Computer-Aided Circuit Simulation and Verification CSE245 – Fall 2004 Professor:Chung-Kuan Cheng
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Computer-Aided Circuit Simulation and Verification
CSE245 – Fall 2004Professor:Chung-Kuan Cheng
Administration• Lectures: 5:00pm ~ 6:20pm TTH HSS 2152• Office Hours: 4:00pm ~ 4:45pm TTH APM 4256• Textbook
Electronic Circuit and System Simulation MethodsT.L. Pillage, R.A. Rohrer, C. Visweswariah, McGraw-Hill
• TA: Zhengyong (Simon) Zhu ([email protected])
Circuit Simulation
Simulator:Solve dx/dt=f(x) numerically
Input and setup Circuit
Output
Types of analysis:– DC Analysis– DC Transfer curves– Transient Analysis– AC Analysis, Noise, Distortion, Sensitivity
Program Structure (a closer look)
Numerical Techniques:–Formulation of circuit equations –Solution of ordinary differential equations–Solution of nonlinear equations–Solution of linear equations
Input and setup Models
Output
CSE245: Course Outline• Formulation
– RLC Linear, Nonlinear Components,Transistors, Diodes– Incident Matrix– Nodal Analysis, Modified Nodal Analysis– K Matrix
• Dynamic Linear System– S domain analysis, Impulse Response – Taylor’s expansion– Moments, Passivity, Stability, Realizability– Symbolic analysis, Y-Delta, BDD analysis
• Matrix Solver – LU, KLU, reordering– Mutigrid, PCG, GMRES
CSE245: Course Outline (Cont’)• Integration
– Forward Euler, Backward Euler, Trapezoidal Rule– Explicit and Implicit Method, Prediction and Correction– Equivalent Circuit– Errors: Local error, Local Truncation Error, Global Error– A-Stable– Alternating Direction Implicit Method
• Nonlinear System– Newton Raphson, Line Search
• Transmission Line, S-Parameter– FDTD: equivalent circuit, convolution– Frequency dependent components
• Sensitivity• Mechanical, Thermal, Bio Analysis
Lecture 1: Formulation
• KCL/KVL• Sparse Tableau Analysis• Nodal Analysis, Modified Nodal Analysis
*some slides borrowed from Berkeley EE219 Course
Formulation of Circuit Equations
• Unknowns– B branch currents (i)– N node voltages (e)– B branch voltages (v)
• Equations– N+B Conservation Laws – B Constitutive Equations
Branch Constitutive Equations (BCE)
Ideal elements
is = AI · ic, v = ?CCCSvs = RT · ic, i = ?CCVSis = GT · vc, v = ?VCCSvs = AV · vc, i = ?VCVSi = is, v = ?Current Sourcev = vs, i = ?Voltage Sourcev = L·di/dtInductori = C·dv/dtCapacitorv = R·iResistorBranch EqnElement
Conservation Laws
• Determined by the topology of the circuit• Kirchhoff’s Voltage Law (KVL): Every circuit
node has a unique voltage with respect to the reference node. The voltage across a branch eb is equal to the difference between the positive and negative referenced voltages of the nodes on which it is incident– No voltage source loop
• Kirchhoff’s Current Law (KCL): The algebraic sum of all the currents flowing out of (or into) any circuit node iszero.– No Current Source Cut
Equation Formulation - KCL
0
1 2
R1G2v3
R3
R4 Is5
=
−− 0
011100
00111
5
4
3
2
1
iiiii
A i = 0
Kirchhoff’s Current Law (KCL)
N equations
Equation Formulation - KVL
0
1 2
R1G2v3
R3
R4 Is5
=
−
−−
00000
101011
0101
2
1
5
4
3
2
1
ee
vvvvv
v - AT e = 0
Kirchhoff’s Voltage Law (KVL)
B equations
Equation Formulation - BCE
0
1 2
R1G2v3
R3
R4 Is5
=
+
−
−
−
−
55
4
3
2
1
5
4
3
2
1
4
3
2
1
0000
00000
01000
00100
0000
00001
siiiiii
vvvvv
R
R
GR
Kvv + i = is B equations
Equation FormulationNode-Branch Incidence Matrix
1 2 3 j B12
i
N
branches
nodes (+1, -1, 0)
{Aij =+1 if node i is terminal + of branch j-1 if node i is terminal - of branch j0 if node i is not connected to branch j
Equation Assembly (Stamping Procedures)
• Different ways of combining Conservation Laws and Constitutive Equations– Sparse Table Analysis (STA)– Modified Nodal Analysis (MNA)
Sparse Tableau Analysis (STA)
1. Write KCL: Ai=0 (N eqns)2. Write KVL: v -ATe=0 (B eqns)3. Write BCE: Kii + Kvv=S (B eqns)
=
−
Sevi
KKAI
A
vi
T 00
00
00N+2B eqnsN+2B unknowns
N = # nodesB = # branches
Sparse Tableau
Sparse Tableau Analysis (STA)Advantages• It can be applied to any circuit• Eqns can be assembled directly from input data• Coefficient Matrix is very sparse
ProblemSophisticated programming techniques and datastructures are required for time and memoryefficiency
Nodal Analysis (NA)1. Write KCL
A·i=0 (N eqns, B unknowns)2. Use BCE to relate branch currents to branch
voltagesi=f(v) (B unknowns → B unknowns)
3. Use KVL to relate branch voltages to node voltages4. v=h(e) (B unknowns → N unknowns)
Yne=ins
N eqnsN unknowns
N = # nodesNodal Matrix
Nodal Analysis - ExampleR3
0
1 2
R1G2v3
R4 Is5
1. KCL: Ai=02. BCE: Kvv + i = is → i = is - Kvv ⇒ A Kvv = A is3. KVL: v = ATe ⇒ A KvATe = A is
Yne = ins
=
+−
−−++
52
1
433
32
32
10
111
111
siee
RRR
RG
RG
R
Nodal Analysis
• Example shows NA may be derived from STA
• Better: Yn may be obtained by direct inspection (stamping procedure)– Each element has an associated stamp– Yn is the composition of all the elements’ stamps
Spice input format: Rk N+ N- Rkvalue
Nodal Analysis – Resistor “Stamp”
−
−
kk
kk
RR
RR11
11N+ N-
N+
N-
N+
N-
iRk
( )
( ) ∑∑
∑∑
=−−
=−+
−+
−+
sNNk
others
sNNk
others
ieeR
i
ieeR
i
1
1KCL at node N+
KCL at node N-
What if a resistor is connected to ground?
