Basics in Systems and Circuits Theory
BSC Modul 3: Two-Port Networks
•
•
•
•
•
Michael E.Auer
Impedance and Admittance Parameters
Hybrid and Transmission Parameters
Interconnected Two-Port Networks
Applications (BJT, FET Small Signal Models)
Frequency Domain Analysis
01.11.2011
BSC03
Basics in Systems and Circuits Theory
BSC Modul 3: Two-Port Networks
• Impedance and Admittance Parameters
•
•
•
•
Michael E.Auer
Hybrid and Transmission Parameters
Interconnected Two-Port Networks
Applications (BJT, FET Small Signal Models)
Frequency Domain Analysis
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Definition of a Port
It is a pair of terminals through which a
current may enter or leave a network.
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
One and Two Port Circuits
One port or two terminal
circuit
Two port or four
terminal circuit
• It is an electrical
network with two
separate ports for input
and output.
• Pairs of identical
currents.
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Impedance Parameters (1)
Assume no independent source in the network  Passive NW
V1 = z11I1 + z12 I 2
V2 = z 21I1 + z 22 I 2
V1   z11 z12   I1 
= [z ]
V  =  z



 2   21 z 22   I 2 
 I1 
I 
 2
where the z terms are called the impedance parameters, or simply z
parameters, and have units of ohms.
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Impedance Parameters (2)
z11 =
V1
I1
z 21 =
and
I 2 =0
V2
I1
I 2 =0
z11 = Open-circuit input impedance
z21 = Open-circuit transfer impedance
from port 1 to port 2 (forward)
z12 =
V1
I2
z 22 =
and
I1 = 0
V2
I2
I1 = 0
z12 = Open-circuit transfer impedance
from port 2 to port 1 (reverse)
z22 = Open-circuit output impedance
6
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Impedance Parameters (3)
z11 =
z12 =
Michael E.Auer
V1
I1
V1
I2
and
z 21 =
I 2 =0
and
I1 = 0
z 22 =
V2
I1
I 2 =0
V2
I2
I1 = 0
•
When z11 = z22, the two-port network is
said to be symmetrical.
•
When the two-port network is passive and
linear, the transfer impedances are equal
(z12 = z21), and the two-port is said to be
reciprocal.
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Impedance Parameters (4)
Example
Determine the Z-parameters of the following circuit.
I1
I2
V2
V1
Answer:
Michael E.Auer
z11 =
60
z=
40
z12 =
V1
I1
V1
I2
and
I 2 =0
and
I1 = 0
 z11
z=
z821
40
Ω

70
01.11.2011
z 21 =
z 22 =
V2
I1
I 2 =0
V2
I2
I1 = 0
z12 
Ω

z 22 
BSC03
Basics in Systems and Circuits Theory
Admittance Parameters (1)
Assume no independent source in the network  passive NW
I1 = y11V1 + y12 V2
I 2 = y 21V1 + y 22 V2
I1   y11 y12  V1 
V1 
= [y]  
I  =  y



 2   21 y 22  V2 
V2 
where the y terms are called the admittance parameters, or simply y
parameters, and they have units of Siemens.
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Admittance Parameters (2)
y11 =
I1
V1
and
y 21 =
V2 = 0
I2
V1
V2 = 0
y11 = Short-circuit input admittance
y21 = Short-circuit transfer admittance
from port 1 to port 2 (forward)
y12 =
I1
V2
and
V1 = 0
y 22 =
I2
V2
V1 = 0
y12 = Short-circuit transfer admittance
from port 2 to port 1 (revers)
y22 = Short-circuit output admittance
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Admittance Parameters (3)
Example
Determine the y-parameters of the following circuit.
