Engineering science lesson 8

Chapter 3- DC and AC theory

3.1 DC electrical principles
3.1.1 Ohm’s and Kirchhoff’s laws
3.1.2 voltage and current dividers
3.1.3 analogue and digital signals
3.1.4 review of motor and generator principles
3.1.5 fundamental relationships (eg resistance,
inductance, capacitance; series C-R circuit, time
constant, charge and discharge curves of
capacitors, L-R circuits)

Electrical current
Electrical current is the rate of flow of
electrical charge through a conductor or
circuit element.
The units are amperes (A), which are
equivalent to coulombs per second (C/s).

Mathematical relationship

dq (t )
i (t ) =
dt
t
q (t ) = ∫ i (t )dt + q (t0 )
t0

Direction of current

The current direction in the circuit elements
(a) Indicating current i1 flows from
a to b
(b) Indicating current i2 flows from
b to a

Voltage
The voltage associated with a circuit
element is the energy transferred per unit
of charge that flows through the element.
The units of voltage are Volts (V), which are
equivalent to joules per coulomb (J/C).

Note:
Relationship between voltage and current is
given by ohms law

Direction of voltage drop

The voltage vab has a
reference polarity that is
positive at point a and
negative at point b

The positive reference for
v is at the head of the
arrow.

Resistor
• A resistor is a circuit element that dissipates
electrical energy (usually as heat)
• Eg: incandescent light bulbs, heating elements
(stoves, heaters, etc.), long wires
• It may be lumped (eg: bulbs) or continuous type
(distribution lines)
• Resistance is measured in Ohms (Ω)
• Demonstration of colour code calculator

Resistance Related to Physical Parameters

ρL
R=
A

ρ is the resistivity of the material used to construct
the resistor (Unit is Ohm-meter)

Resistor construction

Old style carbon resistor: Ceramic cylinder with thin
film layer that is made converted into a special carbon
wire by cutting groves in the cylinder
New style carbon resistor: Ceramic plate with carbon
film layer that is converted into long zig-zag wire with
groves

Questions to think
• Why carbon is used for resistors
• Why did they change in shape
• Why use resistors
• How the power rating of the resistor get
changes
• What is the standard symbol of a resistor
• What is a conductor

Ohms law
• Ohms law: Current through a resistor is
proportional to the voltage applied across it at a
given temperature
• Ohms law establishes a relationship between
voltage and current. It can be mathematically
expressed as
I ∝V
1 V- Voltage
I = V
R I – Current
V = IR R - Resistance

Resistors and Ohm’s Law
a

v = iR
vab = iab R
b

Power and energy
p(t ) = v (t )i (t )
P - Power (J/s or W)
t2

w = ∫ p(t )dt W - energy (J)

t1

p(t) = v(t)i(t)
From ohms law v(t) = i(t)R or i(t) = v(t)/R
p(t) = i2(t) R = v2(t)/R

Example: a 25W Bulb
• If the voltage across a 25W bulb is 120V,
what is its resistance?
R = V2/P = (120V)2/25W = 576 Ω
• What is the current flowing through the
25W bulb?
I = V/R = 120V/576 Ω = 0.208 A

Thought Question
• When measured the resistance of a 25W
bulb, the value got was about 40Ω.
What’s wrong here?

• Answer: The resistance of a wire
increases as the temperature increases.
For tungsten, the temperature coefficient
of resistivity is 4.5x10-3/oK. A light bulb
operates at about 5000oF.

Direct Current (DC) and
Alternating Current (AC)

When a current is constant with time, we say
that we have direct current, abbreviated as dc.

On the other hand, a current that varies with
time, reversing direction periodically, is called
alternating current, abbreviated as ac.

dc and ac current waveforms

.

ac currents can have various waveforms

See the list of circuit elements

Kirchoff’s law
• KIRCHHOFF’S CURRENT LAW (KCL)
The net current entering a node is zero.
Alternatively, the sum of the currents entering a
node equals the sum of the currents leaving a
node.

