ECE 570
Session 2
Computer Aided Engineering for Integrated Circuits
IC 752-E
Formulation of Circuit Equations
Basics of circuit modeling
1. Notation
2. Circuit elements
3. Kirchoff laws
4. Tableau formulation
5. Modified nodal analysis
Supplemental reading: Vladimirescu, Chpt. 2
Ruehli, Chpt. 2 (by K. Singhal and J. Vlach)
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1. Notation
t - independent variable (time)
x - independent variable (space), applicable in distributed elements
Lumped elements
Variables
i = i (t)
v = v (t)
q = q(t)
dependent variable (current)
dependent variable (voltage)
dependent variable (charge)
Φ = Φ ( t ) dependent variable (flux).
Value of a variable at a specific moment of time:
i ( 0 ) , i ( t1 ) - values of current at t = 0 and t = t1 respectively,
v ( 0 ) , q ( 0 ) , Φ ( 0 ) - values of voltage, charge, and flux at t = 0 .
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Parameters such as R, L, C can be functions of dependent variables. This feature
provides mechanism for implementation of nonlinear elements.
3
Distributed elements (used in modeling interconnections)
Notation
i = i ( x , t ) current as dep. var. - function of position and time
v = v ( x , t ) voltage as dep. var. - function of position and time.
Value of dependent variable at specific argument values
i ( x , 0)
v ( x , 0)
initial distributions of current and voltage in a line
i (d , t )
v (d , t )
time varying current and voltage at the position
x=d
.
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i ( a , t1 )
v (a , t1 )
values of current and voltage at the position
x=a
at time t1 .
−1
Parameters: R , L, C , G are specified in proper units ( Ω , H , F , Ω ) per unit
of length.
In some situations (case of non-uniform lossy lines) these parameters can depend on
position and frequency:
Example:
R = R ( x, ω )
or
R = R (ω ) .
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2. Basic Circuit Elements
(ideal components)
I. Lumped elements
A. Resistor
Linear
Symbol
+
i
v
R
-
Voltage controlled
1
i= v
R
Current controlled
v = Ri
6
Nonlinear resistor
Symbol
+
v
-
Voltage controlled
Current controlled
i
R
i = i (v )
v = v (i)
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B. Capacitor
Linear
+
v
i
C
-
Nonlinear
+
v
-
i
C
dq
q = Cv ; i =
dt
IC : v ( 0 )
q = q (v ); i =
dq
dt
dv
i =C
dt
i=
dq dv
dv
= C (v )
dv dt
dt
IC : v ( 0 ) ; C ( v ) − incremental
capacitance
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C. Inductor
i
Linear
+
v
-
L
di
v = L
dt
IC : i ( 0 )
;
d Φ di
di
v =
= L (i )
di dt
dt
Nonlinear
i
;
IC : i ( 0 )
+
v
-
L
dΦ
L (i ) =
di
−
in crem en ta l
in d u cta n ce
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D. Dependent (controlled) sources
Symbol
VCVS
ik
+
+
vc
vk
-
Linear
+
-
Nonlinear
vk = E k vc
v k = v k (v c )
ik = F k ic
ik = ik ( ic )
-
+
CCCS
ic
vk
ik
-
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VCCS
ik
+
+
vc
vk
-
ik = g k v c
ik = ik (v c )
v k = R k ic
v k = v k ( ic )
-
CCVS
ik
+
ic
vk
+
¯
-
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E. Independent sources - excitations (external, time dependent)
Voltage sources
V = V (t )
E = E (t )
+
+
-
-
Current sources
I = I (t )
+
-
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II. Distributed elements
F. Transmission lines - interconnections
Lossless
i1
v1
im
v2
1
1
2
2
.
.
m
.
.
.
m.
vm
x
0
D
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Vector notation:
v1 ( x , t )
v2 ( x , t )
⋅
v = v ( x, t ) =
⋅
⋅
vm ( x , t )
i1 ( x , t )
i2 ( x , t )
⋅
i = i ( x, t ) =
⋅
⋅
im ( x , t )
Model of interconnections (lossless transmission lines)
∂v
∂i
; IC : v ( x , 0 )
= −L
∂x
∂t
∂i
∂v
; IC : i ( x , 0 )
= −C
∂x
∂t
L - matrix of m × m inductances (PUL)
C - matrix of m × m capacitances (PUL)
D - length of interconnections
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Model of lossy lines (in frequency domain)
dV
= − ( R + jω L ) I
dx
dI
= − ( G + jω C )V
dx
V = V ( x ,ω ) = F {v ( x , t )}
where
I = I ( x ,ω ) = F {i ( x , t )}
R = R (ω ) - matrix of resistances representing the conductor losses
L = L (ω ) - matrix of inductances
C = C (ω ) - matrix of capacitances
G = G (ω ) - matrix of conductances representing the dielectric losses.
