Circuit Transient Analysis Using State Space Equations - VLSI Design and Test - page 333

3 Solving Circuits with Reduced SV Equations

The proposed method can be applied to circuits with non-linear elements such
as MOSFETs. After formulating the circuit matrix by the MNA method, apply
algorithm 1 to produce a reduced matrix. Then, it must be rearranged to the set
of explicit ordinary differential equations that is ready for the RK4 integration
method to evaluate. However, the derivation of the A sv matrix is lengthy, for
instance, the entry A sv (1 , 1) with the sum of conductance for the fulladder circuit
shown in Fig.1 is ( G dsp 1 + G dsn 4 + G dsp 2 + G dsn 5 − ( G dsp 2 + G gsp 2 ) ... For

the sake of simplicity, let A sv (1 , 1) be written as G 1 , the sum of conductances,
G 2 = A sv (1 , 2) and so on. For B sv ,let i 1 be the sum of currents relating to the
first state variable, etc. Hence, A sv X 1 = B sv is shown as:
G 1 + C 1
dt

( G dsp 2 ∗

v n +1
1
v n +1
2

= C 1

dt v 1 + i 1
C 2

G 2

(12)

G 4 + C 2
dt dt v 2 + i 2

G 3

The sizes of A sv and B sv depend on the number of dynamic elements attached
to the network. There must be at least one dynamic element in the circuit when
using the approach. For N elements in the circuit, A sv and B sv are N × N and

1. The matrix is then transformed to state variable form dv 1
dt

= f ( v 1 ,t ) , dv 2
dt

N × =

f ( v 2 ,t ). The first equation is written as
G 1 + C 1

dt v n +1

= C 1 + G 2 v n +1 v 1 + i 1

dt ×

(13)

1 2

Rearranging gives

+ C 1

dt v 1 + dv 1 + G 2 v 2 n +1 = C 1

G 1 v n +1

dt v 1 + i 1

(14)

1

The final form of the ODE for v 1 is
dv 1
dt

G 1 v n +1 G 2 v n +1

= i 1 −

−

1 2

(15)

C 1

The second equation can be written in the same way to become
dv 2
dt

G 3 v n +1 G 4 v n +1

= i 2 −

−

1 2

(16)

C 2

From equations (15) and (16), it can be concluded that for the circuit equations
with M capacitors and hence M state variables, the matrix can be written as
dv 1
dt dt = f ( v 2 ,t ) ... dv dt = f ( v M ,t ). The final form of the set of

ODE equations representing the non-linear circuit appears as:

= f ( v 1 ,t ) , dv 2

= i 1 −G 1 ( v n +1

) −G 2 ( v n +1
2

) ···−G M ( v n + M )

dv 1
dt

1

C 1

= i 2 −G M +1 ( v n +1

) −G M +2 ( v n +1
2

) ···−G M + M ( v n + M )

dv 2
dt

1

C 2

(17)

.

= i M −G ( M− 1) ×M +1 ( v n +1

) ···−G ( M− 1) ×M + M ( v n + M )
C M

dv M
dt

1

where n again is the number of time steps. These equations can be solved by the
Runge-Kutta method with Newton Raphson iteration.