• KIRCHHOFF’S VOLTAGE LAW (KVL)
The algebraic sum of the voltages equals zero
for any closed path (loop) in an electrical circuit

KCL (Kirchhoff’s Current Law)
i1(t) i5(t)
i2(t) i4(t)

i3(t)
The sum of currents entering the node is
n zero:

∑ i (t ) = 0
j =1
j

Analogy: mass flow at pipe junction

KVL (Kirchhoff’s Voltage Law)

+ –
v2(t) +
+
v1(t) v3(t)
–
–

• The sum of voltages around a loop is
zero: n

∑ v j (t ) = 0
j =1

• Analogy: pressure drop thru pipe loop

KVL Polarity
• A loop is any closed path through a circuit
in which no node is encountered more
than once
• Voltage Polarity Convention
– A voltage encountered + to - is positive
– A voltage encountered - to + is negative

In applying KVL to a
loop, voltages are added
or subtracted depending
on their reference
polarities relative to the
direction of travel
around the loop

Consider the circuit shown below. Use Ohm’s law,
KVL, and KCL to find Vx

Using KVL, KCL, and Ohm’s
Law to Solve a Circuit

15 V
iy = =3A
5Ω

ix + 0.5ix = i y

ix = 2 A

v x = 10ix = 20 V
Vs = v x + 15

Vs = 35 V

Voltage Dividers
• Resistors in series
provide a mechanism
• The resistors
determine the output
Voltage
• KCL says same
current in R1 and R2
• Vout =
Example: Light dimmer (has a potentiometer
V1 * R2/(R1+R2) which is a variable resistance). You dim the
light by the ratio of resistors dropping the
voltage going to the light bulb

Voltage Division

R1
v1 = R1i = v total
R1 + R2 + R3

R2
v 2 = R2 i = v total
R1 + R2 + R3

R3
v3 = R3i = vtotal
R1 + R2 + R3

Current Dividers
• Resistors in parallel provide a mechanism
• The resistors determine the current in
each path
• I1 * R1 = I2 * R2, I2 = I1 * R1/R2
• I = I1 + I2  I1 = I * R2/(R1+R2)
I1
R1
I
I2 R2

Current Division
v R2
i1 = = itotal
R1 R1 + R2

v R1
i2 = = itotal
R2 R1 + R2

Example Dividers
• Given 10V, Need to
provide 3V, how?
• Resistors in Series
• R2/(R1+R2) = 3/10, choose
R2 = 300 KΩ
• R1 = 700 KΩ
• Why should R1, R2 be
high?
• What happens when we
connect a resistor R3
across R2?

Example Dividers
• Want to divide current into two paths, one
with 30% --how?
• Resistors in parallel
• R2/(R1+R2) = 0.3, Choose R2 = 300 KΩ
• R1 = 700 KΩ
• Why should R1, R2 be high?
• What happens when we connect a resistor
R3 in series with R2?

• Although the following concepts are
very important they are not sufficient
to solve all circuits
– series/parallel equivalents
– current/voltage division
principles

Signal and waveform
• A signal is a physical quantity, or quality, which
conveys information
• The variation of the signal value as a function of
the independent variable is called a waveform
• The independent variable often represents time
• We define a signal as a function of one
independent variable that contains information
about the behavior or nature of a phenomenon
• We assume that the independent variable is
time even in cases where the independent
variable is a physical quantity other than time

Continuous or analog signals
• Continuous signal is a signal that exists
at every instant of time
• In the jargon of the trade, a continuous
signal is often referred to as
continuous time or analog
• The independent variable is a
continuous variable
• Continuous signal can assume any value
over a continuous range of numbers

Discrete-time signals
• A signal defined only for discrete values of
time is called a discrete-time signal or
simply a discrete signal
• Discrete signal can be obtained by taking
samples of an analog signal at discrete
instants of time
• Digital signal is a discrete-time signal
whose values are represented by digits

What is sampling?
• Sampling is capturing a signal at an
instant in time
• Sampling means taking amplitude values
of the signal at certain time instances
• Uniform sampling is sampling every T
units of time
xk = x(kT ) = x(t ) t =0,±T ,±2T ,±3T ,
Sampling
frequency or 1
sampling rate F0 = time step or
T sample interval

Sinusoidal signal
x s (t ) = X s sin( 2πf s t + φ s )
Amplitude Phase in
radian (rad)
xx(t) ==X ssin(2 ππf f st t++φφ) )
s
(t) X sin(2
s
2 s s s s
2

Time in
seconds (s)

0
s
xx

0
s

Frequency in
Hertz (Hz)

-2
-2-0.1 0 0.1 0.2
-0.1 0 0.1 0.2
tt

Modern Capacitors

Ceramic and Electrolyte Capacitors

High Voltage Capacitor Banks

Capacitor

• Capacitors consist of two conductors( insulated from each other) which carry
equal and opposite charges +q and –q.