I1
I2
V1
Answer:
Michael E.Auer
y11 =
V2
 0.75 − 0.5 
y=
S

− 0.5 0.625
01.11.2011
y12 =
I1
V1
I1
V2
and
y 21 =
V2 = 0
and
V1 = 0
 y11
y=
 y 21
y 22 =
I2
V1
I2
V2
V2 = 0
V1 = 0
y12 
S

y 22 
BSC03
Basics in Systems and Circuits Theory
BSC Modul 3: Two-Port Networks
•
Impedance and Admittance Parameters
• Hybrid and Transmission Parameters
•
•
•
Michael E.Auer
Interconnected Two-Port Networks
Applications (BJT, FET Small Signal Models)
Frequency Domain Analysis
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Hybrid Parameters (1)
Assume no independent source in the network  passive NW
V1 = h11I1 + h12 V2
I 2 = h 21I1 + h 22 V2
V1   h11
I  = h
 2   21
h12  I1 
I1 
= [h ]  



h 22  V2 
V2 
where the h terms are called the hybrid parameters, or simply h parameters, and
each parameter has different units, refer above.
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Hybrid Parameters (2)
Assume no independent source in the network
V
h11 = 1
I1
h 21
I
= 2
I1
Michael E.Auer
V2 = 0
V2 = 0
h11= short-circuit input
impedance (Ω)
h21 = short-circuit forward
current gain
01.11.2011
h12
h 22
V1
=
V2
I2
=
V2
I1 = 0
I1 = 0
h12 = open-circuit reverse
voltage-gain
h22 = open-circuit output
admittance (S)
BSC03
Basics in Systems and Circuits Theory
Hybrid Parameters (3)
Example
Determine the h-parameters of the following circuit.
I1
I2
V1
Answer:
Michael E.Auer
V2
4Ω
h= 2
 3
V1
I1
h11 =
h12 =
V1
V2
and
V2 = 0
and
I1 = 0
h11Ω
h=
 h 21
− 23 
1 
9 S
01.11.2011
h 21 =
h 22 =
I2
I1
V2 = 0
I2
V2
I1 = 0
h12 
h 22S
BSC03
Basics in Systems and Circuits Theory
Transmission Parameters (1)
Assume no independent
source in the network
V1 = A11V2 − A12 I 2
I1 = A 21V2 − A 22 I 2
V1   A11
I  = A
 1   21
A12  V2 
V2 
= [T ] 




A 22  − I 2 
I
−
 2
where the A terms are called the transmission parameters, or simply
T parameters, and each parameter has different units.
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Transmission Parameters (2)
V
A11 = 1
V2
A 21 =
I1
V2
V1
A12 = −
I2
A 22 = −
Michael E.Auer
01.11.2011
I1
I2
A11 = open-circuit voltage
ratio
I 2 =0
I 2 =0
A21 = open-circuit transfer
admittance (S)
A12 = negative short-circuit
transfer impedance (Ω)
V2 = 0
V2 = 0
A22 = negative short-circuit
current ratio
BSC03
Basics in Systems and Circuits Theory
BSC Modul 3: Two-Port Networks
•
•
Impedance and Admittance Parameters
Hybrid and Transmission Parameters
• Interconnected Two-Port Networks
•
•
Michael E.Auer
Applications (BJT, FET Small Signal Models)
Frequency Domain Analysis
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Parallel Connection of Two-Port NWs
[y] = [ ya ] + [ yb ]
y11 = y11a + y11b
y12 = y12 a + y12b
y21 = y21a + y21b
y22 = y22 a + y22b
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Series Connection of Two-Port NWs
[z] = [za ] + [zb ]
z11 = z11a + z11b
z12 = z12 a + z12b
z21 = z21a + z21b
z22 = z22 a + z22b
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Cascade Connection of Two-Port NWs
[A ] = [Aa ] ⋅ [Ab ]
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
BSC Modul 3: Two-Port Networks
•
•
•
Impedance and Admittance Parameters
Hybrid and Transmission Parameters
Interconnected Two-Port Networks
• Applications (BJT, FET Small Signal Models)
•
Michael E.Auer
Frequency Domain Analysis
01.11.2011
BSC03
Basics in Systems and Circuits Theory
BJT Y-Parameter Small Signal Model
y12 =
y21 =
Using 2-port y-parameter network,
y22 =
ib = y11vbe + y12vce
ic = y vbe + y22vce
ib
vce
v =0
be
ic
vbe
=
v =0
ce
ic
vce
21
y11 =
=
ib
=
v =0
be
=
∂iB
∂vCE
=0
Q − point
∂iC
∂vBE
∂iB
IC
VT
=
IC
V A +VCE
Q − point
∂iC
∂vCE
=
Q − point
=
IC
The port variables can represent either
β oVT
vbe
∂vBE
Q − point
v =0
ce
time-varying part of total voltages and
currents or small changes in them away
βo is the small-signal commonfrom Q-point values.
