LINEAR CIRCUITS
BACKGROUND PHYSICAL
1
FORCES & FIELDS
2
Let’s start with a familiar force, gravity.
𝑚2
𝑭
This force is a function of the value of the masses,
𝑚1 and 𝑚2 , and the distance between them, 𝑑
𝑚1 𝑚2
𝐹=𝐺
𝑑2
𝑚1
Instead of using this formula, what we can do is draw a
sort of map of the space around the Earth (𝑚1 ) that
shows the magnitude and direction of the force which acts
on on the apple (𝑚2 ) at every point.
3
𝑚2
𝑭
• To this end we simply place the apple (𝑚2 ) at various points
and represent the force that acts on it as a vector.
• A vector is an entity with magnitude and direction which is
perfect to represent our force!
• The “beefier” our vector, the greater the magnitude of the
force.
𝑚1
• Let’s make it more visual!
• (Can you think of a downside of this approach?)
4
Forces acting on a test mass
(apple), at various points
around a large mass (Earth)
5
•
Our map is great but it only shows the force that will act on our
apple i.e. on a specific mass 𝑚2 
•
Ideally, we want a map from which we can easily work out the
direction and magnitude of the force acting on any mass..
•
The direction will be the same for any mass but the magnitude will
change.
•
What if we divide the magnitude of the force by the value of our
test mass 𝑚2 ?
•
Then we can simply work out the force by multiplying this new
quantity by the value of any mass.
6
GRAVITATIONAL FIELD
• So we take the force experienced at every point by our test mass 𝑚2 and we divide it by 𝑚2 . We term this new vector,
“Gravitational Field”, 𝑮𝑬
𝑭
𝑮𝑬 =
𝑚2
• Now the magnitude of the force acting on 𝑚2 is simply 𝐹 = 𝑚2 𝐺𝐸 . More importantly, we can work out the magnitude
of the force acting on any mass 𝑚𝑥 as 𝐹𝑥 = 𝑚𝑥 𝐺𝐸 .
• Since the direction of the field vector is the same as that of the force acting on a test mass, we can fully determine both
magnitude and direction of the force which would act on any mass by means of the gravitational field 𝑮𝑬 .
• Note that Bold characters indicate vectors and standard ones scalar quantities
7
Gravitational field 𝑮𝑬 at
various points in space
around the Earth
8
WORK &
POTENTIAL ENERGY
9
• If we want to take a mass from the surface of the Earth up to a
certain height ℎ , we will have to apply a force which is opposite in
direction to the gravitational force.
• We will also have to continue to apply this force as we move our
mass up until we reach the desired height.
ℎ
• We can say that we are doing work against gravity or better still,
that we are working against the gravitational field to take our
mass to height ℎ.
• This work will require us to expend some energy and hence we see
it as negative work, in that it causes us to lose energy.
10
• But where did this energy, which we expended through doing
negative work, go?
ℎ
• We have now given our apple the “potential” to do something. If we
release it, it will fall back down to Earth at a certain speed. This
will also require energy!
• The energy that we gave to our mass by doing negative work, i.e.
work against the field, is termed “potential energy”.
• This energy gives our mass the potential to do something, like
falling back down to Earth, if allowed to move freely..
11
To Summarise:
• A Field can be employed to represent the effect of natural forces on objects, like gravity on an apple.
• When we act against a field, we do negative work, in the sense that we lose energy. In doing so, we give the object on
which we do the work, potential energy.
• This potential energy, will allow the object, when left unconstrained, to be affected yet again by the forces which
would naturally act upon it.
• In the case of our apple, the potential energy gives it the ability to fall back down to Earth at a certain speed. By the
time it gets back to the Earth’s surface, all its potential energy will have been turned into Kinetic energy, the energy
required for a mass to be accelerated from rest to a specific velocity.
12
APPLICATIONS
13
Water distribution systems rely on storing water
in a reservoir at the highest point in the town. The
water thus acquires potential energy.
It is then piped so it flows back down through a
different route which takes it to people’s houses.
14
CHARGES &
ELECTRIC FIELDS
15
Charges and Forces (1)
Let’s start with the easy stuff..
Opposite charges attract
Like charges repel
16
Charges and Forces (2)
That is to say that forces act on our charges to either bring them together or pull them apart.
Attractive Force
𝑭𝑨
The magnitude of these forces is directly
proportional to the value of the charges and
inversely proportional to the square of the distance
between them.
