Chapter III

DC and AC
ELECTRICAL SYSTEM
 PART I

DC ELECTRICAL CIRCUITS
 Introduction
Electrical circuit
-A system of conductors though w/c current or
electricity flows upon the application of
electrical voltage.
DC circuit
- An electrical circuits in w/c the applied source
is dc. The common conductors in electrical
circuits are silver, copper and aluminum.
 Introduction
Types of electric current:
There are two general types of electric current,
distinguished from each other by the manner in
w/c they vary in magnitude and direction; these
are classified as follows:
-Direct current
- Alternating current
 Direct Current
- Sometimes called a continuous current, an
electric current that flows in one direction. One
in w/c the energy transfer takes place
unidirectionally, w/ changes in value from
instant that are either zero or so small that they
may be neglected.
 Direct Current
Forms of direct current:
Direct current can be classified into three forms
that can be derived from three sources, namely:
Continuous dc w/c is produced by a battery,
unidirectional dc derived from a dc generator
and pulsating dc produced from rectifier circuits
(either half wave or full wave)
 Direct Current
Types of current flow:
Conventional flow (+ to - ) is still currently the
preferred type of current flow. Most books are
still using this flow. Unless otherwise stated or
specified, the current is assumed to be
conventional (from point of higher potential to
point of lower potential).
Electron flow w/c is opposite to the
conventional flow is the actual flow of current.
However, this flow is seldom considered no
matter what current flow is followed the
magnitude of current remains the same.
 The Basic Electrical Circuits
Series Circuit Parallel Circuit
 Direct Current
Ohm’s Law:
-The law that relates the three basic electrical
quantities: current, voltage and resistance. This
law is considered the fundamental law of an
electric circuit. This was formulated in 1826 by
Georg Simon Ohm. One simple statement of the
law is given below:
“ The current flowing in a circuit is directly
proportional to the voltage (applied emf) and
inversely proportional to the resistance.”
Direct Current
Direct Current
Person behind the discovery of
power

James Watt ( 1736 – 1819)

 Person behind the discovery of
power
James Watt (1736 – 1819)
- A Scottish inventor and mechanical engineer is
renowned for his improvements of the steam
engine. Watt was born on January 19, 1736 in
Greenock, Scotland. He worked as a
mathematical – instrument maker at the age of
19 and soon became interested in improving the
steam engines, invented by the English engineers
Thomas Savery and Thomas Newcomen, w/c
where used at the time to pump water from
mines.
 Direct Current
The Joule’s Law:
The Joule’s Law about power dissipation of a
resistance element in a circuit fomulated by
James Prescott Joule. The law states that
“ Electric power is dissipated in a resistance
whenever it carries an electric current. The power
dissipated is directly proportional to the square of
electric current and resistance.”
In symbols,
P α I2
p = I2R
 Circuit Connections
The Series Circuit
-A circuit in w/c components like resistors are
connected end to end so that there is only one
path for current flow.
The Series Circuit
The Series Circuit
 The parallel circuit
Parallel Circuit
-A circuit in w/c one end of each resistance is
joined to a common point and the other
resistance is joined to another common point so
that there are as many paths for current flow as
the number of resistances.
The parallel circuit
The parallel circuit
Simple Theorems
Simple Theorems
 Network Laws and Theorems
Some Important Terms:
A network is defined as the interconnection of
components such as resistors and batteries
forming a complicated circuit.
A branch (b) represents a single element such as
a voltage source or a resistor.
A node (n) is the point of connection between
two or more branches.
A loop (l) is any closed path in a circuit.
A mesh is a loop w/c does not contain any other
loops within it.
 Network Laws and Theorems
The fundamental theorem of network topology
is given by
b=l+n-1
 Network Laws and Theorems
Note:
This network cannot be solved easily without the
following theorems. The first solution is by Kirchoff’s
laws using the famous current and voltage laws.
These laws are considered the fundamental laws of
networks in addition to the Ohm’s law. The second
solution is by the so called nodal method w/c is
utilizing the current law. The third is by Maxwell’s
method involving the second KVL law. The
remaining solutions are using theorems like
superposition theorem, Thevenin’s theorem,
Norton’s theorem and the Millman’s theorem.
 The Kirchoff’s Laws
The Kirchoff’s Current Law or Law of
conservation of current
-”In any electrical network, the algebraic sum of
the currents meeting at a point (or junction) is
zero.” This law was based form the law of
conservation of charge. To make the sum of
entering and leaving currents zero the sign
convention followed is that entering or
incoming current is positive and leaving or
outgoing current is negative.
The Kirchoff’s Laws
Person behind Kirchoff’s Laws

