 have negligible losses and less tolerance
- Less components also means reduction of costs and PCB area
page 13
Impedance adjustment with L-topology
ZE
DRIVER R
D
ZM
EMC-FIL
L0
UD
ZA
RE
CSM
C0
f RES >> f CAR
RA
CA
CPM
R A + jω L A
ZA =
1 + j ω R AC A − ω 2 LAC A
ZA =
ANTENNA
MATCHING
QA >>
(> QSYS )
LA
An equivalent circuit of the loop antenna may
have the above structure.
It can be extracted from measurement.
Complex antenna impedance ZA can be
calculated (over angular frequency ω).
(
0.58 Ω + j 2 ⋅ π ⋅13.56 ⋅106 Hz ⋅1.314 ⋅10−6 H
)
(
F ) − [(2 ⋅ π ⋅13.56 ⋅10 Hz ) 1.314 ⋅10
Z A (@13.56 MHz ) = 0.607 + j114.52
6
1 + j 2 ⋅ π ⋅13.56 ⋅10 Hz ⋅ 0.58 Ω ⋅ 2.35 ⋅10
−12
6
2
−6
H ⋅ 2.35 ⋅10 −12 F
]
page 14
Antenna coupling – NFC antenna only
Coupling affects antenna impedance
significantly
-
mutual inductance LA changes
-
„Loading“ Q changes
Impedance trace is shown in Smith Chart
distance variation to “loading” antenna coupling factor k varies
page 15
Q-factor adjustment for operating frequency
The quality factor of the antenna conductor shall be high… above the intended
Q-factor for operation at the carrier frequency 13.56 MHz.
We can neglect losses in the capacitor (if good components are chosen)
- For one frequency, the serial equivalent circuit can be calculated to an
equivalent parallel circuit for the inductor, using the Q equation:
RA
CA
LA
QA =
ω LA
RSERIAL
≡
RPARALLEL
ω LA
CA
RA
LA
- This way, an external resistor in series to the antenna allows to adjust the
intended QA:
RE
- To note: This can again be recalculated to a “new” antenna
ω LA
RA
RE =
− RA
impedance ZA (which might also
QINTENDED
CA
be a parallel equivalent circuit).
LA
page 16
Impedance adjustment with L-topology
ZE
DRIVER R
D
ZM
EMC-FIL
L0
UD
ZA
CSM
C0
No EMC
R A + jω L A
ZA =
1 + j ω R AC A − ω 2 LAC A
ZM = ZE =
ANTENNA
MATCHING
1 + j ω Z A (CSM + C PM )
ω ( jCSM − ωCSM C PM Z A )
RE
Im{Z } ≡ 0
RA
CA
CPM
CSM (CPM ) =
CPM
f RES ≡ f CAR
LA
Re{Z } ≡ RDESIRED
RA LA
(ω LA )2 RD 1 − ω 2 LA (CA + CPM )
p
=− ±
2
[
]
p2
−q
4
2 ω 2 LAC A − 2
p=
ω 2 LA
q=
R A (1 + R A )
(ω L A )6
−
RA
ω 4 RD L A 4
−
2C A
2
+
C
A
ω 2 LA
page 17
Annex: Exact derivation of reactance matching
with 2 capacitors (I):
- The admittance of the parallel equivalent
antenna circuit and CP is given by
CS
1
1
CA RA LA
Y
=
+
+ sC A + sC P =
UD
A
CP
RA sLA
1
1
=
+
+ s (C A + C P )
RA sLA
- Impedance for the parallel equivalent circuit of
antenna and reactance matching network is...
DRIVER
Z LAST =
RD
MATCHING
ANTENNA
1
1
sR A LA
1
+
= 2
+
YA sC S s R A LA (C A + C P ) + sL A + R A sC S
- This impedance is set equal to the desired real ( reactance matching)
source impedance (e.g. 50 Ω).
RD
≡
s 2 [RA LACS + RA LA (C A + C P )] + sLA + RA
s 3 RA LACS (C A + C P ) + s 2 LAC S + sR ACS
⋅ Nenner
s 3 RD R A LAC S (C A + C P ) + s 2 RD LAC S + sRD R AC S = s 2 [R A LAC S + R A LA (C A + C P )] + sL A + R A
page 18
Annex: Exact derivation of reactance matching
with 2 capacitors (II):
- Transition s j ω and coefficient comparison of real and imaginary parts:
− jω 3 RD R A LAC S (C A + C P ) − ω 2 RD LAC S + jωRD R AC S = −ω 2 [R A LAC S + R A LA (C A + C P )] + jωLA + R A
- Imaginary parts:
− jω 3 RD R A LAC S (C A + C P ) + jωRD RAC S = jωLA
CS ,a =
LA
RD R A − ω 2 RD R A LA (C A + C P )
- Real parts:
− ω 2 RD LAC S = −ω 2 [RA LA (C A + C P )] + R A
C S ,b
− ω 2 RA LA (C A + C P ) + RA
=
ω 2 LA (R A − RG )
- Both solutions for CS are set equal, to solve for CP
page 19
Annex: Exact derivation of reactance matching
with 2 capacitors (II):
- For CS, a = CS, b follows...
