Voltage – current relation of network elements
Table. Voltage –Current relation of network elements
Series and parallel connection of circuit elements SL.
No Circuit
element Symbol in
electric circuit Units Voltage – current
Fig. Series and parallel connection of circuit elements Kirchoff ’s Current Law (KCL)
The algebraic sum of currents at a node in an electrical circuit is equal to zero.
Kirchoff ’s Voltage Law (KVL)
In any closed loop electrical circuit, the algebraic sum of voltage drops across all the circuit elements is equal to EMF rise in the same.
Mesh Analysis
In the mesh analysis, a current is assigned to each window of the network such that the currents complete a closed loop. They are also referred to as loop currents. Each element and branch therefore will have an independent current. When a branch has two of the mesh currents, the actual current is given by their algebraic sum. Once the currents are assigned, Kirchhoff’s voltage law is written for each of the loops to obtain the necessary simultaneous equations. The simultaneous equations obtained can be solved using matrix inversion method or crammer’s rule.
Mesh Analysis (using super mesh)
When two of the loops have a common element as a current source, mesh analysis is not applied to both loops separately. Instead both the loops are merged and a super mesh is formed. Now KVL is applied to super mesh.
Nodal Analysis
Typically, electrical networks contain several nodes, where some are simple nodes and some are principal nodes. In the node voltage method, one of the principal nodes is selected as the
C C C
C
C C C C
=
C C C C
=
L=L L LL L
L
L
L L L L
the reference node. At each of these other principal nodes, a voltage is assigned, where it is understood that this voltage is with respect to the reference node. These voltage are the unknowns and are determined by nodal Analysis. When the node voltages to be found by nodal analysis are more than one, the node voltages can be found from simultaneous equations by matrix inversion method or Cramer’s rule
Nodal analysis (including super node)
When two of the nodes have a common element as a voltage source, nodal analysis is not applied to both the nodes separately. Instead both the nodes are merged and a super node is formed.
Now KCL is applied to super node.
Voltage /Current Source
Ideal vs. Practical voltage source
Here E is the EMF of source and is the internal resistance of the source. For an ideal source, is zero and for a practical source, is finite and small.
Ideal vs. Practical current source
Here I is the current of source and is internal resistance of source. For an ideal current source, is infinite and for a practical source, is finite and large.
Dependent Sources
A source is called dependent if voltage / current of the source depends on voltage / current in some other part of the network. Depending upon the nature of the source, dependent sources can be classified as below.
Voltage Controlled Voltage Source (VCVS )
Voltage Controlled Current Source (VCCS)
Current Controlled Voltage Source (CCVS)
Current Controlled Current Source (CCCS)
Fig. Practical Current Source E
Fig. Practical Voltage Source
Superposition theorem
In a linear bilateral network, the current through or voltage across any element is equal to algebraic sum of currents through (or voltages across) the elements when each of the independent sources are acting alone, provided each of the independent sources are replaced by corresponding internal resistances.
Source conversion theorem
Source conversion theorem states that a voltage source, E in series with resistance, as seen from terminals a and b is equivalent to a current source, I = E/ in parallel with resistance, .
Thevenin’s and Norton’s Theorems
Any linear/bilateral network as viewed from terminals A and B can be replaced by a voltage source in series with resistance. The theorem is mainly helpful to draw the load characteristics (output voltage v/s output current as load resistance is varied).
In the figure shown above, V is Thevenin’s voltage as viewed from terminal A & B and is Thevenin’s resistance as viewed from terminals A & B
Norton’s Theorem
Any linear / bilateral network, as viewed from terminals A and B, can be replaced by a current source in parallel with resistance When source conversion theorem is applied for a Thevenin’s equivalent circuit, Norton equivalent circuit is obtained and vice versa.
Let I = Norton current as between terminals A & B and
Fig Demonstration of Norton’s Theorem A Fig Demonstration of Thevenin’s Theorem
Evaluation of Thevenin’s / Norton’s equivalent circuit Let V open- circuit voltage between terminals A & B, I = short – circuit current between terminals A &B, R = Resistance as viewed from terminals A & B,
Table Evaluating of Thevenin’s and Norton’s equivalent circuits S.
Maximum power transfer theorem (as applied to dc network)
Maximum power transfer theorem in a dc network states a condition on load resistance for which the maximum power is transferred to the load resistance. In a dc network, maximum power is transferred to the load when the load resistance is equal to Thevenin’s ( / Norton’s) resistance as viewed from load terminals.
For maximum power transfer, Also, P and I =
Total power consumed in the circuit =
N/W
I E
Fig. Demonstration of maximum power transfer theorem
A
B
R A
B
Star-Delta transformation
( ) ; ( ) ; ( )
( ) ; ( ) ( )
McMillan Theorem
McMillan theorem can be applied to the circuits of the form shown and is based on nodal analysis.
V= ( ∑ E ∑ I) (∑ )
Substitution theorem
Substitution theorem can be used to get incremental change in voltage/current of any circuit element when a resistance R is changed by R and the same can be found by inserting a voltage source – I in series with R.
N/W R I
N/W -I
R
Fig. Demonstration of substitution theorem V
I I
E E
Fig. Mcmillan Theorem B
Fig. Star(Y) – Delta( ) transformation
A
B C
A
C
Reciprocity theorem
Reciprocity theorem states that in a linear bilateral network, voltage source and current sink can be interchanged.
Following are the conditions to be satisfied to apply reciprocity theorem
Only one source is present
No dependent sources are present
No initial conditions ( zero state )
Circuit which satisfies above conditions is called “ eciprocity network”
+
I V N/W
KI
N/W KV
V
Fig. Demonstration of reciprocity theorem