Gustav Robert Kirchoff (1824 – 1887)

 Person behind Kirchoff’s Laws
Gustav Robert Kirchoff (1824 – 1887)
- A German physicist, born in Konisberg (now
Kaliningrad, Russia) and educated at the
University of Konisberg. He was professor of
physics at the universities of Breslau,
Heidelberg, and Berlin. With the German
chemist Robert Wilhelm Bunsen, Kirchoff
developed the modern spectroscope for
chemical analysis. In 1860, the two scientist
discovered the elements cesium and rubidium
by means of spectrum analysis.
 Person behind Kirchoff’s Laws
Gustav Robert Kirchoff (1824 – 1887)
- Kirchoff conducted important investigations of
radiation heat transfer and also postulated two
rules, now known as Kirchoff’s laws of networks,
concerning the distribution of current in electric
circuits.
 The Nodal Method
Nodal Method
-Method offers the advantage of requiring minimum
number of equations needed to be written to
determine desired quantities. This method is using the
current law alone.
- “ Every junction in the network that represents a
connection of three or more branches is regarded as
node. One node is always considered a reference node
or zero – potential point, then current equations are
written for the remaining junctions using Kirchoff’s
current law; thus; a solution is possible with n – 1
equations, where n is the number of nodes.
 The Nodal Method
Steps to determine node voltages:
1.) Select a node as reference node. Assign
voltages to the remaining n – 1 nodes. These
voltages are referenced with respect to the
reference node.
2.) 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.
3.) Solve the resulting simultaneous equations
to obtain the unknown node voltages.
 The Maxwell’s Loop (Mesh) Method
Mesh Analysis
- This method was first proposed by Maxwell using the
voltage law only. The methohd involves a set of
independent loop currents assigned to as many
meshes as exist in the circuit, and these currents are
employed in connection with appropriate resistances
when the Kirchoff voltage law equations are written.
The arbitrary assumed loop currents may or may not
exist in the various resistors but when determined will
readily yield the desired values by simple algebraic
additions. The scheme offers the advantage that fewer
equations need to be written to solve a given problem.
 The Maxwell’s Loop (Mesh)
Method
Steps to determine mesh currents:
1.) Assign mesh currents to the meshes. A mesh
is a loop w/c does not contain any other loops
within it.
2.) Apply KVL to each of the n meshes. Use
Ohm’s law to express the voltages in terms of
the mesh currents.
3.) Solve the resulting n simultaneous equations
to get the mesh currents.
 The Superposition Theorem
Superposition theorem
- States that “ the voltage across or current
through an element in a linear circuit is the
algebraic sum of the voltages across or currents
through that element due to each independent
source acting alone.” The idea of the theorem
lies on the linearity property.
 The Superposition Theorem
Steps to apply superposition theorem:
1.) Turn off all independent sources except one
source. Find the output (voltage or current) due
to the active source using conventional
techniques like Ohm’s law and voltage and
current division theorems.
2.) Repeat step 1 for each of the other
independent sources.
3.) Find the total contribution by adding
algebraically all contributions due to the
independent sources.
The Thevenin’s Theorem
 The Thevenin’s Theorem
Steps for solving Thevenin’s theorem:
1.) When the value of n ideal voltage source is
set to zero (deactivate it), the voltage of that
branch becomes zero. The voltage source is
replaced with a short circuit.
2.) Conversely, when an ideal current source is
set to zero, the current of that branch becomes
zero. The current source is replaced with an
open circuit.
The Thevenin’s Theorem
Norton’s Theorem
 Norton’s Theorem
Steps in using Norton’s theorem;
1.) As before (Thevenin’s theorem), identify the
load resistance and remove it from the original
circuit.
2.) Then, to find the Norton current, place a
direct wire connection between the load points
and determine the resultant current. Note that
this step is exactly opposite the respective step
in Thevenin’s theorem.
Norton’s Theorem
Source Transformation method
 The Millman Method
The Millman Method
- “When any number of voltage sources of
arbitrary generated voltage and finite internal
resistance different from zero are connected in
parallel, the resulting voltage across the parallel
combination is the ratio of the algebraic sum of
the currents that each source individually delivers
when short circuited to the algebraic sun of the
internal conductance.”
The Millman Method
 Other Theorems
Reciprocity Theorem
-States that “ if an emf in a circuit A produces a
current in circuit B, then the same emf I the
circuit B produces the same current in circuit A.
Compensation Theorem
- In its simplest form, this theorem asserts that “
any resistance R in a branch of a network in w/c
a current I is flowing can be replaced for the
purpose of calculation, by a voltage equal to IR
 Other Theorems
The Maximum Power Transfer Theorem
“ The maximum power transfer to a load
resistor occurs when its has a value equal to the
resistance of the network looking back from the
load terminals with all the sources of voltage
replaced by their internal resistance.” Under the
condition of maximum power transfer, the
efficiency is only 50%.
 Transformations or Conversions
Delta (Δ) to Wye (Y) or Pi to T
“If the delta is to be electrically equivalent to
the star or wye, the resistance between any pair
of terminals on the delta must be equal to the
resistance of the corresponding terminals on the
star or wye.”
Delta (Δ) to Wye (Y) or Pi to T
Wye (Y) to Delta (Δ)
 Transformations or Conversions
Note:
It is advisable to memorize only the general
formula for conversion not the derived formulas
for they are figure – dependent. Different
arrangements of letters would result into
different formulas. The general formula is not
figure dependent so applicable to whatever
arrangement of letters as labels to resistor
elements.
 Sample Problems
1.) By using Superposition, Thevenin’s Theorem,
Millmans’s and Norton’ Theorem, solve for I
and V in the circuit.
AC ELECTRICAL CIRCUITS
 Introduction
Alternating Current (ac)
- Defined as one that continuously varies in
amplitude and periodically reverses in polarity.
This particular source to a circuit was advocated
by Nikola Tesla, a pioneer in ac theory. He is
Serbian – born American electrical engineer and
inventor.
 Properties of Waves