Specification: matrices R , L, C , G and line length
D.
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3. Kirchoff laws
Kirchoff current law - KCL
of currents into a node = 0
Kirchoff voltage law - KVL
of voltages in a loop = 0
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4. Tableau formulation
Equations determining the topology
KCL:
Ai = 0 ; A - incidence matrix, i - branch currents
KVL:
v = AT vn ; v - branch voltages , vn - nodal voltages
Constitutive equations (describe the physics of the elements)
CE:
Yv + Zi = W
; Y - "admittance" matrix,
Z - "impedance" matrix, W - vector of sources
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Note that CE are mixed:
wj
Y1
K1
v+
i=
K2
Z2
wE
Y
Z
⇐ currents
⇐ voltages
W
Y1 - admittance matrix , Z 2 - impedance matrix , K1 , K 2 - dimensionless matrices
w j - vector of current sources, w E - vector of voltage sources
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Matrix formulation - Sparse Tableau Analysis (STA)
Vector of unknowns:
i
v =x
vn
Model equations:
A
0
0
1
0
− AT
Z Y
0
T
i
0
v = 0
vn
W
x
Model in the compact form:
⇐ KCL , dim Z = dim Y = n × b
⇐ KVL , dim A = n × b
⇐ CE
, b - # of branches
wS
Tx = w S
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Modified tableau
Obtained from the STA via the elimination of branch voltages using KVL:
v = AT v n
KCL:
CE:
which yields
Ai = 0
YAT vn + Zi = W
or in the matrix form
A
0
Z YAT
0
i
=
vn
W
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5. Modified Nodal Analysis (MNA)
Based on separation of currents into:
I1 - branch currents of elements having an admittance representation
(these currents are eliminated from the equations)
I2 - branch currents of elements which do not have an admittance representation
(includes branch currents of voltage sources and those that required for
output)
J - independent current sources.
Thus we partition the vector of currents accordingly:
I1
i = I2
J
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The incidence matrix is partitioned such that KCL
( A1
A2
AJ )
I1
I2 = 0
J
( Ai = 0) can be written as
A1 I1 + A2 I 2 = − AJ J
The vector of branch voltages is partitioned appropriately:
such that the KVL ( v
V1
A1T
V2 = A2T vn
VJ
AJT
= AT v n
(1)
v T = (V1 V2 VJ )
) can be written in the form
V1 = A1T v n
or else V2 = A2T v n
VJ = AJT v n
(2)
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Constitutive equations are also written in the partitioned form:
1. for elements with admittance representation
2. for elements without admittance representation
Y1V1 = I1
K1V1 + K 2V2 + Z 2 I 2 = W2
(3)
(4)
Note: the vector W2 contains entries representing the voltage sources.
Substitution of (3) and (2a) into (1) yields
T
AY
A
1 1 1 vn + A2 I 2 = − AJ J
(5)
Substitution of (2b) into (4) yields
K1 A1T vn + K 2 A2T vn + Z 2 I 2 = W2
(6)
The equations (5) and (6) define the modified nodal analysis (MNA) with v n , I2 as
unknowns.
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In the matrix form the MNA is defined as follows:
A1Y1 A1T
K1 A1T + K 2 A2T
A2
Z2
vn
− AJ J
=
I2
W2
⇐ KCL
⇐ CE
MNA matrix - interpretation of block matrices
A1Y1 A1T =Y
MNA admittance matrix,
Z 2 - impedance matrix of impedance defined branches
K 1 A1T + K 2 A2T =C
dimensionless matrix in constitutive equation
for "non-admittance" branches,
A2 - incidence matrix for "non-admittance" branches
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MNA equations with block matrices
Y A2
C Z2
vn
− AJ J
=
I2
W2
The entries of the MNA matrix are determined via inspection of the net list
following some simple rules, which are suitable for computer implementation.
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