emitter current gain of the BJT.
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
BJT Hybrid Small Signal Model
h11 =
h12 =
h21 =
h22 =
Michael E.Auer
vce
ib
=
v =0
ce
vbe
vce
=
i =0
b
ic
ib
=
v =0
ce
ic
vce
01.11.2011
=
i =0
b
∂vCE
∂iB
= rBE
Q − point
∂v BE
∂vCE
= µ
Voltage-feedback
Q − point
∂iC
∂iB
Input impedance
= β
Current gain
Q − point
∂iC
∂vCE
=
Q − point
1
rCE
Output admittance
BSC03
Basics in Systems and Circuits Theory
MOSFET Hybrid Small Signal Model (1)
y11 =
Using 2-port y-parameter network,
ig = y11vgs + y12vds
id = y21vgs + y22vds
y12 =
y21 =
ig
v gs
01.11.2011
v =0
ds
ig
vds
v gs
∂vGS
=
v
gs
=0
id
The port variables can represent either
i
y22 = d
time-varying part of total voltages and
vds
currents or small changes in them away
from Q-point values.
Michael E.Auer
=
∂iG
=
v =0
ds
=
v
gs
=0
=0
Q − point
∂iG
∂vDS
=0
Q − point
∂iD
∂vGS
=
Q − point
∂iD
∂vDS
2I D
VGS −VTN
=
Q − point
ID
1
λ
+VDS
BSC03
Basics in Systems and Circuits Theory
MOSFET Hybrid Small Signal Model (2)
Transconductance:
2I D
gm = y21 =
= 2K n I D
VGS −VTN
Output resistance:
•
•
Since gate is insulated from channel by
gate-oxide input resistance of transistor
is infinite.
Small-signal parameters are controlled
by the Q-point.
Michael E.Auer
01.11.2011
1+ λVDS
1
1
ro =
=
≅
y22
λI D
λI D
BSC03
Basics in Systems and Circuits Theory
BSC Modul 3: Two-Port Networks
•
•
•
•
Impedance and Admittance Parameters
Hybrid and Transmission Parameters
Interconnected Two-Port Networks
Applications (BJT, FET Small Signal Models)
• Frequency Domain Analysis
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Transfer Function (1)
•
The transfer function H(ω) of a circuit is the frequency-dependent
ratio of a phasor output Y(ω) (an element voltage or current ) to a
phasor input X(ω) (source voltage or current).
Y(ω )
H (ω ) =
= | H (ω ) | ∠φ
X(ω )
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Transfer Function (2)
Four possible transfer functions:
H(ω ) = Voltage gain =
Vo (ω )
Vi (ω )
H(ω ) = Transfer Impedance =
H(ω ) =
H(ω ) = Current gain =
Michael E.Auer
Vo (ω )
I i (ω )
Y(ω )
= | H(ω ) | ∠φ
X(ω )
I o (ω )
I i (ω )
H(ω ) = Transfer Admittance =
01.11.2011
I o (ω )
Vi (ω )
BSC03
Basics in Systems and Circuits Theory
Transfer Function (3)
Example
For the RC circuit shown below, obtain the transfer function Vo/Vs and
its frequency response.