𝑭𝑨
Repulsive Force
𝑭𝑹
𝑭𝑹
𝑞1 𝑞2
𝐹=𝑘 2
𝑑
17
Charges and Forces (3)
Now let’s consider the case of a big bunch of fixed negative charges (somewhat stuck together), whose total charge adds up to a value
− 𝑄.
If we place a small positive charge +𝑞𝑡 near −𝑄, it will experience an attractive force which will pull it towards the bunch of like
charges.
−𝑄
𝑭
𝑑
+𝑞𝑡
𝑞𝑡 𝑄
𝐹=𝑘 2
𝑑
The magnitude of this force is a function of the value
of the charges and the distance between them.
18
Electric Forces & Fields (1)
• Just as we did for our gravitational force, we can place a positive test charge (+𝑞𝑡 ) at various points and represent the
force that acts on it as a vector.
• The “beefier” our vector, the greater the magnitude of the force.
−𝑄
𝑭
+𝑞𝑡
𝑑
19
Forces acting on a positive
test charge 𝑞𝑡 , at various
points around our bunch
of negative charges
20
Electric Forces & Fields (2)
• Our map is great but it only shows the force which will act on a charge of value +𝑞𝑡 
• Ideally we want a map from which we can easily work out the direction and magnitude of the force acting on any
charge.
• The direction of course will be the same for any positive charge but the magnitude will change according to its value..
• What if we divide the magnitude of the force by the value of our test charge 𝑞𝑡 ?
• Then we can simply work out the force on any charge by multiplying this new quantity by its value.
21
Electric Forces & Fields (3)
• So we take the force measured at every point with our test charge 𝑞𝑡 and we divide it by the value of 𝑞𝑡 .
𝑭
𝑬=
𝑞𝑡
• Now the magnitude of the force acting on 𝑞𝑡 is simply 𝐹 = 𝑞𝑡 𝐸. More importantly, we can define the force acting on any
charge 𝑞𝑥 as 𝐹𝑥 = 𝑞𝑥 𝐸.
• Since the Electric Field vector has the same direction as the force acting on a positive charge, the direction of the force
acting on a positive charge will simply be that of the Electric Field.
• Note that the direction of the force acting on a negative charge would instead be in opposite direction to the field.
22
Electric filed 𝑬 at various points in
space around our bunch of charges.
23
ELECTRIC FIELDS & POTENTIALS
24
Electric Fields & Potentials (1)
𝑬
+𝒒𝒕
𝑭
• If masses 𝑚1 and 𝑚2 in the gravitational example, are replaced by
a big bunch of negative charges and a small positive charge
respectively, then, when it comes to potential energy, we have a
similar situation as the gravitational field.
−𝑸
• There will be an electric field which will act to bring positive and
negative charges together and we will need to act against this field, i.e.
do negative work, to take them apart.
25
Electric Fields & Potentials (2)
𝑃𝑉𝐸2
2
𝑃
𝑉𝐸1
1
−𝑸
∆𝑃
∆𝑉𝐸
• The further apart we take them, the more potential energy they
acquire.
• Rather than talking of potential energy in electric circuits, we talk
about potential energy per unit charge and we call this Voltage or
Potential. This is indicated by the letter V and measured in Volts.
26
Electric Fields & Potentials (3)
So what we can do is separate positive and negative
charges, much in the same way as we took some of the
water away from the pond and into the reservoir.
∆𝑉
This will create a voltage difference ∆𝑉.
Note that the colour of the charges represents their potential energy: the dark
red is associated with the highest potential and the pink with the lowest.
We then give the charges a different path (circuit)
through which to flow to come together again under the
action of the electric field.
27
Water distribution systems rely on storing water
in a reservoir at the highest point in the town. The
water thus acquires potential energy.
It is then piped so it flows back down through a
different route which takes it to people’s houses.
28
ELECTRIC CURRENT
29
• We also define a measure akin to water flow-rate in the reservoir example, which we term electric current. This measures the amount of
charge (∆𝑄) which crosses a specific volume over an interval of time (∆𝑡), mathematically
∆𝑄
𝐼=
∆𝑡
• While we use pipes to channel water, in the case of charges, we use conductors. These allow positive charges to flow through them, from a
point of higher potential to a point of lower potential.
• Perfect conductors allow charges to flow unopposed and hence very, very fast! These do not exist in practice but a good physical
approximation would be copper wires. They can be seen as very large diameter pipes in the water analogy.
