Basic Laws • Circuit Theorems • Methods of Network Analysis • Non-Linear Devices and Simulation Models

• Basic Laws • Circuit Theorems • Methods of Network Analysis • Non-Linear Devices and Simulation Models
EE Modul 1: Electric Circuits Theory
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
• Current, Voltage, Impedance • Ohm’s Law, Kirchhoff's Law • Circuit Theorems • Methods of Network Analysis

EE Modul 1: Electric Circuits Theory
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Electric Charges

• Charge is an electrical property of the atomic particles of which matter consists, measured in coulombs (C).   • The charge e on one electron is negative and equal in magnitude to 1.602 × 10-19 C which is called as electronic charge. The charges that occur in nature are integral multiples of the electronic charge. 
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Electric Current (1)

• Electric current i = dq/dt. The unit of ampere can be derived as 1 A = 1C/s. • A direct current (dc) is a current that remains constant with time. • An alternating current (ac) is a current that varies sinusoidally with time. 
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Electric Current (2)
The direction of current flow:
Positive ions Negative ions
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Electric Current (3) Example 1 A conductor has a constant current of   5 A.  How many electrons pass a fixed point on the conductor in one minute? Solution Total no. of charges pass in 1 min is given by   5 A = (5 C/s)(60 s/min) = 300 C/min Total no. of electrons pass in 1 min is given by
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Electric Voltage • Voltage (or potential difference) is the energy required to move a unit charge through an element, measured in volts (V). 

• Mathematically,                                     (volt)

– w is energy in joules (J) and q is charge in coulomb (C). 

• Electric voltage, vab, is always across the circuit element or between two points in a circuit.  vab > 0 means the potential of a is higher than potential of b.  vab < 0 means the potential of a is lower than potential   of b.

Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
• Power is the time rate of expending or absorbing energy, measured in watts (W).
 • Mathematical expression:
 Power and Enegy (1)
i
+
– v
i
+
– v
Passive sign convention  P = +vi      p = –vi absorbing power                          supplying power

iv
dt dq
dq dw
dt dwp ⋅ =⋅==
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Power and Enegy (2) • The law of conservation of energy      ∑ =0 p

• Energy is the capacity to do work, measured in joules (J).    • Mathematical expression   ∫ ∫ == t t t t vidtpdtw 0 0
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Active Elements Passive Elements
Independent sources
Dependant sources
• A dependent source is an active element in which the source quantity is controlled by another voltage or current. 

• They have four different types: VCVS, CCVS, VCCS, CCCS.  Keep in minds the signs of dependent sources. 

Circuit Elements (1)
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Circuit Elements (2) Example Obtain the voltage v in the branch shown below for i2  = 1A.

Solution Voltage v is the sum of the currentindependent 10-V source and the current-dependent voltage source vx. 

Note that the factor 15 multiplying the control current carries the units Ω.

Therefore,  v = 10 + vx = 10 + 15(1) = 25 V
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
• Current, Voltage, Impedance • Ohm’s Law, Kirchhoff's Laws,  • Circuit Theorems • Methods of Network Analysis

BSC Modul 1: Electric Circuits Theory Basics
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
• Ohm’s law states that the voltage across a resistor is directly proportional to the current I flowing through the resistor.  • Mathematical expression for Ohm’s Law  is as follows:    R = Resistance 

• Two extreme possible values of R:  0 (zero)  and ∞ (infinite)  are related with two basic circuit concepts:  short circuit and open circuit.
Ohm‘s Law (1)
Riv ⋅=
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Ohm‘s Law (2)
v i
R G = = 1
R vRiivp 2 2 = ⋅=⋅=
• Conductance is the ability of an element to conduct electric current; it is the reciprocal of resistance R and is measured in siemens. (sometimes mho’s)



• The power dissipated by a resistor:     
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Branches, Nodes, Loops (1)
• A branch represents a single element such as a voltage source or a resistor.  • A node is the point of connection between two   or more branches.  • A loop is any closed path in a circuit. 

• A network with b branches, n nodes, and l independent loops will satisfy the fundamental theorem of network topology:  1 −+= nlb
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Branches, Nodes, Loops (2)
Example 1 
How many branches, nodes and loops are there?
Original circuit Network schematics or graph
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Branches, Nodes, Loops (3)
Example 2 
How many branches, nodes and loops are there?
Should we consider it as one branch or two branches?
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Kirchhoff’s Current Law (1)
• Kirchhoff’s current law (KCL) states that the algebraic sum of currents entering a node  (or a closed boundary) is zero.
0
1 =∑ = N n niMathematically,

Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Kirchhoff’s Current Law (2)
I + 4 - (-3) -2 = 0 ⇒  I = -5A
This indicates that the actual current for I is flowing in the opposite direction.
We can consider the whole enclosed area as one “node”.
• Determine the current I for the circuit shown in the figure below.
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Kirchhoff’s Voltage Law (1)
• Kirchhoff’s voltage law (KVL) states that the algebraic sum of all voltages around a closed path (or loop) is zero. 
Mathematically,  0 1 =∑ = M m nv
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Kirchhoff’s Voltage Law (2)
Example • Applying the KVL equation for the circuit of the figure below.
va- v1- vb- v2- v3 = 0 V1 = I ∙R1 ; v2 = I ∙ R2 ; v3 = I ∙ R3   ⇒ va-vb = I ∙(R1 + R2 + R3)
321 RRR vvI b a ++ − =
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Series Circuit and Voltage Division (1) • Series: Two or more elements are in series if they are cascaded or connected sequentially and consequently carry the same current.  • The equivalent resistance of any number of resistors connected in a series is the sum of the  individual resistances. 


• The voltage divider can be expressed as ∑ = =+⋅⋅⋅++= N n nNeq RRRRR 1 21
v
RRR Rv
N
n n + ⋅⋅⋅++ = 21
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Series Circuit and Voltage Division (2)
Example 
    10V and 5Ω are in series.
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Parallel Circuit and Current Division (1) • Parallel: Two or more elements are in parallel if they are connected to the same two nodes and consequently have the same voltage across them.     • The equivalent resistance of a circuit with N resistors in parallel is: 

• The total current i is shared by the resistors in inverse proportion to their resistances. The current divider can be expressed as:  Neq RRRR 1111 21 +⋅⋅⋅++=
n
eq
n n R Ri R vi ⋅ ==
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Parallel Circuit and Current Division (2)
Example 
    2Ω, 3Ω and 2A are in parallel
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
• Current, Voltage, Impedance • Ohm’s Law, Kirchhoff's Laws,  • Circuit Theorems • Methods of Network Analysis

BSC Modul 1: Electric Circuits Theory Basics
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Linearity Property (1) It is the property of an element describing a linear relationship between cause and effect.  A linear circuit is one whose output is linearly related (or directly proportional) to its input.
Homogeneity (scaling) property v = i R  →  k v = k i R

Additive property    
v1 = i1 R and v2 = i2 R   →  v = (i1 + i2) R = v1 + v2
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Linearity Property (2)
28
Example By assume Io = 1 A for IS = 5 A, use linearity to find the actual value of Io in the circuit shown below.
Answer: Io = 3A
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Superposition Theorem (1)
It states that the voltage across (or current through) an element in a linear circuit is the algebraic sum of the voltage across (or currents through) that element due to EACH independent source acting alone.
The principle of superposition helps us to analyze a linear circuit with more than one independent source by calculating the contribution of each independent source separately. 
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Superposition Theorem (2)
We consider the effects of the 8A and 20V sources one by one, then add the two effects together for final vo.
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Superposition Theorem (3) Steps to apply superposition principle

1. Turn off all independent sources except one source. Find the output (voltage or current)     due to that active source using nodal or        mesh analysis.

2. Repeat step 1 for each of the other independent sources.

3. Find the total contribution by adding algebraically all the contributions due to the independent sources.
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Superposition Theorem (4)
Two things have to be keep in mind:

1. When we say turn off all other independent sources:  Independent voltage sources are replaced by 0 V (short circuit) and   Independent current sources are replaced by 0 A (open circuit). 

2. Dependent sources are left intact because they are controlled by circuit variables. 
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Superposition Theorem (5) Example

Use the superposition theorem to find v in the circuit shown below. 3A is discarded by open-circuit
6V is discarded by short-circuit
Answer: v = 10V
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Source Transformation (1)
• An equivalent circuit is one whose v-i characteristics are identical with the original circuit. 
 • It is the process of replacing a voltage source vS in series with a resistor R by a current source iS in parallel with a resistor R, or vice versa.
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Source Transformation (2)
vs open circuit voltage is short circuit current 
(a) Independent source transform

(b) Dependent source transform
Remarks:

• The arrow of the current source is  directed toward the positive terminal of the voltage source.

• The source transformation is  not possible when R = 0 for voltage source and R = ∞ for current source. 
+ +

+ +

-

-

s
s
i vR    =
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Source Transformation (3)
Example

Find vo in the circuit shown below using source transformation. 
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Thevenin’s Theorem (1)
It states that a linear two-terminal circuit (Fig. a) can be replaced by an equivalent circuit (Fig. b) consisting of a voltage source VTH in series with a resistor RTH, 
 where  • VTh is the open-circuit voltage at the terminals.

