function of time and which does not depend on what is connected to its terminals.
Voltage and Current Sources
Independent Voltage Source
The ideal voltage source is represented as
Resistors and Ohm's Law
This symbol may be used to represent any type of independent voltage source, i.e. a
voltage source which has a specified voltage as a function of time and which does
not depend on what is connected to its terminals.
For d.c. voltages, i.e. for voltage sources which do not depend on time, the
representation for a battery may be used as in (i) below. Here, terminal "a" is
always at a potential that is volts higher that the potential at terminal "b". In
electronic circuits, the same power supply may also be shown as in (ii) below,
although in this second form it is necessary to indicate where the zero reference
voltage (or earth) is located.
Independent Current Source
They may be a little uneasiness with the concept of a current source, if only because
while you can see and buy a battery as a voltage source, there is no equivalent
simple package for a current source. However, as a circuit element, the current
source is just as important as the voltage source and will be found in many circuit
representations.
Current is defined as the flow rate of charge
Since electrons carry a negative charge, the conventional current is in a direction
that is opposite to the electron flow. All currents shown in a circuit are shown as
conventional current flow.
The unit of current, representing coulomb/second, is the ampere (abbreviated as A).
The ideal current source is represented as
Apply a constant voltage source to a block of
conducting material. Electrons, with their negative
charge, will flow towards the positive potential of the
source.
Thus the conventional current flow will be in the
opposite direction, as shown, with the current flowing
through the material from the positive to the negative
terminals of the source.
If the current is proportional to the applied voltage, then the conducting material
behaves like a linear resistor. This linear relationship is expressed as Ohm's Law,
namely
V = R.I or more generally as v(t) = R.i(t)
The linear resistance relationship may be expressed on either a (V, I) diagram or an
(I, V) diagram as shown below.
The unit of resistance, R, is the ohm ( ), while the unit for conductance, G, is
siemens (S).
The Current and Voltage Sign Convention for Ohm's Law
Consider a voltage source (a d.c. source in this example) attached to a resistor.
The complete circuit is shown in (a). In (b), the voltage and current for the resistor
only are shown, i.e. the source of the voltage to the left is not shown. This resistor,
representing an element somewhere within a circuit, is now redrawn without change
in (c). Here, we see that Ohm's Law applies if the current is flowing inwards at the
positive terminal and energy will be dissipated in the resistor.
Example
Consider the following situation which may occur within a circuit. If R = 2 and I
= –2A, what is the value of the voltage, V, as shown in the diagram?
This symbol may be used to represent all types of independent current sources, i.e.
current sources which have a specified current (including a constant current) as a
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Two branches of a circuit are in parallel if
i) there have common nodes at each end of the two
branches and
ii) there are no other nodes within either branch.
In this circuit, there will be the same voltage, V, across
both resistors. Therefore the currents in each resistor are
In this example, the current is flowing in at the terminal with negative potential.
Thus, by convention, Ohm's Law is expressed as
Applying Kirchhoff's Current Law at node A gives
Kirchhoff's Current Law and Parallel Resistors
Consider a connection in a circuit where two or more circuit elements meet at a
point. This connection point is called a node.
The combination of two parallel resistors will give an equivalent resistance, R, as
seen by the source.
Thus
Kirchhoff's Current Law states that:
The algebraic sum of the currents flowing into a node
at any instant in time is zero. Thus
What is the formula for three parallel connected resistors?
This law was originally formulated as an experimental law, but also may be seen as
a consequence of the conservation of charge and the definition of current. If the law
was not true, then it would be possible to obtain a continual build up of charge at a
point.
The definition of the law may be extended to
apply to a closed surface.
Whether applying Kirchhoff's Current Law at
node A or to the surface S,
In terms of the conductances for parallel resistors,
Current Division through Parallel Resistors
The fraction of the input current which passes through one
of the resistors is given by
It is important to note that two "apparent nodes", A and B in the figure below, when
joined together by a short circuit, are at the same potential and may be considered as
one node.
Exercises
i) What happens to the percentage split between the two parallel resistors if
?
ii) For two parallel conductances, show that
An Equivalent Element for Parallel Resistors
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Example
How much current flows through the 4.7k resistor ?
Applying K.V.L. to the loop gives
If an equivalent resistance satisfies the condition
, then for a series connection of resistors
Voltage Division for Series Connected Resistors
When a voltage source is applied to two series-connected resistors as illustrated in
the figure above, the voltage across each resistor may be determined as follows:
Kirchhoff's Voltage Law and Series Resistors
Starting at any point in a circuit, a loop is formed by passing through elements and
returning to the starting node, never passing through any other node more than once.
Kirchhoff's Voltage Law (K.V.L.) states that:
At any instant in time, around any loop in a circuit,
the algebraic sum of the voltages is zero. Thus
Note that the larger voltage appears across the larger resistor. It is also important to
note that if a further resistor is connected across the terminals then, in evaluating
the voltage across these terminals, the parallel combination of the new resistor with
must be used in place of .
Dependent Sources
An independent source, for example a 3V battery or a 2A current source, has a
specified value which does not depend on other parameters in the circuit. On the
other hand a dependent source, which may be either a voltage or a current source,
has a value which is controlled by a specified parameter within the circuit.
The concept of dependent sources is important. They are found, for example, in
circuit models as they are used in amplifier applications. Dependent (or controlled)
sources are represented by a diamond-shaped symbol as illustrated below.
Consider the application of Kirchhoff's Voltage
Law to this loop which forms a part of a more
complex circuit.
Moving clockwise from A around the loop 4V
or
The latter expression is probably preferred, since it leads to equations with the
coefficients of the unknown quantities on the left hand side and constants on the
right hand side.
Note: When Kirchhoff's Voltage Law is used in the solution
of circuits, the equations will all be written in terms of circuit currents.
Note that the value of a each source depends on a variable, in each case the variable
here is a current, which must be shown elsewhere in the circuit.
Example
Series Connected Resistors
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Thus a circuit element provides power to the remainder of the circuit if the current is
flowing out from the more positive terminal of the element, while a circuit element
absorbs power if the current is flowing in at the more positive terminal of the
element.
Calculate the value of the current, .
Applying Kirchhoff's Current Law at the upper node gives
Instantaneous Power
It takes a certain amount of work or energy to drive a current through a resistive
element. This energy is absorbed by the element.
Consider a term formed by the product of voltage times current.
Thus at any instant in time,
For d.c. circuits,
. The unit for power is the watt (W).
watt.
Through the use of Ohm's Law, the power may be expressed in one of three forms,
namely
Power and the Sign Convention
Consider the connection of a physical resistor to a voltage source, e.g. to a battery.
The voltage source will provide power which will be absorbed by the resistor. Now
consider the voltages and currents for each component in this simple circuit as
illustrated below.
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