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CIRCUITS and
SYSTEMS – part I
Prof. dr hab. Stanisław Osowski
Electrical Engineering (B.Sc.)
Projekt współfinansowany przez Unię Europejską w ramach Europejskiego Funduszu Społecznego. Publikacja dystrybuowana jest bezpłatnie
Lecture 4
Methods of analysis of complex
circuits
The basic method - Kirchhoff’s
equations
In this method we describe the circuit by the set of equations
following from Kirchhoff’s laws.
At b branches there is b unknown currents and the same number
of equations. We formulate (n-1) independent KCL equations (n –
the number of nodes) and the rest (b-n+1) equations follows from
KVL. The KVL equations should be written for arbitrary chosen
(independent) meshes of the circuit (avoid choosing the mesh
containing ideal current source).
Thevenin theorem
Any linear circuit (from the point of view of the terminals AB) may
be replaced by an equivalent circuit formed by series connection of
the ideal voltage source UAB and the impedance ZAB .
The voltage source is equal to the voltage across the terminals
AB of the original circuit. The Thevenin impedance ZAB is the
equivalent resistance from the terminals AB after elimination of all
independent sources in the original circuit (voltage sources – short
circuit, current sources – open circuit).
Thevenin theorem (cont.)
The graphical interpretation of Thevenin equivalence
Analysis of circuit using Thevenin theorem
1. Eliminate the branch of the current under interest
2. Replace the original circuit by the Thevenin equivalent circuit
(the circuit below)
3. Calculate the needed current from the equation
U AB
I
Z  Z AB
Example
Using Thevenin theorem calculate the current I of the circuit below.
Assume e(t)=14.1sin(ωt), R0=7.5Ω, R1=5Ω, R2=5Ω, XL=ωL=5Ω,
XC=1/(ωC)=10 Ω, XC0=1/(ωC0)=10 Ω.
Solution
The parameters of the Thevenin equivalent circuit are calculated for
the circuits presented below (after elimination of the branch R0, C0).
The subcircuits for calculation a) impedance ZAB, b) voltage UAB
Thevenin equivalent parameters
Equivalent impedance ZAB
Z AB 
R1R2
Z Z
55
j 5  ( j10)
 L C 

 2,5  j10
R1  R2 Z L  Z C 5  5
j 5  j10
Equivalent voltage UAB
I1 
E
1
R1  R2
I2 
E
 2j
jX L  jX C
U AB  R1 I1  Z C I 2  15
Equivalent Thevenin circuit
Current I
I
Z AB
U AB
 15
 15
 j 26


 1,34e
j 26
 R0  jX C 0 2,5  j10  7,5  j5 11,18e
Norton theorem
Any linear circuit (from the point of view of the terminals AB) is
equivalent to the parallel connection of the ideal current source IZ
and the impedance ZAB.
The current source is equal to the short-circuit current of the
terminals AB. The Thevenin impedance ZAB is the equivalent
resistance from the terminals AB after elimination of all
independent sources in the circuit (voltage sources – short circuit,
current sources – open circuit). It is the same impedance as for
Thevenin theorem.
Norton theorem (cont.)
The graphical interpretation of Norton equivalence
Equivalence of Thevenin and Norton
circuits
Both equivalent circuits should results in the same external current in
the branch attached to AB. Hence the following equivalnce rules
hold.
Nodal method
Nodal method (known also as nodal potential method) allows to
simplify the calculations of all currents in the circuit.
One node is treated as the reference one (the ground node). The
voltages of other independent nodes measured with respect to the
ground one are treated as the nodal potentials. All branch currents
are expressed through these potentials. Application of KCL to all
(N) independent nodes leads to the set of N equations with
respect to nodal potentials. Its solution enables to calculate all
branch currents.
.
Nodal method (cont.)
Any RLC circuit may be described by th enodal equation in
the matrix form
YV  I zr
 Y11 Y12
Y
Y22
21

Y
 ...
...

YN 1 YN 2
... Y1N 
... Y2 N 
. ,
... ... 

... YNN 
V1 


V
V   2 ,
 ... 


V
 N 
I zr
 I zr1 
I 
  zr 2 
 ... 