• RTh is the input or equivalent resistance at the terminals when the independent sources are turned off. 
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Thevenin’s Theorem (2) Example

Using Thevenin’s theorem, find the equivalent circuit to the left of the terminals in the circuit shown below. Hence find i.
Answer: VTH = 6V, RTH = 3Ω, i = 1.5A
6 Ω
4 Ω
(a)
RTh
6 Ω
2A
6 Ω
4 Ω
(b)
6 Ω 2A
+ VT h −
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Norton’s Theorem (1) It states that a linear two-terminal circuit can be replaced by an equivalent circuit of a current source IN in parallel with a resistor RN, 

Where   • IN is the short circuit current through   the terminals.    • RN is the input or equivalent resistance   at the terminals when the independent   sources are turned off.
  The Thevenin’s and Norton equivalent circuits are related by a source transformation.
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Example

Find the Norton equivalent circuit of the circuit shown below.
Answer:  RN = 1Ω, IN = 10A
2 Ω
(a)
6 Ω
2vx +  −
+ vx −
+ vx − 1V

+ − i x
i
2 Ω
(b)
6 Ω 10 A
2vx +  −
+ vx −
Isc
Norton’s Theorem (2)
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
• Current, Voltage, Impedance • Ohm’s Law, Kirchhoff's Laws • Circuit Theorems • Methods of Network Analysis

EE Modul 1: Electric Circuits Theory
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Things we need to know in solving any resistive circuit with current and voltage sources only:
Number of equations • Ohm’s Law     b  • Kirchhoff’s Current Laws  (KCL) n-1 • Kirchhoff’s Voltage Laws  (KVL)  b – (n-1)
Introduction
Number of branch currents and branch voltages = 2b (variables)
Problem: Number of equations!
mesh = independend loop
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Mesh Analysis (1)

1. Mesh analysis provides a general procedure for analyzing circuits using mesh currents as the circuit variables. 

2. Mesh analysis applies KVL to find unknown currents.

3. A mesh is a loop which does not contain any other loops within it (independent loop). 
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Mesh Analysis (2) Example – circuit with independent voltage sources 
Note:  i1 and i2 are mesh current (imaginative, not measurable directly) I1, I2 and I3 are branch current (real, measurable directly) I1 = i1;  I2 = i2;  I3 = i1 - i2
Equations: R1∙i1 + (i1 – i2) ∙ R3 = V1 R2 ∙ i2 + R3 ∙(i2 – i1) = -V2 reordered: (R1+ R3) ∙ i1 - i2 ∙ R3 = V1 - R3 ∙ i1 + (R2 + R3)∙i2 = -V2
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Mesh Analysis (3)
Formalization: Network equations by inspection.
   
   
−
=   
   
⋅   
   
+− −+
2
1
2
1
323
331 )( )(
V V
i i
RRR RRR
General rules: 1. Main diagonal: ring resistance of mesh n 2. Other elements: connection resistance between meshes n and m • Sign depends on direction of mesh currents! 
Impedance matrix
Mesh currents Excitation
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Mesh Analysis (4)
Example: By inspection, write the mesh-current equations in matrix form for the circuit below.
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Nodal Analysis (1) It provides a general procedure for analyzing circuits using node voltages as the circuit variables. 
Example
3
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Nodal Analysis (2) Steps to determine the node voltages:     1. Select a node as the reference node. 2. Assign voltages v1,v2,…,vn-1 to the remaining n-1 nodes. The voltages are referenced with respect to the reference node.  3. Apply KCL to each of the n-1 non-reference nodes. Use Ohm’s law to express the branch currents in terms of node voltages.  4. Solve the resulting simultaneous equations  to obtain the unknown node voltages. 
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Nodal Analysis (3)
v1 v2
Example
3
Apply KCL at node 1 and 2
G1
G3
G2
G1∙v1 + (v1 – v2) ∙ G3 = 1A G2 ∙ v2 + G3 ∙(v2 – v1) = - 4A
reordered: (G1+ G3) ∙ v1 - v2 ∙ G3 =  1A - G3 ∙ v1 + (G2 + G3)∙v2 = - 4A
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Nodal Analysis (4)
Formalization: Network equations by inspection.
   
   
−
=   
   
⋅   
   
+− −+
A2 A1
)(
)(
2
1
323
331
v v
GGG GGG
General rules:

1. Main diagonal: sum of connected admittances at node n 2. Other elements: connection admittances between nodes n and m • Sign: negative! 
Admittance matrix
Node voltages Excitation
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Example: By inspection, write the node-voltage equations in matrix form for the circuit below. Nodal Analysis (5)
Electrical Engineering – Electric Circuits Theory
Michael E.Auer     24.10.2012             EE01
Summary