4 | DIRECT
CURRENT
SERIES &
PARALLEL
CIRCUITS
“
You will never always be motivated.
You have to learn to be DISCIPLINED.
DC-CURRENT SERIES CIRCUITS
A series circuit is a circuit in which
there is only one path for current
to flow along.
The current I is the same in all
parts of the circuits.
I
𝑰𝑺𝑶𝑼𝑹𝑪𝑬 = 𝑰𝑹𝟏 = 𝑰𝑹𝟐 = 𝑰𝑹𝟑
The total resistance in the circuit is
equal to the sum of the resistance
of all the parts of the circuit.
𝑹𝒕𝒐𝒕𝒂𝒍
𝑹𝑻𝒐𝒕𝒂𝒍 = 𝑹𝟏 + 𝑹𝟐 + 𝑹𝟑
𝑽𝒕𝒐𝒕𝒂𝒍
The total voltage across a series
circuit is equal to the sum of the
voltages across each resistance
of the circuit.
𝑽𝑻𝒐𝒕𝒂𝒍 = 𝑽𝟏 + 𝑽𝟐 + 𝑽𝟑
𝑷𝒕𝒐𝒕𝒂𝒍
The total power across a series
circuit is equal to the sum of the
power across each resistance of
the circuit.
𝑷𝑻𝒐𝒕𝒂𝒍 = 𝑷𝟏 + 𝑷𝟐 + 𝑷𝟑
PROBLEM 1
A series circuit has a 50-Ω, a 75Ω, and a 100- Ω resistor in series
shown on the figure below. Find the
total resistance of the circuit.
PROBLEM 2
A series circuit has a 6V across
R1, 30V across R2, and 54V
across R3. What is the total
voltage across the circuit?
PROBLEM 3
A resistor of 45-Ω and a bell of
60- Ω are connected in series shown
below. What voltage is required
across this combination to produce a
current of 0.3A?
PROBLEM 4
A 95V battery is connected in
series with three resistors: 20-Ω,
50-Ω, and 120- Ω. Find the
voltage across each resistor.
PROBLEM 5
In the circuit shown below, find
the total power Pt and power
dissipated by R1, R2 and R3.
POLARITY OF VOLTAGE DROPS
The polarity of the voltage drop is
determined by the direction of
conventional current from positive
to a more negative potential.
The nearer to the positive terminal of
the voltage source is more positive;
while the point nearer to the
negative terminal of the applied
voltage is more negative.
V1 = 10V
A
•
+
V1, V2, V3 are
voltage drops.
B
•
+
V2 =
25V
-
•D
-
V3 = 60V
+
•C
 Va = 95V
 Vb = 85V
 Vc = 60V
 Vd = 0V
CONDUCTORS
• Conductor is a material having many free electrons.
• Generally most metals are good conductors but top 3 good electrical
conductors are COPPER, SILVER, and ALUMINUM.
• Conductors have very low resistance. Copper wire has a typical resistance of
<1Ω for 10feet.
• Function of a wire conductor is to connect a source of applied voltage to a
load resistance with minimum IR voltage drop in the conductor.
WIRE
MEASUREMENT
RESISTIVITY
TEMPERATURE
COEFFICIENT
WIRE MEASUREMENT
• Standard wire sizes corresponds to the American Wire Gauge (AWG).
• Gauge numbers specify the size of round wire in terms of its diameter and
cross-sectional circular area.
 Cross-sectional area of round wire is measure in CIRCULAR MILS (CMIL or CM) where 1
mil is one-thousandth of an inch (0.001 inch).
cmil = CM = 𝒅𝟐
COPPER WIRE
TABLE
WIRE MEASUREMENT
1. Gage numbers increase (1 to 40),
the diameter & circular area
decrease.
Gauge# 5
Gauge# 12
Gauge# 15
2.
The circular area doubles for every
3 gauge sizes.
3.
The higher the gauge number and
the smaller the wire, the greater the
resistance of the wire for any given
length.
RESISTIVITY
• Resistance of a given length (of any conductor) depends upon the resistivity of
the material, length of the wire, and the cross-sectional area of the wire.
R=
𝒍
ρ
𝑨
R = resistance of the conductor, Ω
l = length of the wire, ft
A = cross-sectional area of the wire, CM
Ρ = specific resistance or resistivity, CM.Ω/𝑓𝑡 2
PROBLEM 5
What is the resistance of 500-ft
of No.20 and No.23 copper
wires?
