Download Electric circuits

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 1
The basic laws of electrical circuits
Basic notions
• Carriers of electricity: electrons and protons of atom
• Electric current: the ordered movement of electrical
charges q in time, measured as i=dq/dt. Denoted by the
letter i. Its unit is amper (A)
• Electric voltage: the difference of potentials between
two points of the conducting media (circuit). Denoted
by the letter u. The unit of voltage is volt (V).
• Electric circuits: the connection of electrical elements
enabling the flow of current in such connection.
Basic notions (cont.)
• Branch – one or more circuit elements connected together
of two external terminals accessible for connection to
other elements.
• Node – the terminal of the circuit enabling to connect the
next branches. The nodes are separated by the branch.
• Mesh – the set of circuit branches forming closed way
(loop) for the current.
• Element – the smallest part of the circuit of strictly
defined function.
Basic notions (cont.)
• Passive elements - the electric elements able to either
accumulate or dissipate the energy. They don’t generate
energy. To this set belong: resistor, capacitor and
inductor.
• Active element (sources): the elements generating the
electrical energy. It is usually generated by converting
from other types of energy (mechanical, solar, nuclear,
etc). This set is formed by independent and controlled
sources of either voltage or current type.
• Linear element – the circuit element described by the
linear relation between its voltage and current signals.
• Nonlinear element - the element described by the
nonlinear relation between its voltage and current signals.
Resistor
Graphical symbol of linear resistor
Mathematical description (Ohm’s law)
uR  Ri R
R – resistance
G = 1/R – conductance
The unit of resistance is om () and of conductance
siemens (S).
Inductor
Graphical symbol of inductor
Mathematical description
  LiL
uL 
d
di
L L
dt
dt
•  – flux linkage (unit: Weber = Vs)
• L – self-inductance (inductance) , (unit: Henr = Ωs)
Capacitor
Graphical symbol of capacitor
Mathematical description
q  CuC
duC
dq
iC 
C
dt
dt
• q – charge (unit kulomb = As)
• C – capacitance (unit: farad = As/V)
Independent sources
Graphical symbols of a) voltage, b) current source
Current-voltage characteristics of :
a) voltage source, b) current source
Independent sources (cont.)
• Voltage on the terminals of the ideal voltage source is
independent on the current flowing through it.
• Internal resistance of the ideal voltage source (R=dv/di) is equal
zero (short circuit).
• Current of the ideal current source is independent on the voltage
(load of the source).
• Internal resistance of the ideal current source (R=dv/di) is equal
infinity (open circuit).
Controlled sources - description
• Voltage controlled voltage source
u2  au1
• Current controlled voltage source
u 2  ri1
• Voltage controlled current source
i 2  gu1
• Current controlled current source
i2  bi1
Controlled sources – circuit structures
Graphical symbols of controlled sources
Kirchhoff’s laws
• Current law (KCL)
 ik  0
i1  i2  i3  i4  i5  0
k
• Voltage law (KVL)
u
k
k
0
u1  u 2  u3  u 4  e  0
Example 1
 KCL equations:
i L1  i L 2  iC  0
i L 2  i R1  i R 2  0
i L1  i
 KVL equations:
u C  u L 2  u R1  0
u R1  u R 2  e  0
Example 2
Determine the currents and voltages in the circuit at following values
of parameters:
R1=1, R2=2, R3 = 3, R4 = 4, e = 10V, iz1 = 2A, iz2 = 5A.
Equations of the circuit
• KCL and KVL equations
i z1  i1  i 2  i 4  0
i2  i4  i z 2  i3  0
u R1  u R 2  e  u R 3  0
uR2  e  uR4  0
• Equations including Ohm’s law
i1  i 2  i 4  i z1
i 2  i3  i 4  i z 2
R1i1  R2 i 2  R3i3  e
R2 i 2  R4 i 4  e
• Solution:
i1 = 3,187A, i2 = 0,875A, i3 = 3,812A oraz i4 = -2,062A.
Series connection of resistors
Resistors connected in series
Circuit equation
u  ( R1  R2  ...  RN )i
Equivalent resistance
R  R1  R2  ...  RN
Parallel connection of resistors
Resistors connected in parallel
Circuit equation
i  (G1  G2  ...  GN )u
Equivalent conductance
G  G1  G2  ...  GN
Equivalent resistance for 2 resistors
R1 R2
R
R1  R2
Wye and delta connections
Connection of resistors a) delta i b) wye
Wye-delta transformation
R1 R2
R12  R1  R2 
R3
R 2 R3
R23  R2  R3 
R1
R3 R1
R31  R3  R1 
R2
Delta-wye transformation
R12 R31
R1 
R12  R23  R31
R23 R12
R2 
R12  R23  R31
R31 R23
R3 
R12  R23  R31
Example
Determine the equivalent input resistance Rwe of the circuit.
Assume: R1=2Ω, R2=4Ω, R3=3Ω, R4=2Ω, R5=4Ω, R6=5Ω, R7=5Ω,
R2=10Ω.
The succeeding stages
R23  3  4 
3 4
 10
4
Rz1 
3 4
R35  3  4 
 10
4
R25  4  4 
44
 13,33
3
R we
Rz 2 
R1  R23
 1,667
R1  R23
R4  R35
 1,667
R4  R35
R z3  R z1  R z2  3,333
Rz 4 
3,333 13,33
 2,667
3,333  13,33
R z5  R 6  R z4  R 7  12,667
12,667  10
R z 5 R8


 5,588
R z 5  R8 12,667  10