State Equations for Networks

By Mallesham Devasane|Updated : December 28th, 2016

Equations (integro-differnetial) of dynamic circuits can then be transformed into one scalar differential equation of the second or higher order. State equations play an important role in the study of the dynamic behavior of a circuit.

Differential equations of a circuit may also be written as a set of first-order differential equations, or when expressed in matrix form it results in a first-order vector differential equation.

stateequation

The set of differential equations written in the form of first order vector differential equation, such equation is called as state equation.

Advantages of using state equations:

  • It can be represented using matrix.
  • It can be extended to non-linear and other networks.
  • It can be easily programmable for computers.
Procedure for writing down the state equations as follows:
  1. Select the currents through inductors and voltages across capacitors as the state variables.
  2. Write the loop (node-pair) equations for all loops (node-pairs) that contain (are connected to) at least one storage element (that is, an inductor or capacitor)
  3. If there are n storage elements and only m (m<n) loop (node-pair) equations, then there will be an additional (n-m) relationships between the variables we have chosen. Total n equations will be there.

Example-1: Write the State equations for the following circuit using Mesh Analysis

1

Solution:

Mesh Equations:

2

Differentiate the second and third equations (transform integral to differential), the above equations can be written as:

3

Rewrite the above equations as follows:

4

Substitute third equation in first and second for i2.

5

Now, we can define the x1, x2 and x3 as following:

6

Write the state equations as following:

7

Rearrange the above equations as below:

8

In Matrix form, we can represent the above equations as following:

9

Example-2: Write the State equations for the following circuit using Nodal Analysis.

10

Solution: Write the nodal equations for the given circuit as following.

13

Rearranging the above equations, we can obtain the following.

12

Two port Network Parameter equations

1. Z-parameters

  • Port variables: 
    • (Dependent): V1, V2
    • (Independent): i1, i2
  • Equations:
    • V1 = Z11i1 + Z12i2
    • V2 = Z21i1 + Z22i2
  • Condition of reciprocity: Z12 = Z21
  • Condition of symmetry: Z11 = Z22
2. Y-parameters
  • Port variables: 
    • (Dependent): i1, i2
    • (Independent): V1, V2
  • Equations:
    • i1 = Y11V1 + Y12V2
    • i2 = Y21V1 + Y22V2
  • Condition of reciprocity: Y12 = Y21
  • Condition of symmetry: Y11 = Y22
3. h-parameters
  • Port variables: 
    • (Dependent): 1, i2
    • (Independent): I1, V2
  • Equations:
    • V1 = h11i1 + h12V2
    • I2 = h21i1 + h22V2
  • Condition of reciprocity: h12 = –h21
  • Condition of symmetry: h11h22 – h12h21 = 1
4. ABCD or transmission parameters
  • Port variables: 
    • (Dependent): V1, i1
    • (Independent): V2,  i2
  • Equations:
    • V1 = AV2 + B(–i2)
    • i1 = CV2 + D(–i2)
  • Condition of reciprocity: AD – BC = 1
  • Condition of symmetry: A = D
5. g-parameters 
  • Port variables: 
    • (Dependent): i1, V2
    • (Independent): V1, i2
  • Equations:
    • i1 = g11V1 + g12i2
    • V2 = g21V1 + g22i2
  • Condition of reciprocity: g12 = –g21
  • Condition of symmetry: g11 g22 – g12 g21 = 1
6. Inverse transmission parameters
  • Port variables: Express in terms of
    • (Dependent): V2, i2
    • (Independent): V1, i1
  • Equations:
    • V2= A′V1 + B′(–i1)
    • i2= C′V1 + D′(–i2)
  • Condition of reciprocity: A′D′ – B′C′ = 1
  • Condition of symmetry: A′ = D′

Interconnection between Two Ports Network

In certain applications, it becomes necessary to connect the two-port networks together.

  • Cascaded Network

image001

image002

  • Series Network

image003

image004

  • Parallel Network

image005

image006

  • Series-Parallel Network

image007

  • Parallel-Series Network

image008

Key Points

  • Hybrid parameters are used in constructing models for transistors.
  • Transmission parameters are used in the analysis of power transmission lines.
  • The correlation between the two ports network parameters and equivalent T and Z-networks.

Comparison between T and Z Networks

image009

  • The correlation between the image impedances and the ABCD constants are

image014

For symmetric networks, since A = D

image015

It is the characteristic or iterative impedance of the line. A third parameter called the image transfer impedance φ, along with the two independent image impedances is defined to describe a network.

The image transfer impedance φ, is given by

image016

For symmetrical and reciprocal networks:

 image017

where y is the propagation constant of the transmission line.

Posted by:

Mallesham DevasaneMallesham DevasaneMember since Oct 2015
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Comments

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Ajay Lokare
Sir please provide network notes in pdf format...
Amruta Wandre
PDF format please😊
Swapnil Bande
Dynamic logic circuits are usually faster than static counterparts, and require less surface area, but are more difficult to design. Dynamic logic has a higher toggle rate than static logic
...Read More
Tanu

TanuDec 29, 2016

Can anyone please explain the state equation concept??

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tags :ESE & GATE ECNetworksGATE EC OverviewGATE EC NotificationGATE EC Apply OnlineGATE EC Eligibility CriteriaGATE EC Selection Process

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