EE1101: Circuit Theory

1
EE1101 Circuit Theory

• Circuit Elements
• Alternating Quantities

2
Circuit Elements R, L, C

• R, L C are the basic circuit elements.
• Each one has special properties (physical
  elements) and behaves differently when subjected
  to an electrical signal. E.g. if a d. c. source
  is connected R will dissipate heat, L will be a
  short-circuit C will be an open circuit.

3
R, L, C (cont.)

• Used alone or better in conjunction with others
  to have a desired circuit behaviour
• When connected together in series or parallel and
  subjected to various sources e.g. d. c. or a. c.
  with varying frequency, the responses are of
  engineering interests as will be evident in a. c.
  circuit analyses.

4
R, L, C (cont.)

• It is best to define the impedances of R, L C
  as complex expressions.
• Impedance of R R j0
• Impedance of L 0 jwL
• Impedance of C 0- j/wC

5
R, L, C circuits Series R-L circuit

• Series R-L Circuit

R
i

V
Vsin(wt)
L
-
6
R, L, C circuits Parallel- Series circuit

• Parallel-Series Circuit

i
L
R
Isin(wt)
C
7
R, L, C circuits Parallel- Series circuit

• Exercises
• 1. Determine the expressions for the currents in
  the circuits above.
• 2. Determine the expressions for the different
  impedances total impedance.
• 3.Find arg.Z mod.Z
• 4.Assume a source of varying freq (w) sketch
  arg.Z vs. w indicate critical pts.

8
R, L, C circuits Parallel- Series circuit

• 5. Assume the source to be the reference phasor
  determine the expressions for the various
  voltages currents.
• 6.Sketch the currents voltages vs. w indicate
  all the critical points.

9
Volt-Ampere Relationships

• Basic Assume V I are d. c. values. Hence have
  no phase difference i.e. V is in phase with I.
• Hence
• Power VI (Watts)
• Resistance V/I (Ohms)
• Conductance I/V (mohs)

10
Volt-Ampere Relationships

• Time-varying
• We have so far assumed that the magnitude (phase
  frequency) of any given current or voltage is
  constant.
• In some cases this may not be true either by
  design or due to some influence.
• However, the basic laws still apply.

11
Volt-Ampere Relationships

• Either the magnitude, freq or phase may change
  with time (in our case predictably).
• Consider V I given by
• v V(t)sin(w(t))
• i Isin(w(t))
• Clearly, V w as well as v i vary with time.

12
Sine wave is a Periodic Function

• But, unless stated otherwise, we always assume
  that V w are constant. So that
• v Vsin(wt).
• Or
• v(t) Vsin(wt)
• where
• V peak value.

13
Sine wave is a Periodic Function
V
v Vsin(wt)
T
PeriodT Frequencyw
14
Periodic Functions

• It is noted that even if v w were constant V
  I would be variable.
• But in this case there would certain predictions
  of the values at any time.
• It is also true that the shapes of V I repeat
  themselves after _at_ given time known as the
  period. Hence V I are called periodic functions.

15
Periodic Functions

• v Vsin(wt)
• i Isin(wt)
• These are called sinusoidal functions with period
  (T).
• w 2pf
• Where f is the frequency.
• N.B. f 1/T

16
Average RMS Values

• It is important to understand average values and
  Root Mean Square (RMS) values because they form
  the basis of power in electrical engineering.
• Clearly, the average values of a.c. signals are
  ZERO! We can easily show this by integrating v
  i over a full period.

17
Average RMS Values

• The average values of v I are
• v Vsin(wt)
• i Isin(wt)

18
Average RMS Values

• This would worry us because it may seem that a.c.
  currents are useless when passed through a
  resistor.
• But practice shows that when an a.c. current
  passes through a resistor it dissipates energy in
  the form of heat.
• So we use power to understand the effects of a.c.
  signals.

19
Power Dissipated in Resistor, R

• We know that power dissipated in a resistor is
  given by
• The plot of P is

P
wt
20
Power Dissipated in Resistor, R

• The plot of P clearly shows that it is always
  positive hence its average value is positive.
  an assurance that P causes heat/energy to be
  dissipated in R.
• Let us confirm mathematically

21
Power Dissipated in Resistor, R

• Consider the power dissipated in resistor R when
  the source is a.c. or d.c. Assume that the same
  power is dissipated.

i
I
d.c.
a.c.
R
r
r
R
22
Average RMS Values

• The average power dissipated in R is given by

23
Average RMS Values

• Hence it is clear d.c. current which will
  dissipate equal power to R equals square root of
  the mean of the squares of the a.c. current.
  Hence the name Root of the Mean of Squares or
  RMS.
• Exercise If the a.c. current is given by
• iIsin(wt) find the relationship between the
  d.c. and a.c. currents.

24
Average RMS Values

• Solution

25
Average RMS Values
I
d.c. equivalent value
0.707 I
iIsin(wt)
-I