Let vs = Vm cosωt.
time domain circuit
Michael E.Auer
frequency domain circuit
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Transfer Function (4)
Solution:
The transfer function is
1
V
1
j ωC
H (ω ) = o =
=
Vs R + 1/ jω C 1 + jω RC
,
The magnitude is
H(ω ) =
1
1 + (ωRC ) 2
The phase is φ = − arctan ωRC
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
The Decibel Scale (1)
Historically, the bel was used to measure the ratio of two
levels of power or power gain.
P2
G = Number of bels = lg
P1
(unit less)
The decibel (dB) provides a unit of less magnitude:
GdB
Michael E.Auer
P2
=10 ⋅ lg
P1
01.11.2011
BSC03
Basics in Systems and Circuits Theory
The Decibel Scale (2)
GdB
P2
V22 R2
=10 ⋅ lg =10 ⋅ lg 2
P1
V1 R1
For R1 = R2
Michael E.Auer
GV / dB
V2
= 20 ⋅ lg
V1
GI / dB
I2
= 20 ⋅ lg
I1
01.11.2011
BSC03
Basics in Systems and Circuits Theory
The Decibel Scale (3)
V2 / V1
10-2
10-1
0,5
1
2
10
102
103
104
Gv / dB
-40
-20
-6
0
6
20
40
60
80
GP / dB
-20
-10
-3
0
3
10
20
30
40
Gv / Np
-4,6
-2,3
-0,7
0
0,7
2,3
4,6
6,9
9,2
GP / Np
-2,3
-1,15
-0,35
0
0,35
1,15
2,3
3,45
4,6
GNp
Michael E.Auer
P2
=10 ⋅ ln
P1
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Bode Plots (4)
Bode Plots are semilog plots of the magnitude (in decibels) and
phase (in degrees) of a transfer function versus frequency.
Polar form of transfer function H:
H = H ∠ φ = H ⋅ e jφ
ln H = ln H + ln e jφ = ln H + jφ
In a Bode Plot H is plotted in decibels (dB) versus frequency
H dB = 20 ⋅ lg H
and Φ is plotted in degrees.
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Bode Plots (5)
Examples
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Possible First-order Low and High Pass Filter
lowpass
Michael E.Auer
highpass
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Low Pass Filter (1)
The transfer function is
1
Vo
1
j ωC
=
=
H(ω ) =
Vs R + 1/ jω C 1 + jω RC
The magnitude is H (ω ) =
The phase is
1 ,
1 + (ω / ωo ) 2
ω
φ = − arctan
ωo
ωo =1 τ =1/RC
Michael E.Auer
= corner frequency ωc
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Low Pass Filter (2)
Bode plot
H
Michael E.Auer
H dB
01.11.2011
BSC03
Basics in Systems and Circuits Theory
High Pass Filter (1)
The transfer function is
H(ω ) =
Vo
jω L
1
=
=
Vs R + jω L 1 + R
jω L
,1
H (ω ) =
1+ (
The magnitude is
The phase is
ωo 2
)
ω
ω
φ = − tan
ωo
−1
ωo = R/L
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
High Pass Filter (2)
Bode plot
H
Michael E.Auer
H dB
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Series Resonance (1)
Resonance is a condition in an RLC circuit in which the
capacitive and inductive reactance are equal in
magnitude, thereby resulting in purely resistive
impedance.
Resonance frequency:
1
)
Z = R + j (ω L −
ωC
Michael E.Auer
01.11.2011
1
ωo =
or
LC
1
fo =
2π LC
BSC03
Basics in Systems and Circuits Theory
Series Resonance (2)
1
Z = R + j (ω L −
)
ωC
The features of series resonance:
The impedance is purely resistive, Z = R;
• The supply voltage Vs and the current I are in phase;
• The magnitude of the transfer function H(ω) = Z(ω) is minimum;
• The inductor voltage and capacitor voltage can be much more than
the source voltage.