• Also note that we assume that positive charges are the mobile carriers and that negative charges are fixed.. This assumption is termed
“conventional current”. We will see later that, in actual facts, it is negative charges (electrons) which are the mobile carriers.
30
In this animation, we are using a perfect conductor to allow positive charges, which have been separated from negative charges and hence
acquired potential energy, to flow back down to the negative charges.
∆𝑉
Once the positive charges reach the negative charges, they have lost all of their potential energy. Some additional work against the field
(negative work) will need to be done to yet again give the positive charges some potential energy.
31
• Often in circuits, charges will flow through circuit elements called resistors. The internal construction of these, makes it harder for the
charges to flow. Within resistors, charges will be colliding with obstacles and lose energy as they do. This lost energy will manifest as
heat.
• When charges flow through resistors two things happen:
• charges lose potential energy, i.e. they drop some voltage and
• they take longer to flow through and thus a lower rate of flow of charge (electric current) will be observed.
𝐼=
∆𝑄
∆𝑡
The electric current is lower in this than
in the case of the perfect conductor,
since charges take longer to flow back
down to the negative terminal
32
• In the previous example we only had a transient current in that we gave the positive charges potential energy, they flowed back
down through the resistor but then there were no more charges to sustain continuous current flow.
• If we use a battery however, this device will continually supply charges at its positive terminal as they get whisked away through
the resistor. This continuous supply of charges, which would entail keeping the voltage at the positive terminal of the battery pretty
constant, allows us to establish a continuous current.
∆𝑉
• In an actual battery, electro-chemical processes take place
which maintain a constant voltage difference between its
terminals, i.e. there is always a constant supply of charges at
the top terminal with the same potential energy.
33
Resistors & Resistance
The physical property that resistors have to resist current is known as resistance and is represented by the letter 𝑅.
The resistance of any material with a uniform cross-sectional area 𝐴, is a function of its length ℓ and a parameter which is
characteristic of the material, the resistivity 𝜌.
ℓ
ℓ
𝑅=𝜌
𝐴
𝜌
𝐴
34
• It stands to reason that the amount of current that flows through a resistor is proportional to the voltage applied across it. The
more potential energy the charges have, the faster they will flow, the more charge will go through a set volume over a given time.
• It also stands to reason that, if a resistor slows down the flow of charge, the current through it will be inversely proportional to its
resistance.
• As it happens, there is a very simple linear equation which allows us to express these intuitive physical facts in a mathematical
form, “Ohm’s law”:
𝐼
∆𝑉
𝑅
∆𝑉
𝐼=
𝑅
35
If we have two resistors connected in series, what will happen is that charges will drop some voltage (potential energy) through the first
one and then some more through the second one.
𝑅1
∆𝑉
𝑅2
The battery maintains a constant voltage between its terminals and this allows a constant flow of charges through the circuit.
36
Since the rate of flow of charges is constant and continuous, if we measure the voltage at point A, after charges have gone through 𝑅1 ,
this will also be constant. This is because charges always lose the same amount of energy through 𝑅1 and, at any instant in time, the
same amount of charge flows through 𝑅1 .
𝑅1
A
∆𝑉
𝑅2
37
• When we have two resistors connected in series, the electric current through them is the same.
• This stems from the principle of conservation of charge which you will be taught in Fields and Devices. This states that charge is
neither created or destroyed and thus remains constant in a closed system.
• This means that all charge that flows through 𝑅1 must also flow through 𝑅2 . That is to say that the electric current through 𝑅1 is the
same as the electric current through 𝑅2 .
𝐼
∆𝑉
𝑅1
𝐼
𝑅2
𝐼
38
Series resistors – Water analogy (1)
We can yet again resort to a water flow analogy to make
things clearer.
If we have a pump in a loop of water, the water will begin to
flow at a certain rate. This rate will be determined by the
natural resistance of the water.
39
Series resistors – Water analogy (2)
If we now place an obstacle to the flow of water in the form of a sand
filter, this will resist the pump which is attempting to push the water
through the pipe.
The resistance provided by the sand filter to the circling water depends
on the properties of the sand and will decrease the flow rate (i.e. the
speed of circulation) .
40
Series resistors – Water analogy (3)
If the sand in the filter is replaced with a courser sand, it will offer less
resistance to the water flow as there will be larger gaps between the
sand particles. Consequently, the water will be able to pass through the
filter more easily.
It will, therefore, be easier for the pump to circulate the water and the
flow rate will be higher.
41
Series resistors – Water analogy (4)
1
2
Suppose that a single pipe has
two identical filters in series.