One cycle of a sinusoidal wave

Properties of Waves
Properties of Waves
Equations of Alternating Emf or
Current
Equations of Alternating Emf or
Current
Average Value of Alternating Emf
or Current
Effective (or RMS) value of AC
(Root Mean Square)
Form factor and Peak factor
 Sample Problems
1. A current is given by 1 = 22.62sin377t,
determine
a. max. value
b. rms value
c. Frequency
d. radians through w/c its vector has gone when
t = 0.01 sec
e. no. of degrees
f. Value of current at instant in d.
 Sample Problems
2. A 50 – cycle Ac has a maximum instantaneous
value of 42.42 A. It crosses the zero axis in a
positive direction when time is 0, determine
a.time when current first reaches a value of 30A.
b. time when current, after having gone through
its maximum positive value, reaches a value of
36.7 A.
c. value of current when time is 1/20 sec.
 Sample Problems
3. An emf is given by 170 sin314.2t, determine
a.max. value
b. rms value
c. frequency
 Sample Problems
4. The figure shows the square tapped 500 cycle
emf wave having a maximum value of 20 V,
determine
a.Average value of a half wave
b. Rms value
Addition of Alternating current (or
voltages) / Phase Relations
Addition of Alternating current (or
voltages) / Phase Relations
Addition of Alternating current (or
voltages) / Phase Relations
 Sample Problems
1. Two alternators A and B are connected to the
same 2, 300 V bus bars. Alternator A delivers
150 A and alternator B delivers 200 A. The
current of alternator A leads that of B by 20°,
determine
a. Total current I, to the load
b. Phase angle between I and current of
alternator A.
Sample Problems
 Alternating Current Circuits
Alternating current power
P = ei
Where
e = instantaneous voltage in volts
i = instantaneous current in ampere
P = instantaneous power in watts
 Alternating Current Circuits
Case I. Voltage and current are In – Phase
 Alternating Current Circuits
Let e = Emsinωt
i = Imsinωt
P = ei
= (Emsinωt)(Imsinωt) = EmImsin2ωt
Note:
If current is changed from + to -, it changed its
direction
When P is +, it means the source is delivering energy
to the load
When p is -, it means that the load is returning energy
back in the source.
 Alternating Current Circuits
P = EI watts
Where:
P – average power
E – rms Voltage
I – rms Current
 Alternating Current Circuits
Case II. Voltage and Current are In –
Quadrature:
Assumption: e leads I by 90°
Alternating Current Circuits
 Alternating Current Circuits
Case III. Voltage and Current differ in Phase by θ.
Assumption: e leads I by θ
Alternating Current Circuits
Alternating Current Circuits
 Alternating Current Circuits
Hence,
P = Scosθ
Where
S – EI (apparent power in VA)
 Sample Problems
1. At light load, a 10 Hp 220 V, 60 cycle single
phase motor takes 29.5 A at 220 V, and the
power factor is 0.44. Determine watts input.
2. Near rated load, the input is 45.2 A at 220 V
and power factor is 0.83. If motor efficiency at
this load is 0.86, determine its output in
horsepower.
Series Circuits
Circuits Containing Resistance Only