ω 2 LA 2 RA − ω 2 LA 2 RD = RD RA 2 − ω 2 RD RA 2 LA (C A + C P ) − ω 2 RD RA 2 LA (C A + C P ) + ω 4 RD RA 2 LA 2 (C A + C P )2
- Sort according the power of CP:
2
[
]
[2ω R R L C − 2ω R R L ]+
[R R − ω L (R − R ) − 2ω R R
2
2
C P ω 4 RD R A L A +
+ CP
+
2
4
D
2
D
A
2
A
A
A
D
2
2
2
2
A
A
A
2
2
A
D
D
A
2
2
L A C A + ω 4 RD R A L A C A
2
]
- Resolve the characteristic equation, cancel:
2
2
2ω 2 LAC A − 2
RD R A ω 4 R A LA C A − 2ω 2 R A LAC A + 1 − ω 2 (R A − RD )
p=
q=
2
ω LA
ω 4 RG RA 2 LA 2
(
C P a ,b
p
=− ±
2
)
p2
−q
4
- Knowing CP allows to calculate CS using one or the other equation.
Note: only positive, real values have a physical meaning!
page 20
Antenna coupling – no EMC filter
2 RPA
+ TX
½ C1
1
+ VPA
0
2 RQ
k, M
LANT
½ C2
CANT
0V
RANT
GND
distance 80 … 3 mm
page 21
Reasoning for extending the network
Due to Standard (e.g. ISO/IEC14443) requirements for modulation timing, the
allowable Q-factor for the operated system is rather low (~limited bandwidth).
For efficient H-field emission, a higher reactive current is desirable a 2nd
resonance frequency can increase the bandwith.
Furthermore, this 2nd resonance (LC low-pass) can suppress unwanted
harmonics emission ( name EMC filter)
This comes on the expense of
more signal distortions…
page 22
Impedance adjustment with L-topology
ZE
DRIVER R
D
ZM
EMC-FIL
L0
UD
C0
ZA =
R A + jω L A
1 + j ω R AC A − ω 2 LAC A
ZM =
1 + j ω Z A (CSM + C PM )
ω ( jCSM − ωCSM C PM Z A )
ZE =
Z M + j ω L0
j ω C0 Z M − ω 2 L0C0
ZA
ANTENNA
MATCHING
CSM
RE
CPM
f RES ≡ f CAR
RA
CA
LA
Im{Z } ≡ 0
Re{Z } ≡ RDESIRED
page 23
Impedance adjustment with L-topology
For the calculation of CP and CS we follow a more systematic approach and recalculate a real and imaginary impedance for every step:
1st step: starting from right side, we
calculate an antenna impedance ZA
2
RA X CA
Re(Z A ) = 2
= Ra
2
RA + ( X CA + X LA )
2
2
2
X R + X L + X CA X LA
Im(Z A ) = CA A 2 CA A
= Xa
2
RA + ( X CA + X LA )
page 24
Impedance adjustment with L-topology
2nd step: starting from the left side, we calculate the EMC filter impedance and the
adjustment network impedance ZM
X C0 = −
1
ω C0
X L 0 = ω L0
2
R0 X C 0
Re(Z M ) = 2
= Rm
2
R0 + ( X C 0 + X L 0 )
2
Im(Z M ) = − jX C 0
2
X L 0 + X C 0 X L 0 + R0
= Xm
2
2
(
)
R0 + X C 0 + X L 0
page 25
Impedance adjustment with L-topology
This way, we bring the matching condition in the middle of the network.
3rd step: Solving condition for the parallel and serial capacitance (target impedance):
X P1, 2
Rm
=
Ra − Rm
2


R
X
a
a

⋅ X a ± Ra ⋅
+
−1 


Rm Rm Ra


2
1
ω XP
CS = −
1
ω XS
2
R + Xa + XaXP
XS = Xm − XP ⋅ a 2
2
Ra + ( X a + X P )
CP = −
Finally, we need to sort out solutions which have no physical meaning…
- only positive capacitance values have a representation
- …not all antennas can be matched (if we violate a pre-condition, it is not
possible…
Note: In practice, the impedance must be checked by measurement, as
parasitics may cause deviations from the ideal conditions
page 26
Antenna coupling – with EMC filter
+
2 RPA 2 L0
+ TX
1
½ C0
V
PA
0
½ C1
2 RQ
LANT
½ C2
0V
k, M
CANT
RANT
GND
distance 80 … 3 mm
page 27
Which loads can be matched?