….Only contributes to the
diagonal
Spice input format: Gk N+ N- NC+ NC- Gkvalue
Nodal Analysis – VCCS “Stamp”
−
−
kk
kk
GGGG
NC+ NC-N+
N-
N+
N-
Gkvc
NC+
NC-
+
vc
-
( )( ) ∑∑
∑∑=−−
=−+
−+
−+
sNCNCkothers
sNCNCkothers
ieeGi
ieeGi KCL at node N+
KCL at node N-
Spice input format: Ik N+ N- Ikvalue
Nodal Analysis – Current source “Stamp”
−=
k
k
II
N+ N-N+
N-
N+
N-
Ik
Nodal Analysis (NA)Advantages• Yn is often diagonally dominant and symmetric• Eqns can be assembled directly from input data• Yn has non-zero diagonal entries• Yn is sparse (not as sparse as STA) and smaller than
STA: NxN compared to (N+2B)x(N+2B)
Limitations• Conserved quantity must be a function of node
variable– Cannot handle floating voltage sources, VCVS, CCCS,
CCVS
Modified Nodal Analysis (MNA)
• ikl cannot be explicitly expressed in terms of node voltages ⇒ it has to be added as unknown (new column)
• ek and el are not independent variables anymore ⇒ a constraint has to be added (new row)
How do we deal with independent voltage sources?
ikl
k l+ -
Ekl
=
−
−
klkl
l
k
Ei
e
e
011
1
1k
l
MNA – Voltage Source “Stamp”
ik
N+ N-+ -
Ek
Spice input format: Vk N+ N- Ekvalue
kE00
0-11-100100N+
N-
Branch k
N+ N- ik RHS
Modified Nodal Analysis (MNA)
How do we deal with independent voltage sources?Augmented nodal matrix
MSie
CBYn =
0
Some branch currents
MSie
DCBYn =
In general:
MNA – General rules
• A branch current is always introduced as and additional variable for a voltage source or an inductor
• For current sources, resistors, conductors and capacitors, the branch current is introduced only if:– Any circuit element depends on that branch current– That branch current is requested as output
MNA – CCCS and CCVS “Stamp”
MNA – An example
Step 1: Write KCLi1 + i2 + i3 = 0 (1)-i3 + i4 - i5 - i6 = 0 (2)i6 + i8 = 0 (3)i7 – i8 = 0 (4)
0
1 2
G2v3R4 Is5R1
ES6- +
R8
3
E7v3
- +4
+ v3 -R3
MNA – An exampleStep 2: Use branch equations to eliminate as many branch currents as possible1/R1·v1 + G2 ·v3 + 1/R3·v3 = 0 (1)- 1/R3·v3 + 1/R4·v4 - i6 = is5 (2)i6 + 1/R8·v8 = 0 (3)i7 – 1/R8·v8 = 0 (4)
Step 3: Write down unused branch equationsv6 = ES6 (b6)v7 – E7·v3 = 0 (b7)
MNA – An exampleStep 4: Use KVL to eliminate branch voltages from previous equations1/R1·e1 + G2·(e1-e2) + 1/R3·(e1-e2) = 0 (1)- 1/R3·(e1-e2) + 1/R4·e2 - i6 = is5 (2)i6 + 1/R8·(e3-e4) = 0 (3)i7 – 1/R8·(e3-e4) = 0 (4)(e3-e2) = ES6 (b6)e4 – E7·(e1-e2) = 0 (b7)
MNA – An example
=
−−−
−
−
−+−
+−++
06
00
0
001077000110
101100
011100
0100111
0000111
5
7
6
4
3
2
1
88
88
433
32
32
1
ES
i
iieeee
EE
RR
RR
RRR
RG
RG
R
s
MSie
CBYn =
0
Modified Nodal Analysis (MNA)
Advantages• MNA can be applied to any circuit• Eqns can be assembled directly from input data• MNA matrix is close to Yn
Limitations• Sometimes we have zeros on the main
diagonal
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