Unit-2.0
Circuit Element Theory
Dr. Anurag Srivastava
Associate Professor
ABV-IIITM, Gwalior
Circuit Theory
Overview Of Circuit Theory;
Lumped Circuit Elements; Topology Of
Circuits; Resistors; KCL and KVL;
Resistors in Series and Parallel; Energy
Storage Elements; First-Order Circuits
2
Objectives
To commence our study of circuit theory.
n To develop an understanding of the
concepts of Lumped circuit elements;
topology of circuits; resistors; KCL and
KVL; resistors in series and parallel;
energy storage elements; and first-order
circuits.
n
1
Overview of Circuit Theory
Electrical circuit elements are idealized
models of physical devices that are defined
by relationships between their terminal
voltages and currents. Circuit elements can
have two or more terminals.
n An electrical circuit is a connection of
circuit elements into one or more closed
loops.
n
Overview of Circuit Theory
n
n
A lumped circuit is one where all the terminal
voltages and currents are functions of time only.
Lumped circuit elements include resistors,
capacitors, inductors, independent and
dependent sources.
A distributed circuit is one where the terminal
voltages and currents are functions of position
as well as time. Transmission lines are
distributed circuit elements.
Overview of Circuit Theory
Basic quantities are voltage, current, and
power.
n The sign convention is important in
computing power supplied by or absorbed
by a circuit element.
n Circuit elements can be active or passive;
active elements are sources.
n
2
Overview of Circuit Theory
n
n
n
n
Current is moving electrical charge.
Measured in Amperes (A) = Coulomb/s
Current is represented by I or i.
In general, current can be an arbitrary function
of time.
n
n
Constant current is called direct current (DC).
Current that can be represented as a sinusoidal
function of time (or in some contexts a sum of
sinusoids) is called alternating current (AC).
Overview of Circuit Theory
Voltage is electromotive force provided by
a source or a potential difference between
two points in a circuit.
n Measured in Volts (V): 1 J of energy is
needed to move 1 C of charge through a 1
V potential difference.
n Voltage is represented by V or v.
n
Overview of Circuit Theory
The lower case symbols v and i are usually
used to denote voltages and currents that
are functions of time.
n The upper case symbols V and I are usually
used to denote voltages and currents that
are DC or AC steady-state voltages and
currents.
n
3
Overview of Circuit Theory
n
Current has an assumed direction of flow;
n
n
n
Voltage has an assumed polarity;
n
n
n
currents in the direction of assumed current flow have
positive values;
currents in the opposite direction have negative values.
volt drops in with the assumed polarity have positive values;
volt drops of the opposite polarity have negative values.
In circuit analysis the assumed polarity of voltages are
often defined by the direction of assumed current flow.
Overview of Circuit Theory
n
n
Power is the rate at which energy is being
absorbed or supplied.
Power is computed as the product of voltage
and current:
p(t ) = v(t )i(t ) or P = VI
n
Sign convention:
n
n
positive power means that energy is being absorbed;
negative power means that energy is being supplied.
Overview of Circuit Theory
i(t)
Rest of
circuit
+
v(t)
• If p(t) > 0, then the circuit
element is absorbing power
from the rest of the circuit.
• If p(t) < 0, then the circuit
element is supplying power
to the rest of the circuit.
Circuit element under
consideration
4
Overview of Circuit Theory
If power is positive into a circuit element,
it means that the circuit element is
absorbing power.
n If power is negative into a circuit element,
it means that the circuit element is
supplying power.
n Only active elements (sources) can supply
power to the rest of a circuit.
n
Active and Passive Elements
n
Active elements can generate energy.
n
n
Passive elements cannot generate energy.
n
n
Examples of active elements are independent and
dependent sources.
Examples of passive elements are resistors,
capacitors, and inductors.
In a particular circuit, there can be active
elements that absorb power – for example, a
battery being charged.
