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Basic Circuit Elements
Electric circuits consist of two basic types of elements. These are the active elements and the passive
elements.
An active element is capable of generating electrical energy, i.e. generating electrical energy from a
non-electrical form of energy. Examples of active elements are voltage sources such as batteries,
generators, and current sources. Most sources are independent of other circuit variables but some
elements are dependent, such as transistors and operational amplifiers which require dependent
sources. Active elements maybe ideal voltage sources or current sources. In such cases the
generated voltage or current would be independent of the rest of the circuit.
A passive element is one which doesn’t generate electricity but either consumes it or stores it.
Resistors, inductors, capacitors are simple passive elements. Diodes, transistors etc. are also passive
elements. Passive elements may either be linear or non-linear. Linear elements obey a straight line
law. For example a linear resistor has a linear voltage vs current relationship which passes through
the origin (V = R I). A linear inductor has a linear flux vs current relationshipwhich passes through the
origin (φ = k I) and a linear capacitor has a linear charge vs voltage relationship which passes through
the origin (q = C V). Resistors, capacitors and inductors may be linear or non-linear while diodes and
transistors are always non-linear.
Branch - A branch represents a single element such as a resistor or a battery. A branch is a two
terminal element.
Node – A node is the point connecting two branches. A node is usually indicated by a dot ( ) in a
circuit.
Loop or Mesh – A loop is any closed path in a circuit, formed by starting at a node, passing through a
number of branches and ending up once again at the original node.
same node
loop
branch
node
Resistance - R
Non-inductive Resistor
Resistance is measured in Ohms (Ω). The relationship between voltage and current is given by v = R i,
or i = G v, G = Conductance = 1/R
R = ρl / A where ρ is the resistivity, l is the length and A is the cross section of the material.
Power loss in a resistor = R i2. Energy dissipated in a resistor w = ∫ R. i2 dt
There is no storage of energy in a resistor.
Aluminium
Carbon
Germanium
Silicon
Paper
Mica
Glass
Teflon
Insulators
Gold
Semi Conductors
Copper
Resistivity (Ωm)
Conductors
Silver
Usage
Material
1.64 x
10-8
1.72x
10-8
2.45 x
10-8
2.8 x
10-8
4x
10-5
4.7 x
10-2
6.4 x
102
1.64 x
10-8
1010
5x
1011
3x
1012
Inductance - L
Not commonly used
Inductance is measured in Henry (H).
The relationship between voltage and current is given by n = N. dφ/dt = L. di/dt
L = (N2μA)/l for a coil; where μ is the permeability, N the number of turns, l the length and A cross
section of core.
Energy stored in an inductor = ½ L i2
No energy is dissipated in a pure inductor. However as practical inductors have some wire resistance
there would be some power loss. There would also be a small power loss in the magnetic core (if
any).
Capacitance – C
Capacitance is measured in Farads (F).
The relationship between voltage and current is given by i = dq/dt = C. dv/dt
C = ε A / d for a parallel plate capacitor; where ε is the permittivity, d the spacing and A the cross
section of dielectric.
Energy stored in a capacitor = ½ C v2
No energy is dissipated in a pure capacitor. However practical capacitors also have some power loss.
Fundamental Laws
The fundamental laws that govern electric circuits are Ohm’s law and Kirchoff’s laws.
Ohm’s Law
Ohm’s law states that the voltage v across a resistor is directly proportional to the current i flowing
through it.
v α i,
v = R. I, where R is the proportionality constant.
A short circuit in a circuit element is when the resistance (and any other impedance) of the elements
approaches zero [the term impedance is similar to resistance but is used in alternating current
theory for other components]
An open circuit is when the resistance (and any other impedance) of the element approaches
infinity.
In addition to Ohm’s law we need the Kirchoff’s voltage law and Kirchoff’s current law to analyse
circuits.
Kirchoff’s Current Law
Kirchoff’s first law is based on the principle of conservation of charge, which requires that the
algebraic sum of the charges within a closed system cannot change. Since charge is the integral of
currnet, we have irchoff’s current law that states that the algebraic sum of the currents entering a
node (or a closed boundary) is zero.
∑𝑖 = 0
i1
i2
i3
i5
i4
i1 + i2 - i3+ i4 – i5 = 0
Kirchoff’s Voltage Law
Kirchoff’s second law is based on the principle of conservation of energy, which requires that the
potential difference taken round a closed path must be zero.
Kirchoff’s voltage law states that the algebraic sum of all voltages around a closed path (or loop) is
zero.
∑𝑣 = 0
v1
v2
-v1 + v2 + v3+ v4 = 0
loop
v4
Depending on the convention you may also
write
v3
v1 - v2 - v3- v4 = 0
Note: v1 , v2 .... may be voltages across either active elements or passive elements or both and may be
obtained using Ohm’s law.
Series Circuits
v1
i1
R1
v
v2
i2
i
R2
R
When elements are connected in series, from Kirchoff’s current law, i1 = i2 = i and from Kirchoff’s
voltage law v1 + v2 = v. Also from Ohm’s law, v1 = R1 i1, v2 = R2 i2, v = R i
Parallel Circuits
v1
i1
v2
i2
v1
R1
i
R
R2
When elements are connected in parallel, from Kirchoff’s current law, i1 + i2 = i and from Kirchoff’s
voltage law v1 = v2 = v. Also from Ohm’s law, v1 = R1 i1, v2 = R2 i2, v = R i
Network Theorems
Complex circuits could be analysed using Ohm’s law and Kirchoff’s laws directly, but the calculations
would be difficult and time consuming. To handle the complexity, some theorems have been
developed to simplify the analysis. It must b emphasised that these are applicable to circuits with
linear elements only.
Superposition Theorem
In a linear circuit, if independent variable x1 gives y1 and x2 gives y2
then, an independent variable k1x1 + k2x2 would give k1y1 + k2y2
where k1 and k2 are constants.
y
y2
y1
In the special case, when input is x1 + x2 the output would be y1 + y2.
x1
x2
x
The Superposition theorem states that the voltage across (or current through) an element in a linear
circuit is the algebraic sum of all the voltages across (or current through) that element due to each
independent source acting alone i.e. with all other sources replaced by their internal impedance.
Any
Linear
Bilateral
Network
Linear
Bilateral
Network
Linear
Linear
Bilateral
Bilateral
Network
Network
Thevenin’s Theorem
Any linear active bilateral single-port (a component may be connected across 2 terminals of a port)
network can be replaced by an equivalent circuit comprising of a single voltage source EThevenin and a
series resistance RThevenin (or impedance ZThevenin, in general).
Linear
Active
Bilateral
Network
RThevenin
Port
EThevenin
If the port is kept open circuit (current zero), then the open circuit voltage of the network must be
equal to the Thevenin’s equivalent voltage source. If all the sources within the network are replaced
by their internal resistances (or impedances), then the impedance seen into the port from outside
will be equal to the Thevenin’s resistance (impedance).
Norton’s Theorem
Any linear active bilateral single-port network can be replaced by an equivgalent circuit comprising
of a single current source INorton and a shunt conductance GNorton (or admittance YNorton in general).
Norton’s theorem is the dual theorem of Thevenin’s theorem where the voltage source is replaced
by a current source.
Linear
Active
Bilateral
Network
Port
INorton
GNorton
If the port is kept on short circuit (voltage zero), then the short circuit current of the network must
be equal to the Norton’s equivalent current source. . If all the sources within the network are
replaced by their internal conductance (or admittances), then the admittance seen into the port
from outside will be equal to the Norton’s conductance (or admittance).