I
 zrN 
Automatic creation of nodal description
Elements of the main diagonal Yii of the admittance matrix Y are
equal to the sum of all admittances connected to ith node. The
admittance Yij is equal to the common admittance joining node ith
with jth, taken with minus sign (all valid strictly for passive RLC
circuit only).
The elements of the current excitation vector Izr are formed by the
sum of current sources attached to the proper node. The current
entering the node is taken with plus sign and that leaving the node
with minus sign. Only current sources are allowed. The voltage
sources should be transformed to current ones by applying TheveninNorton eqiuvalence.
.
Solution of circuit
1) Form the nodal equation in the form
YV  I zr
2) Solve for nodal potentials V
V  Y 1I zr
3) Calculate branch currents using KVL for each branch at tknown
nodal potentials
Example of nodal analysis
Circuit diagram under analysis
Nodal description
Nodal matrix equation
YV  I zr
 Y2
 Y
 2
 0
 Y2
Y2  Y3  Y4
 Y4
0
I z1  I z 2
 V1  

 V    E Y  I  I 
 Y4
z2
z4 
 2   3 3
Y4  Y5  Y6  V3   I z 4  I z 6  E5Y5 
Solution
Vector of node potentials
V  Y 1I zr
Branch currents
I 2  Y2 (V1  V2 )
I 3  Y3 (V2  E3 )
I 4  Y4 (V2  V3 )
I 5  Y5 (V3  E5 )
I 6  Y6V3
Remarks of nodal analysis
Nodal potential method requires solution of N equations , where N is
the number of independent nodes (always smaller than number of
branches).
For passive RLC networks nodal desription is formed automatically.
When circuit contains controlled sources we apply 2 steps.
1.In the first step treat controlled sources like independent ones and
form automatically the nodal type description
2.In the second step express all controlled sources through node
voltages and move these terms to the left side of equations, forming
the final nodal matrix equation.
Mesh method
In mesh (loop) analysis we assume the independent meshes covering
the whole circuit. Each mesh is associated with a mesh current Ioi
circulating in the mesh (usually of the same direction for all meshes).
Example of choosing the meshes and mesh currents in the circuit
Mesh matrix description
Matrix mesh equation
ZI o  E
 Z11 Z12
Z
Z 22
21

Z
 ...
...

Z N1 Z N 2
Z1N 
 Eo1 
 I o1 
E 
I 
... Z 2 N 
o
2
 , I   o2 
. , E
o
 ... 
 ... 
... ... 



 
... Z NN 
E
 oN 
 I oN 
...
Mesh description results from application of KVL to all
meshes of the circuit. The real current of the branch is the
superposition of mesh currents adjacent to this particular
branch.
Automatic creation of mesh description
Elements Zii on the main diagonal of Z are the sum of impedances
existing in the ith loop. The off-diagonal element Zij is equal to the
impedance common to loop ith and jth, taken with minus sign (at
assumption that all mesh currents have the same directions).
The kth element of the excitation vector E is equal to the sum of
voltage sources existing in kth mesh. If the direction of source is
identical with mesh current it is taken with plus sign, in opposite case
with minus sign.
Example of mesh analysis
Circuit diagram
Mesh description
Since the circuit contains 3 independent loops the mesh desription is
composed of 3 equations and is of the form
ZIo=E
 Z1  Z 2  Z 3

 Z3

 Z1

 Z3
Z 3  Z 4  Z5
 Z5
 Z1
  I o1   E1  E3 
I    E  E 
 Z5
4 
 o2   3
Z1  Z5  Z 6   I o 3   E1  E6 
Solution
Mesh currents
I o  Z 1E
Branch currents
I 1  I o 3  I o1
I 2  I o1
I 3  I o1  I o 2
I 4  I o2
I 5  I o3  I o2
I 6   I o3
Remarks of mesh analysis
Mesh method requires solution of N equations , where N is the
number of independent loops (always smaller than number of
branches).
For passive RLC networks mesh desription is formed
automatically.
When circuit contains controlled sources we apply 2 steps.
1.In the first step treat controlled sources like independent and
form automatically the mesh type desription
2.In the second step express all controlled sources through
mesh currents
and move these terms to the left side of
equations, forming the final mesh matrix equation.
Superposition principle
The time response of the linear circuit at many excitations is the sum
of time responses to each source acting independently in turn, while
the others are replaced in the circuit by their internal resistances (zero
for voltage source and infinite for current source)
Illustration of superposition theorem