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Parallel Resonance
It occurs when imaginary part of Y is zero.
1
1
Y = + j (ω ( −
)
R
ωL
Resonance frequency:
ωo =
Michael E.Auer
1
1
or f o =
2π LC
LC
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Passive Filters Overview
• A filter is a circuit that is
designed to pass signals
with desired frequencies
and reject or attenuate
others.
• Passive filter consists of
only passive element R,
L and C.
• There are four types of
filters.
Michael E.Auer
01.11.2011
Low Pass
High Pass
Band Pass
Band Stop
BSC03
Basics in Systems and Circuits Theory
Cut-off Frequencies
We set:
1 2
R + (ωL −
) =R⋅ 2
wC
2
Mid-band frequency
ω o = ω1ω 2
Bandwidth
ω1 = ω− 45°
R
R 2 1
( ) +
=−
2L 2L
LC
ω2 = ω+ 45°
R
R 2 1
=+
( ) +
2L 2L
LC
B = ω2 − ω1
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Quality Factor
Q=
Inductive or capacitive component ωo L
1
=
=
resistive component
R
ωo CR
Relationship between B, Q and ω0
B=
R ωo
=
= ωo2CR
L Q
• The quality factor is the ratio of its
resonant frequency to its bandwidth.
• If the bandwidth is narrow, the quality
factor of the resonant circuit must be high.
• If the band of frequencies is wide, the
quality factor must be low.
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Series and Parallel Filter Comparison
Characteristic
Series circuit
Parallel circuit
1
LC
1
LC
ωo L
1
or
R
ωo RC
R
or ωo RC
ωo L
ωo
Q
B
ω1, ω2
Q ≥ 10, ω1, ω2
Michael E.Auer
ωo
ωo
Q
Q
ωo 1 + (
ω
1 2
) ± o
2Q
2Q
ωo ±
01.11.2011
B
2
ωo 1 + (
1 2 ωo
) ±
2Q
2Q
ωo ±
B
2
BSC03
Basics in Systems and Circuits Theory
Second-order Filters
LC Bandstop Filters
RC Bandpass Filter
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Parallel-T-Filter
Vi
Vo
T
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Active Filters (1)
Active filters consist of combinations of resistors,
capacitors and OpAmps.
1. Smaller, less expensive, no inductors;
integrated circuit realization possible!
2. Can provide amplifier gain in addition to
providing the same frequency response as
passive RLC filters.
3. Can be combined with buffer amplifiers
(voltage followers) to isolate each stage of a
complex filter.
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Active Filters (2)
General first-order active filter
Zf
Vo
H (ω ) =
=−
Vi
Zi
Michael E.Auer
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Active First-order Lowpass Filter
Rf
1
H (ω ) = −
⋅
Ri 1 + jωC f R f
Gain
corner frequency
Michael E.Auer
Filter function
ωc =
1
τf
=
1
C f Rf
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Active First-order Highpass Filter
H (ω ) = −
j ωC i R f
1 + jωCi Ri
H (∞) = −
Rf
Ri
corner frequency
Michael E.Auer
ωc =
1
τi
=
1
Ci Ri
01.11.2011
BSC03
Basics in Systems and Circuits Theory
Active Bandpass Filter
H (ω ) = −
Rf
Ri
⋅
1
1 + jωC1 R
⋅
j ωC 2 R
1 + j ωC 2 R
Upper corner frequency
ω2 =
1
C1 R
Lower corner frequency
ω1 =
1
C2 R
Passband gain
H (ω0 ) = K =
Rf
Ri
⋅
ω2
ω1 + ω2
Center frequency
ω0 = ω1 ⋅ ω2
Michael E.Auer
01.11.2011
BSC03