For the 2 capacitor network, the area in the Smith Chart is following…
CS
CP
FORBIDDEN
AREA
Reference: Electronic Applications of the Smith
Chart by Phillip H. Smith, 1969, p. 124
page 28
Which loads can be matched?
For the network with EMC-filter, it is slightly more difficult
We need to calculate a target impedance (ZT) first,
which depends on our desired input impedance
ZIN, and the (loaded) antenna impedance (ZA).
ZIN − jXL0
ZT =
1+ jXC0 ( jXL0 − ZIN )
Ztarget (ZT)
ZANT
VPA +
2
+
+
TX1
L0
0V
VPA
CANT
GND
VPA +
2
LA
C2
C0
C0
TX2
L0
C2
C1
LA
LTP
CTP RTP
LANT
LB
RPA
k, M
RQ
C1
VCARD_AC
RPA
IPCD_ANT
ZIN
RANT
RQ
Differential EMC-Filter
+ lumped component impedance adjustment network
PCD antenna
equivalent circuit
LB
PICC antenna
equivalent circuit
VCARD_DC
PICC equivalent
circuit
page 29
Which loads can be matched?
Antenna impedance ZA
(including coupling)
Region, which can be
matched to desired
network input impedance
+ CS
Adjustable range for
parallel cap CP
+ CP
- CP
- CS
Adjustable range for
serial cap CS
Target impedance ZT
Network input impedance
ZIN (@ carrier frequency)
Frequency trace of ZIN
page 30
Which loads can be matched?
The region of network input impedances ZIN at carrier frequency, which can be
adjusted to an intended resistance (“matched”), is limited by two circles.
For the parallel capacitor, from short to ZT
For the serial capacitor, from ZT to open.
The CP limit circle is given by
- xT, yT…. coordinates of ZT
- c…center, r… radius
(− 1 + x
c=
2
+ yT
2( xT − 1)
2
T
)
r = 1− c
The CS limit circle is given by
(− 1 + x
c=
2
+ yT
2( xT + 1)
T
2
)
r = 1+ c
Impedance adjustment possible, iff
Re(Z A ) ≤ Re(ZT )
Im(Z A ) ≥ 0 (inductive load )
page 31
The resonant coupling system
Transponder equivalent circuit properties
L, R, C
VMID
TXB
GND
TX A
RXA
EMC-Fil.
L0 C0
CS
Receive path
CP
RAS
LA RA
CA
CAS
RC
CC
RAS
Antenna
RQ
Assemb.
Chip
Proximity Chip
RXB
Matching network
NFC Chip analogue front end
Reader equivalent circuit properties
L, R, C
Antenna
VMID
coupling k
Resonant system properties
k, fRES, Q
Standard defines properties
at the Air Interface
page 32
Coupling: 0 %
Distance: -- mm
Matching-Impedance
How near-field coupling affects the air interface
L0
Cs
Cp
C0
EMC-FIL
110 %
100 %
90 %
CA
MATCHING
LA
RA
ANTENNA
Start of t1
End of t 3
60 %
End of t 4
Start of t2
0
5%
t
End of t1 and t2
Start of t3 and t4
t4
t2
t1
t3
page 33
Coupling: 6 %
Distance: 25 mm
Matching-Impedance
How near-field coupling affects the air interface
L0
C0
EMC-FIL
t1 in τC
37.80
t2 in τC
31.12
t3 in τC
2.45
Cs
Cp
CA
MATCHING
t4 in τC
1.52
a
0.007
LA
RA
ANTENNA
HOVS
1.076
page 34
Coupling: 13 %
Distance: 15 mm
Matching-Impedance
How near-field coupling affects the air interface
L0
C0
EMC-FIL
t1 in τC
37.77
t2 in τC
31.47
t3 in τC
2.12
Cs
Cp
CA
MATCHING
t4 in τC
1.34
a
0.007
LA
RA
ANTENNA
HOVS
1.12
page 35
Coupling: 21 %
Distance: 10 mm
Matching-Impedance
How near-field coupling affects the air interface
L0
C0
EMC-FIL
t1 in τC
37.80
t2 in τC
29.41
t3 in τC
1.99
Cs
Cp
CA
MATCHING
t4 in τC
1.34
a
0.010
LA
RA
ANTENNA
HOVS
1.16
page 36
Coupling: 38 %
Distance: 5 mm
Matching-Impedance
How near-field coupling affects the air interface
L0
C0
EMC-FIL
t1 in τC
37.78
t2 in τC
24.97
t3 in τC
1.61
Cs
Cp
CA
MATCHING
t4 in τC
1.04
a
0.020
LA
RA
ANTENNA
HOVS
1.15
page 37
Thank you for your
Audience!
Please feel free to ask questions...
page 38