Let i = Imsinωt
e = Ri
e = RImsinωt
but
Em = RIm
e = Emsinωt
Note:
In purely resistive circuit, the current is in phase
with the voltage.
Circuits Containing Resistance Only
Sample Problems
 Sample Problems
2. An electric flat iron whose heating element is
practically a pure resistance takes 480 watts when
connected across 115 V d – c mains. Determine
a.Power that it takes from 120 V, 60 cycle mains.
b. Its resistance.
c. Equations of ac voltage, current and power
wave, zero time being when voltage is going
through zero and increasing positively.
d. max. value of power
d. Instantaneous power when t = 1/480 sec.
Circuits Containing Inductance
Only
Circuits Containing Inductance
Only
Circuits Containing Inductance
Only
 Sample Problems
1. A pure inductance takes 4 A from 120 V (rms),
60 cycle mains. Determine
a. Equation of voltage and current waves, zero
time being when current is going through zero
and increasing positively.
b. max. instantaneous power
c. Average power
d. Max. energy stored in inductance
e. Rate at w/c emf of self – induction is changing
when t = 1/240 sec.
 Sample Problems
2. A reactor of 200 W is desired for a 1000 –
cycle telephone circuit. Determine:
a.Its inductance
b. Current that it takes from 50 volt, 796 cycle
supply.
Circuits Containing Capacitance Only
Circuits Containing Capacitance Only
Circuits Containing Capacitance Only
Circuits Containing Capacitance Only
Circuits Containing Capacitance
Only
 Sample Problems
1. A capacitance of 40μF is connected across a 230 V, 60
cycle supply. Determine:
a.Current
b. Equations of emf and current waves, zero time being
when emf is crossing zero axis in a positive direction.
c. Max instantaneous current
d. Equation of power wave
e. max. instantaneous power
f. Average power
g. max. energy stored in a capacitor
h. max. rate of change of current.
 Sample Problems
2. It is desired to obtain 43.5 A at 2,300 V, 60
cycles, by means of capacitors. Determine
a.Necessary capacitance in micro farads
b.KVA rating of capacitors
c.KVA rating at 2,300 V, 25 cycles.
Circuits Containing Resistance and
inductance in Series
Circuits Containing Resistance and
inductance in Series
Circuits Containing Resistance and
inductance in Series
Circuits Containing Resistance and
inductance in Series
 Sample Problems
1. The corrected readings of a voltmeter, ammeter and
wattmeter when connected to measure the voltage,
current and power of circuit known to consist only of
resistance and inductance coil in series as follows.
volts:118, A: 3.27, power: 320 W
The frequency is 60 cycles. Determine:
a.Power factor
b. Circuit phase angle
c. Resistance
d. Inductance
e. Circuit phase angle.
f. Voltage across resistor and across inductance coil
 Sample Problems
2. The current in a circuit known to consists only
of resistance and inductance in series. The current
produced is 8.31 A when the circuit is connected
across 120 V, 25 cycle mains; when connected
across 120 V, 60 cycle mains, the current is 5.30 A.
Determine the resistance and the inductance.
Resistance and Capacitance in Series
Resistance and Capacitance in Series
Resistance and Capacitance in Series
 Sample Problem
1. A current of 2A at 60 cycles flows in a circuit w/
a resistor and a capacitor in series. The voltage
across the resistor is 60V and that across the
capacitor is 90.8 V. Determine:
a. Circuit voltage
b. Power
c. Power factor
d. capacitance
 Sample Problem
2. A circuit w/a 50μF capacitor and an adjustable
resistor in series is connected across 120V, 60
cycle mains. To what value of ohms must the
resistor be adjusted for the circuit to take 80
watts?
Resistance, Inductance and
Capacitance in series
Resistance, Inductance and
Capacitance in series
Resistance, Inductance and
Capacitance in series
Resistance, Inductance and
Capacitance in series