TEMPERATURE COEFFICIENT
• The temperature coefficient of resistance, , indicates how much the resistance
changes for a change in temperature.
• A positive value means R increases with temperature; a negative means R
decreases; and a zero means R is constant (not varying with changes in
temperature.
𝑹𝒕 = 𝑹𝒐 + 𝑹𝒐 ( T)
R = resistance of the conductor, Ω
l = length of the wire, ft
A = cross-sectional area of the wire, CM
Ρ = specific resistance or resistivity, CM.Ω/𝑓𝑡 2
PROBLEM 5
A tungsten wire has a 10-Ω
resistance at 20ºC. Find its
resistance at 120ºC.
DC-CURRENT PARALLEL CIRCUITS
A parallel circuit is a circuit in
which two or more components are
connected across the same voltage
source.
The voltage V is the
same in all parts of the
circuits.
V1
V2
V3
𝑽𝑺𝑶𝑼𝑹𝑪𝑬 = 𝑽𝑹𝟏 =
𝑽𝑹𝟐 = 𝑽𝑹𝟑
I
I1
I2
I3
The total current 𝑰𝑻 is
equal to the sum of
all branch currents.
𝑰𝑺𝑶𝑼𝑹𝑪𝑬 = 𝑰𝑹𝟏 +𝑰𝑹𝟐
+ 𝑰𝑹𝟑
𝑷𝟏
𝑷𝑻
𝑷𝟐
𝑷𝟑
The total power 𝑃𝑇
equals the sum of the
individual values of
power in each
branch.
𝑷𝑻 = 𝑷𝟏 +𝑷𝟐 +𝑷𝟑
PROBLEM 6
Two lamps each drawing 2A and
third lamp drawing 1A are
connected in parallel across a 10-V
line. What is the total current?
Two branches 𝑅1 and 𝑅2 across a
110-V power line draw a total line
current of 20A. Branch 𝑅1 takes
12A. What is the current 𝐼2 in
branch 𝑅2 ?
PROBLEM 7
I
𝐼1
𝐼2
PROBLEM 8
A parallel circuit consists of a coffee
maker, a toaster, and a frying pan
plugged into a kitchen appliance
circuit on a 120-V line. What
currents will flow in each branch of
the circuit and what is the total
current drawn by all appliances?
𝑹𝑻 = ?
The total resistance in
parallel is given by
the general
reciprocal formula:
𝟏
𝑹𝑻
=
𝟏
𝑹𝟏
+
𝟏
𝑹𝟐
+
𝟏
𝑹𝟑
𝑹𝑻 = ?
Total resistance of
EQUAL resistors in
parallel is equal to
the resistance of 1
resistor divided by
the number of
resistors.
𝑹
𝑹𝑻 =
𝑵
SIMPLIFIED FORMULAS
𝑹𝑻 = ?
Total resistance of
NONEQUAL resistors
in parallel is equal
to:
𝑹𝟏 𝑹𝟐
𝑹𝑻 =
𝑹𝟏 + 𝑹𝟐
SIMPLIFIED FORMULAS
OPEN CIRCUIT
X
Open in main line
OPEN CIRCUIT
X
Open in parallel
branch
SHORT CIRCUIT
Open in parallel
branch
PROBLEM 9
Find the current in each parallel
branch below. If the resistor in the
second branch burns out, causing an
open circuit, find the new branch
currents.
X
CONDUCTANCES IN PARALLEL
• Conductance is the opposite of resistance.
• The less the resistance, the higher the conductance.
• The symbol for conductance is G and its units is SIEMENS (S).
𝟏
𝑮=
𝑹
𝑮𝑻 = 𝑮𝟏 + 𝑮𝟐 + 𝑮𝟑 + ⋯ . +𝑮𝒏
Find 𝐼1 and 𝐼2 expressing it with
𝐼𝑇 , 𝑅1 and 𝑅2 .
PROBLEM 10
𝐼𝑇
𝐼1
𝐼2
PROBLEM 11
Find 𝑅𝑇 and the 𝑉3 of the circuit?
BOARD WORK 1
Find 𝑅𝑇 and 𝐼4 of the circuit?
𝐼𝑇 = 10A
PROBLEM 12
Find 𝑅𝑇 and the 𝑉5 of the circuit?
PROBLEM 13
Find 𝑅𝑇 of the circuit?
𝐼𝑇 = 10A
END