STATE VARIABLE ANALYSIS
State Variables:
The state variables are the minimal set of system variables that are required to represent the total
behavior of system. Every variable of this minimal set is termed as a state variable. The no. of state variable
is equal to the no. of energy storage elements.
Advantages:
i.
ii.
iii.
At any instant, the internal state of the system can be determined as it is a time domain approach.
For the system where Laplace transforms are not applicable, state variable can be applied.
It is a time domain approach.
Formulation of State Equations:
It can be seen that after writing KVL, KCL and v-i relationship equations and some algebraic
manipulations it is possible to write state equations for a network. We must choose the particular combination
and the particular order in which these equations are invoked. The decision is made on the basis of the following
considerations.
1. The inductor currents and capacitor voltages are not eliminated but kept as state variables the final
equations are to contain integrals. The integrals are the one that arise from the substitution of i =
1
t
∫ V dt + i(t0) inductor current in KCI.. and the substitution of
L 0
2. For capacitors, the initial voltages can be independently specified, and for inductors, the initial
currents can be independently specified. in most cases.
3. The above considerations are non topological and may be used to write state equations orks. However, the
following methods are used to write state equations in a way for more complicated networks.
These methods are :
1. Equivalent source method.
2. Network topological method
1. Equivalent Source Method:
Here the equivalent voltage or current sources replace the capacitors or inductors,
respectively the circuit is reduced to a resistive networks.
Case: 1
Network does not Contain Degeneracies
In this case we are assuming that a given network does not consist of degeneracies. It means the
network does not contain all-inductor cut-sets and ,all-capacitor loops. The capacitors are. replaced by voltage
sources and inductors by current sources. As a result, a pure resistive network with voltage and current sources is
obtained. On applying superposition theorem to resistive network, the current through the capacitors and the
voltage across the inductors are obtained.
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STATE VARIABLE ANALYSIS
Case 2
Network Consists of Degeneracies
In this case, first we need to select the state variables as the method is not applicable directly due
to the presence of degeneracies in it. In such a network, the capacitor voltage is not a part of all capacitor loop
where other voltages across the capacitor are already selected as state variables i.e., the capacitor voltage is
chosen as a state variable only when it is specified independently in the solution. Likewise, the inductors
current is also chosen as state variables only if it cannot form. All inductor onsets as the other current across
inductors are chosen as state variables. These non selected inductor current and capacitor voltages as state
variables are known as excess elements. These excess elements contain voltages and currents which can define
(in case of excess capacitor) as algebraic sum of state variable voltages and their voltage sources. Similarly, the
current through the excess inductor is defined as the 'algebraic sum of state variable currents and their Current
sources. The equivalent Source network is then obtained by replacing the excess capacitors by current
sources and excess inductors by voltage sources. On eliminating the above variables, final state variables
obtained.
Problem: Write the state equations for the network given as shown network.
Solution:
The given network contains no degener Hence, the order of complexity is equal to the nurr energy
storage elements.
Order of complexity = Number of energy storage elements = 2.
The equivalent source diagram is obtained by replacing the capacitor with a voltage source and inductor
with a current source as shown in fig.
Now, applying superposition theorem tofigure i.e., keeping one source at.a time and short circuit
remaining voltage sources and open circuiting the c sources.
Case: 1
When the voltage source Vc is acting, alone the network is represented is shown in fig.
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STATE VARIABLE ANALYSIS
Assume the voltage is flowing reverse direction then the voltage across the inductor is
given by,
VL1=-Vc
Now, the current through the capacitor is given by,
𝑖c1 = −
𝑖𝑐1 = −
𝑣𝑐
𝑅
𝑣𝑐
= −𝑣𝑐
1
Case: 2
When the. Current source is acting alone, and then the network is represented as shown in figure
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STATE VARIABLE ANALYSIS
KLR COLLEGE OF ENGINEERING & TECHNOLOGY, PALONCHA
STATE VARIABLE ANALYSIS
Network Topological Method:
Network Topological Method
Equivalent source method is used to obtain the state equation for simple networks. However, this
method becomes tedious in case of complicated circuits containing more number of loops. Therefore,
another method i.e., the network topological method is preferred. In this method, the state equations are
formulated using network graph theory.
Procedural Steps for Network Topological Method
Step 1
A tree is selected such that all independent voltage sources, capacitors and controlled voltages are
placed in the tree. And all the independent current sources, inductors and controlled current sources are
placed in the co-tree.
Step 2
In the network diagram all the capacitor voltages in the tree and all the inductor currents in co-tree
must be assigned with polarities and directions respectively. If a resistive branch is taken as twig or link,
then represent the twig voltages of the tree or link currents of co-tree in terms of state variables.
If it is not possible then assign a new voltage variable or new current variable to that resistive branch.
Step 3
𝑑𝑣𝑐
Using KCL, capacitor current𝐶 𝑑𝑡 is written as algebraic sum of link currents at that node (or
super node) on either end of the capacitor.
Step 4
𝑑𝑖
Using KVL, inductor voltage𝐿 𝑑𝑡is written as algebrak sum of twig voltages Constituiing the loop such
that no new variable is introduced.
Step 5
Similarly. if a new current variable is assigned to resistor (linki, then using KVL, iR R is set equal to
algebraic sum of twig voltages around the loop containing the resistor link.
Step 6
Equations obtained in step(5) are solved to obtain vn and iR in terms of state variables and the source
quantities.
Step 7
Now the values of vn and in obtained in step(6) are substituted in the equations obtained in steps(3)
and (4), which eliminates the new variables and gives the equations in terms of state variables.
Problem: write the state equations for the circuit as shown in fig.
Solution:
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STATE VARIABLE ANALYSIS
We can draw the normal tree for the circuit given as,
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STATE VARIABLE ANALYSIS
Solution of State Equations:
The two ways alternatively required for the representation of the same physical system are the transfer
function and the state variable methods. As such simultaneous representations required to obtain from one
type to the other in the system model. The certain considerations in order to obtain a transfer function
representation from a given set of state variables.
Consider the state variable equations for single variable systems,
X=AX+BU…………………..(1)
Y=CX………………………..(2)
Applying Laplace transform on both sides of the above equations, we get,
s X(s) — X(0) = AX(s) + BU(s) ... (3)
Y(s) = CX(s)
... (4)
Consider the initial conditions i.e., X(0) = 0, because it is assumed zero in transfer function
representation.
sX(s)= AX(s) + BU(s)
[(s 1 — A)] X(s)= BU(s)
X(s) = 1 — AA-1 BU(s)
... (5)
Substitute equation (5) in the output equation i.e., equation (4), we get,
Y(s) = C[s/ — Ari BU(s)
Transfer function of the system is given by the relation,
Problem: Obtain the transfer function representation for a system represented by Matrices.
Solution:
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STATE VARIABLE ANALYSIS
KLR COLLEGE OF ENGINEERING & TECHNOLOGY, PALONCHA
STATE VARIABLE ANALYSIS
KLR COLLEGE OF ENGINEERING & TECHNOLOGY, PALONCHA
STATE VARIABLE ANALYSIS
Analysis of simple networks with state variable approach:
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STATE VARIABLE ANALYSIS
KLR COLLEGE OF ENGINEERING & TECHNOLOGY, PALONCHA
STATE VARIABLE ANALYSIS
KLR COLLEGE OF ENGINEERING & TECHNOLOGY, PALONCHA