The first filter provides a
resistance as does the second
filter.
42
Series resistors – Water analogy (5)
The resistance of the two filters should not
depend on their separation and
consequently, the resistance of the two
filters should be the same as a single filter
twice the length.
1
1+2
2
The flow rate through both filters will be the
same and determined by the sum of their
individual resistances.
43
Series resistors – Water analogy (6)
Similarly, one could assume that there
exists a filter with finer sand which could
provide an equal resistance to the two
filters.
1
3
2
44
• Similarly to the sand filter examples, the current flowing through two resistors in series is the same and is determined by the
combination of both their resistances.
• For the purpose of calculating the values of the current flowing through the circuit, the two resistors could be replaced by a single
resistor 𝑅𝑆 = 𝑅1 + 𝑅2 . Ohm’s law could then be applied.
𝐼
∆𝑉
𝑅1
𝐼
𝐼
𝑅2
∆𝑉
𝑅𝑆
∆𝑉
∆𝑉
𝐼=
=
𝑅𝑆 𝑅1 + 𝑅2
𝐼
• In general, if a circuit comprises of a single loop, the electric current through all elements in that loop will be the same.
45
Voltage References (1)
• So we said that we assume that electric current is made up of positive charges flowing from a point of greater potential to a point of lower
potential.
• Since potential is a relative measure between two points, we may very well assign a value of zero to the lowest potential point so that, at
any point 𝑋, the voltage will be 𝑉𝑋 − 0V.
𝐼
A 𝑉𝐴
• If we set point C as a reference and
assign a value of zero to the voltage at
that point then
𝑅1
B 𝑉𝐵
𝑉𝐴 − 𝑉𝐶 = 𝑉𝐴 = ∆𝑉
𝑅2
∆𝑉
𝑉𝐵 − 𝑉𝐶 = 𝑉𝐵
C 𝑉𝐶
46
Voltage References (1a)
𝐼
𝑅1
A 𝑉𝐴
𝑉𝐵 − 0V
𝑉𝐴 − 0V
∆𝑉
B 𝑉𝐵
C
𝑅2
𝑉𝐶 = 0
47
Voltage References (2)
• The 0V reference is often called ground or Earth and we will see why shortly.
• In the case of the battery, the point to which we would usually assign zero volts for instance, would be the negative terminal.
• At this point we should highlight that the battery, and any device which uses energy to provide a constant voltage, and thus a
constant current over time, is termed a DC power supply.
• The constant current which is produced by the DC power supply is termed DC current.
• We will see later why we use the term power instead of voltage for our DC supply.
48
Voltage References (3)
• In addition to batteries, we have DC supplies which plug into
the mains.
• There are many of these in our labs, as shown in the picture.
• In the case of these power supplies, the 0V reference is the Earth!
• The surface of the Earth is indeed negatively charged, so it provides the point
of lowest potential. We assign a value of 0V to this potential and call it Earth
or ground reference.
49
Voltage References (4)
A
• So the DC supply provides a
voltage which is 𝑉𝑆 Volts above
the voltage of the Earth (0V).
𝑉𝑆
B
• We connect the positive terminal
of this supply to one end of the
circuit (A) and we connect the
other end of the circuit (B) to
ground.
• This way positive charges can flow
through the circuit from the point
of highest potential (A) to the
point of lowest potential (B).
50
Voltage References (5)
• Since we have only one planet Earth, both
the negative terminal of the DC supply and
the other end of the circuit are connected
to it.
A
𝑉𝑆
B
• This doesn’t mean however that the
negative terminal of the power supply and
point B are directly connected!
• They are like two pipes partially submerged
in an ocean, one drawing water and the
other dumping water.
• The water dumped by one pipe, due to the
enormity of the ocean, cannot be directly
picked up by the other pipe!
51
Voltage References (6)
Usually in circuits, instead of using a picture of the
Earth to indicate the ground reference, we use either of
the symbols shown in the figures.
Note that, although we are using separate ground
references for each end of the circuit, these represent
exactly the same voltage value, 0V.
52
Voltage References (7)
• Often, different points of the circuit are connected
together to a common ground reference.
• Make no mistakes though! Although these points are
connected together, they are also connected to the
ocean of charges that is the Earth.
• This means that no current can flow between them.
Current can only flow into and out of the Earth, as
shown in the figures.
