3.3 State Equations
115
3.3.1 State Equations of Linear RLC Circuits
The SEs of linear circuits containing ideal (linear) resistors, capacitors, inductors,
and independent voltage and current sources can be written by means of an effective
procedure based on considering the storage elements connected to a resistive
multiport network and using the hybrid representation of the multiport resistive
network. The procedure is illustrated in the case where there is one or two storage
elements, but can be easily generalized to any number of storage elements. From
classical circuit theory the state variables are chosen as the capacitor voltages and
inductor currents.
3.3.1.1 First-Order Linear Circuits
Consider a circuit with one capacitor connected to a two-terminal element N
containing linear resistors and independent voltage or current sources, as shown
in Fig. 3.7a. Assume N is voltage-controlled and consider the circuit obtained by
replacing N with its Norton equivalent [1], where G eq is the equivalent conductance
and i sc (t) is the short-circuit current (Fig. 3.7b).
Accounting for the reference directions of currents and voltages, and the CR of
the linear capacitor and N, we have i = G eq v + i sc (t) = G eq v C + i sc (t) = −i C =
−Cdv C /dt, hence the circuit satisfies the first-order linear SE in normal form
dv C
dt
= −
G eq
C
v C −
i sc (t)
C
.
This should be solved using the initial condition v C (t 0 ) = v C 0 .
The dual circuit in Fig. 3.8 where the linear capacitor is replaced with a linear
inductor can be analyzed similarly. If N is current-controlled, by replacing it with
its Thevenin equivalent [1], we obtain v = R eq i + v oc (t) = R eq i L + v oc (t) =
−v L = −Ldi L /dt, where R eq is the equivalent resistance and v oc (t) is the open
circuit voltage of N. Hence, we obtain the first-order linear SE in normal form
di L
dt
= −
R eq
L
i L −
v oc (t)
L
.
This should be solved with the initial condition i L (t 0 ) = i L 0 .
3.3.1.2 Second-Order Linear Circuits
Consider first a two-capacitor configuration. By extracting the capacitors we can
always redraw this configuration as in Fig. 3.9a, where the two-port network N
contains the linear resistors and the independent voltage and current sources.
115
3.3.1 State Equations of Linear RLC Circuits
The SEs of linear circuits containing ideal (linear) resistors, capacitors, inductors,
and independent voltage and current sources can be written by means of an effective
procedure based on considering the storage elements connected to a resistive
multiport network and using the hybrid representation of the multiport resistive
network. The procedure is illustrated in the case where there is one or two storage
elements, but can be easily generalized to any number of storage elements. From
classical circuit theory the state variables are chosen as the capacitor voltages and
inductor currents.
3.3.1.1 First-Order Linear Circuits
Consider a circuit with one capacitor connected to a two-terminal element N
containing linear resistors and independent voltage or current sources, as shown
in Fig. 3.7a. Assume N is voltage-controlled and consider the circuit obtained by
replacing N with its Norton equivalent [1], where G eq is the equivalent conductance
and i sc (t) is the short-circuit current (Fig. 3.7b).
Accounting for the reference directions of currents and voltages, and the CR of
the linear capacitor and N, we have i = G eq v + i sc (t) = G eq v C + i sc (t) = −i C =
−Cdv C /dt, hence the circuit satisfies the first-order linear SE in normal form
dv C
dt
= −
G eq
C
v C −
i sc (t)
C
.
This should be solved using the initial condition v C (t 0 ) = v C 0 .
The dual circuit in Fig. 3.8 where the linear capacitor is replaced with a linear
inductor can be analyzed similarly. If N is current-controlled, by replacing it with
its Thevenin equivalent [1], we obtain v = R eq i + v oc (t) = R eq i L + v oc (t) =
−v L = −Ldi L /dt, where R eq is the equivalent resistance and v oc (t) is the open
circuit voltage of N. Hence, we obtain the first-order linear SE in normal form
di L
dt
= −
R eq
L
i L −
v oc (t)
L
.
This should be solved with the initial condition i L (t 0 ) = i L 0 .
3.3.1.2 Second-Order Linear Circuits
Consider first a two-capacitor configuration. By extracting the capacitors we can
always redraw this configuration as in Fig. 3.9a, where the two-port network N
contains the linear resistors and the independent voltage and current sources.