• If the capacitor is charged then there is a potential difference V between the
two conductors

• The material between the plates is insulating. It has no free charge; charge
does not pass through the insulator to move from one plate to another.

• The charge q is proportional to the potential difference V
• q =CV
• The proportionality constant C is called the capacitance of the capacitor. Its
value depends on the geometry of the plates and not the charge or potential
difference. The unit of capacitance is FARAD

Factors Affecting Capacitance
 Area – directly proportional to
plate area, ‘A’
 Spacing – inversely proportional
to plate spacing, ‘d’
 Dielectric-dependent on the
dielectric as

A
C = ε ( Farad )
d
ε = permittivity of dielectric ( F / m )

Capacitors in Parallel

But V1=V2=V
Total charge ie. Q = Q1+Q2
= C1V+C2V = V(C1+C2)
=VCeq
Where Ceq=C1+C2

Capacitors in Series

V1+V2=V, Q/C1+ Q/C2 =V
Q(1/C1 + 1/C2) =V, i.e. 1/Ceq = 1/C1 +1/C2

Therefore Ceq = (C1C2)/ (C1+C2)

Voltage-Current Relationship

q(t ) = CVc (t )
dq (t ) dVc (t )
ic (t ) = =C
dt dt
dVc (t )
∴ ic (t ) = C
dt
t
1
Vc (t ) = ∫ ic (t )dt + Vc (t0 )
C t0

Energy Stored in a Capacitor
t
w(t ) = ∫ v(t )i (t ) dt
to
t
 dv 
= ∫ v  C  dt
to 
dt 
cancelling differential time and changing
the limits to the corresponding
voltages, we have
v(t ) 1 2 1 q 2 (t )
=∫ Cv dv = Cv (t ) = v(t )q (t ) =
0 2 2 2C

CAPACITORS – DC
Stores charge: Q (Coulombs) I =∆Q/∆T
Flow of charge is Current: I (Amperes)
I
dVC
I =C
dt
1
VC = ∫ idt
C

The capacitor charges
linearly till the voltage across
it reaches the applied
voltage after which the
driving force is lost and the
capacitor ‘blocks’ DC.
Example: Time delay circuit

RC CIRCUIT – DC

VC (t ) = V (1 − e −t / RC )
- VC +

- VR + This is similar but the
capacitor charges non-linearly
till the voltage across it
reaches the applied voltage
after which the driving force is
lost. Time constant τ=RC is
τ the time in which the
capacitor is charged to 67%

RC CIRCUIT – DC
Vo
After a capacitor has charged to
- VC(t) +
I V0, it discharges if there is a
resistance in the external circuit
(otherwise it retains the charge :
Vo use in DRAMs). The discharge is
non-linear VC (t ) = V0 e − t / RC

Time constant
= RC
Example: Discharge the defibrillator
capacitor into the heart

• We will return to Capacitors in the section ‘Impedance’ to consider
their frequency response.

Relationship Between Electricity
and Magnetism
• Electricity and magnetism are different facets
of electromagnetism
– a moving electric charge produces
magnetic fields
– changing magnetic fields move electric
charges
• This connection first elucidated by Faraday,
Maxwell

Magnetic Fields from Electricity
 A static distribution of charges produces an
electric field
 Charges in motion (an electrical current) produce
a magnetic field
 electric current is an example of charges (electrons) in
motion

Faraday’s Law
Faraday’s Law :A voltage is induced in a coil whenever its
flux linkages are changing
Induced EMF produced by a changing Magnetic Flux!