53
𝐼
A
B
𝐼
𝑉𝐵
𝑉𝑆
𝐼
𝑉𝑆
𝑅1
𝑉𝑆 − 𝑉𝐵
𝑉𝑆 − 0
C
𝑉𝐵 − 0
𝑅2
𝑉𝐺𝑁𝐷 = 0𝑉
54
To indicate a voltage difference across a circuit element, we usually draw an arrow pointing to the point of higher potential,
as shown.
Note that we use conventional current which is the rate of flow of positive charges.
Since, as they flow through a resistor, charges lose potential energy, the voltage at point B is lower than the voltage at point
A. The voltage and current arrows will therefore point in opposite directions for every resistor.
𝐼
B
𝑉𝐵
𝑉𝑆
𝐼
𝑉𝑆
𝑅1
A
𝐼
𝑉𝑆 − 𝑉𝐵
𝑉𝑆 − 0
𝑉𝐵 − 0
𝑅2
When it comes to our power supply however,
voltage and current arrows are in the same
direction!
The concepts which underpin this difference are
pivotal in circuits analysis and will be discussed
in the next few slides.
𝑉𝐺𝑁𝐷 = 0𝑉
55
Voltage, Current & Power in Electric Circuits (1)
When carrying out circuit analysis, it is essential to draw voltage and current arrows for each circuit element and do so in a
sensible and consistent manner.
There are two main types of circuit elements that we may encounter:
• Passive elements (e.g. resistors, capacitors, inductors)
• Active elements (e.g. sources, transistors, amplifiers)
In the Linear Circuits unit, we will only be concerned with passive elements and a specific type of active elements termed
sources. Batteries and power supplies are examples of DC voltage sources for instance.
In order to understand the difference between these two types of elements we must introduce the concepts of energy and
power.
56
Voltage, Current & Power in Electric Circuits (2)
Elements like resistors, absorb energy. They take some of the potential energy of charges flowing through them and
transform it into heat. An element that absorbs energy is termed a passive element.
Often in electric circuits we prefer to talk about energy per unit time which we term power (𝑃).
∆𝐸
𝑃=
∆𝑡
So we define passive elements as elements that absorb power.
Conversely we define sources as elements that are able to produce power (and supply it to passive elements).
57
Voltage, Current & Power in Electric Circuits (3)
If we use conventional current, i.e. we assume that electric current is the flow of positive charges, then we have seen how the
arrow representing the voltage drop across a resistor is in opposite direction to the current through it.
𝐼
𝑅1
𝑉𝑅
The power dissipated in the resistor may then simply be calculated as:
𝑃 = 𝑉𝑅 𝐼
Note that, by convention, the power absorbed or dissipated by an element is a positive quantity.
58
Voltage, Current & Power in Electric Circuits (4)
Again assuming that we use conventional current, when it comes to a voltage source, voltage and current arrows must point
in the same direction.
𝐼
𝑉𝑆
This is because the voltage source is internally doing work to keep the voltage difference
between its positive and negative terminals constant. It can only achieve this by continually
pushing positive charges from the negative terminal up to the positive one. The flow of
charges though it (i.e. the current) must therefore be directed from the point of lower to the
point of higher potential.
𝐼
We may use the same formula as before to calculate the power BUT, since the arrows are in the same direction now, we must
add a negative sign!
𝑃 = −𝑉𝑆 𝐼
Note that, by convention, the power produced by a source is negative.
Voltage, Current & Power in Electric Circuits (5)
The concept of power stems from the concept of Energy and we know that energy is neither created nor destroyed. It stands
to reason therefore that if we add up power produced and power dissipated in a circuit we should get zero!
𝐼
𝐼
𝑉𝑆
𝑉𝑆 − 0
The power dissipated in the resistor
is
𝑃𝑅 = 𝑉𝑆 𝐼
𝑉𝑆 − 0
It is positive because voltage and
current arrows are in opposite
directions.
The power produced by the source is
𝑃𝑆 = −𝑉𝑆 𝐼
It is negative because voltage and
current arrows are in the same
direction.
𝑉𝐺𝑁𝐷 = 0𝑉
The total power in the circuit adds up to zero
𝑃𝑅 + 𝑃𝑆 = 0
Voltage, Current & Power in Electric Circuits (6)
In some textbooks, the points of higher and lower voltage are indicated using a plus and a minus sign, as shown below. I am
personally not a fan of this notation and would strongly encourage you to stick with arrows to indicate the polarity of the
voltage.
𝐼
𝐼
𝑉𝑆
𝑽𝑺
𝑉𝐺𝑁𝐷 = 0𝑉