 Sample Problems
1. A voltage of 220 V at 60 cycles is impressed on
a circuit having a 50 Ω resistor, 25 μF capacitor
and 0.2 H inductor in series. Determine;
a. Impedance
b. Current
c. Voltage across resistor, inductor and inductor
d. Total power
e. Power factor and power factor angle
 Sample Problems
2. A series circuit consists of a 12 – Ω, a 24 – Ω
inductive reactor and an adjustable capacitor.
To what two values can the capacitive reactance
be adjusted in order that the circuit may take
620W from 120 V – 25 cycle mains?
Parallel Circuits
Resistance and Inductance in parallel
Resistance and Inductance in parallel
Resistance and Inductance in parallel
 Sample Problems
1. A 24 – Ω resistor and a 0.0796 H inductor are
connected in parallel across a 115 V, 60 cycle
mains. Determine:
a. Current in resistor
b. Current in inductor
c. Total current
d. Power factor
e. Power factor angle
Resistance and Capacitance in Parallel
Resistance and Capacitance in Parallel
Resistance and Capacitance in
Parallel
 Sample problems
A circuit consists of a resistor and a 53 μF
capacitor in parallel across 120 V, 60 cycle
mains. The total current is 4A. Determine ohms
of resistor.
Resistance, Inductance and
Capacitance in Parallel
Resistance, Inductance and
Capacitance in Parallel
Resistance, Inductance and
Capacitance in Parallel
Resistance, Inductance and
Capacitance in Parallel
Resonance
Series Resonance
 What is Resonance?
Resonance
-The condition in a series RLC circuit where the
current is maximum and hence impedance is
minimum.
-At resonance, the circuit behaves like a purely
resistive circuit.
-The capacitive reactance equals inductive
reactance.
- Voltage across inductance equals voltage
through capacitance.
What is Resonance?
Resonance Curve
 Note:
Tuning of a circuit
-Refers to the shape of resonance curve
-When the tuning is sharp, the circuit is more
selective.
Selectivity
-Refers to the sharpness of tuning of the circuit.
Quality Factor
- Measure of selectivity of a circuit.
Quality factor
Resonance
Bandwidth
Quality factor
 Examples
1. A series circuit consists of 12 ohms resistor, a 24
ohms inductive reactor and adjustable
capacitor. To what value should the capacitor
be adjusted in order that the current be a
maximum? Determine
a.) voltage across resistance, inductance,
capacitance.
b.) Power
c.) Quality factor
 Examples
2. An RLC series circuit consists of a resistance of
1000 ohms, an inductance of 100 mH and a
capacitance of 10μμF. If a voltage of 100 v is
applied across the combination, find;
a.) the resonant frequency
b.) Q – factor of the circuit
c.) the half – power frequencies
Resonance in RLC Parallel Circuits
Resonance in RLC Parallel Circuits
 Examples
1. A 0.02 H inductor, a 200 ohm resistor and an
unknown capacitor are connected in parallel
across a 100 v, 120 Hz supply. Determine:
a.) capacitance to make circuit antiresonant
b.) current to inductor and to capacitor
c.) power
DC ELECTRICAL TRANSIENTS
 Introduction

When a dc or an ac circuit is switched either to

apply or remove a source or a component, the
current and voltage in the circuit require a
transition period for them to change values
from initial to final. This transition period is
called the transient period w/c is the time
required for currents and voltages to reach the
steady state values.
 Two States of Transient Circuit

Transient circuit has two states: namely

transient state w/c is a temporary phenomenon
occurring in a dc network prior to reaching
steady state condition and steady state w/c is a
condition wherein circuit values remain
essentially constant.
Two States of Transient Circuit
Response of L and C to a voltage
source
DC Transient Circuits
RL Transient circuit
RL Transient circuit
RC Transient
RC Transient
The RLC Transient Circuit