Self Inductance
d λ di di
e = v(t ) µ µ =L
dt dt dt
di
∴ v(t ) = L
dt

t
1
i ( t ) = ∫ v ( t ) dt + i ( t0 )
L t0

Inductances in Series

v(t ) = v1 (t ) + v2 (t ) + v3 (t )
di (t ) di (t ) di (t )
v (t ) = L1 + L2 + L3
dt dt dt
di (t )
v (t ) = Leq
dt

Inductances in Parallel
i (t ) = i1 (t ) + i2 (t ) + i3 (t )
di 1 1 1
= v(t ) + v(t ) + v(t )
dt L1 L2 L3
di (t )
v (t ) = Leq
dt

Energy stored in an inductor

To compute power, p(t)
p(t ) = v(t )i (t )
di di
= L i (t ) = Li
dt dt
To compute energy, w(t) t
di
w(t ) = ∫ p (t )dt = ∫ Li dt
t0
dt
i (t ) i (t )
i 
2

1 2 = ∫ Lidi = L  
 2 0
w(t ) = Li (t ) 0

2

Transients
• The time-varying currents and voltages resulting from the
sudden application of sources, usually due to switching, are
called transients. By writing circuit equations, we obtain
integro-differential equations.

Mathematical Model - Discharging

dvC ( t ) vC ( t )
C + =0
dt R
vC ( t ) = Ke st

dvC ( t )
RC + vC ( t ) = 0
dt
RCKse + Ke = 0
st st

vC ( t ) = Vi e −t RC

Mathematical Model - Charging
dvC ( t ) vC ( t ) Vs
C + =
dt R R
vC ( t ) = A + Ke st

dvC ( t )
RC + vC ( t ) = Vs
dt
RCKse + A + Ke = Vs
st st

vC ( t ) = Vs − Vs e −t τ

Mathematical Model – RL
Circuit
R
t=0 di
L + R ⋅ i = Vs
Vs i(t) L v(t) dt
i( t ) = K1 + K 2 e st

sLK 2 e st + RK 2 e st + RK1 = Vs

i( t ) =
Vs
R
(
1 − e −t τ
)
L
τ=
R

Step by step solution procedure
• Circuits containing a resistance, a source, and an
inductance (or a capacitance)

1. Write the circuit equation and reduce it to a first-
order differential equation.
2. Find a particular solution. The details of this step
depend on the form of the forcing function.
3. Obtain the complete solution by adding the
particular solution to the complementary solution

Use of sinusoidal waveforms
Sinusoidal waveforms are of special interest for a number
of reasons:

 it is a natural form occurring in an oscillator circuit; also
the form of voltage induced in a turn (coil) of wire
rotated in a magnetic field, ie. a generator
 it is the form of voltage used for both distribution of
electricity and for communications
 all periodic waveforms can be represented as a series of
sine waves using fourier analysis.

Coil rotating in a magnetic field
For uniformity, we express
sinusoidal function using cosine
function rather than the sine
function. The functions are related
by the identity

 π
sin ( θ ) = cos  θ − ÷
 2
π
cos θ = sin(θ + )
2

Induced voltage and resulting current in a coil
rotating in a magnetic field is sinusoidal

Sinusoidal Waveform
Vm cos ( ωt + θ )
Vm is the peak value

ω is the angular frequency
in radians per second

θ is the phase angle

T is the period
1
Frequency f =
T 2π
Angular frequency ω=
T
ω = 2πf

Root Mean Square Values
T 2
1
v 2 ( t ) dt V
Vrms =
T ∫ Pavg = rms
0 R T
1 v 2 (t )
Pavg = ∫ dt
T T0 R
1 Pavg = I 2
R
I rms = ∫ i ( t ) dt
2
rms  1 T 
2

T  ∫ v (t )dt 
2
0
RMS Value of a Sinusoid  T 0
 

Pavg =
Vm Im R
Vrms = I rms =
2 2
The rms value for a sinusoid is the peak value divided by the square root
of two. This is not true for other periodic waveforms such as square
waves or triangular waves!

Power in AC Circuits
• Instantaneous power v ( t ) = Vm cos ( ωt + θ v ) i ( t ) = I m cos ( ωt + θi )
p ( t ) = v ( t ) × i ( t ) = Vm I m cos ( ωt + θ v ) cos ( ωt + θi )
1 1
= Vm I m cos ( θv − θi ) + Vm I m cos ( 2ωt + θ v + θi )
2 2

 V  I 
• Average power P = p ( t ) =  m ÷ m ÷cos ( θ v − θ i )
 2  2 

• Power Factor PF = cos ( θ v − θ i )

 V  I 
• Reactive Power Q =  m ÷ m ÷sin ( θ v − θi )
 2  2 

 V  I 
• Apparent Power =  m ÷ m ÷
 2  2 

Engineering science lesson 8

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Engineering science lesson 8