Independent and Dependent
Sources
An independent source (voltage or
current) may be DC (constant) or timevarying; its value does not depend on other
voltages or currents in the circuit.
n A dependent source has a value that
depends on another voltage or current in the
circuit.
n
5
Independent Sources
vs (t )
is (t )
Voltage Source
Current Source
Dependent Sources
+
v=f(vx)
+
v=f(ix)
-
-
Voltage
Controlled
Voltage Source
(VCVS)
Current
Controlled
Voltage Source
(CCVS)
Dependent Sources
I=f(Vx)
Voltage
Controlled
Current Source
(VCCS)
I=f(Ix)
Current
Controlled
Current Source
(CCCS)
6
Passive Lumped Circuit Elements
n
Resistors
R
n
Capacitors
C
n
Inductors
L
Topology of Circuits
A lumped circuit is composed of lumped
elements (sources, resistors, capacitors,
inductors) and conductors (wires).
n All the elements are assumed to be
lumped, i.e., the entire circuit is of
negligible dimensions.
n All conductors are perfect.
n
Topology of Circuits
A schematic diagram is an electrical
representation of a circuit.
n The location of a circuit element in a
schematic may have no relationship to its
physical location.
n We can rearrange the schematic and have
the same circuit as long as the connections
between elements remain the same.
n
7
Topology of Circuits
n
Example: Schematic of a circuit:
“Ground”: a
reference point
where the voltage
(or potential) is
assumed to be zero.
Topology of Circuits
n
Only circuit elements that are in closed loops
(i.e., where a current path exists) contribute to
the functionality of a circuit.
This circuit
element can be
removed without
affecting
functionality. This
circuit behaves
identically to the
previous one.
Topology of Circuits
A node is an equipotential point in a circuit. It
is a topological concept – in other words, even if
the circuit elements change values, the node
remains an equipotential point.
n To find a node, start at a point in the circuit.
From this point, everywhere you can travel by
moving only along perfect conductors is part of a
single node.
n
8
Topology of Circuits
A loop is any closed path through a circuit in
which no node is encountered more than once.
n To find a loop, start at a node in the circuit.
From this node, travel along a path back to the
same node ensuring that you do not encounter any
node more than once.
n A mesh is a loop that has no other loops inside
of it.
n
Topology of Circuits
If we know the voltage at every node of a
circuit relative to a reference node (ground),
then we know everything about the circuit –
i.e., we can determine any other voltage or
current in the circuit.
n The same is true if we know every mesh
current.
n
Topology of Circuits
N1
N2
M1
N3
N4
M2
N0
• In this example there
are 5 nodes and 2
meshes.
• In addition to the
meshes, there is one
additional loop
(following the outer
perimeter of the circuit).
9
Resistors
A resistor is a circuit element that
dissipates electrical energy (usually as heat).
n Real-world devices that are modeled by
resistors: incandescent light bulb, heating
elements (stoves, heaters, etc.), long wires
n Parasitic resistances: many resistors on
circuit diagrams model unwanted
resistances in transistors, motors, etc.
n
Resistors
i(t)
The
Rest of
the
Circuit
n
n
n
+
R
v(t)
v(t ) = Ri (t )
-
Resistance is measured in Ohms (W)
The relationship between terminal voltage and current
is governed by Ohm’s law
Ohm’s law tells us that the volt drop in the direction of
assumed current flow is Ri
KCL and KVL
Kirchhoff’s Current Law (KCL) and Kirchhoff’s
Voltage Law (KVL) are the fundamental laws of
circuit analysis.
n KCL is the basis of nodal analysis – in which
the unknowns are the voltages at each of the
nodes of the circuit.
n KVL is the basis of mesh analysis – in which
the unknowns are the currents flowing in each of
the meshes of the circuit.
n
10
KCL and KVL
n
KCL
n The sum of all currents
entering a node is zero,
or
n The sum of currents
entering node is equal
to sum of currents
leaving node.
i1(t)
i5(t)
i2(t)
i4(t)
i3(t)
n
å i (t ) = 0
j
j =1
KCL and KVL
n
KVL
n The sum of voltages
around any loop in a
circuit is zero.
-
n
åv
j
(t ) = 0
j =1
+ v2(t) -
v1(t)
+
v3(t)
-
+
KCL and KVL
n
In KVL:
n
n
n
A voltage encountered + to - is positive.
A voltage encountered - to + is negative.
Arrows are sometimes used to represent voltage
differences; they point from low to high voltage.
+
v(t)
-
≡
v(t)
11
Resistors in Series
A single loop circuit is one which has only
a single loop.
n The same current flows through each
element of the circuit - the elements are in
series.
n
Resistors in Series
Two elements are in series if the current that
flows through one must also flow through
the other.
Series
R1
R2
Resistors in Series
Consider two resistors in series with a
voltage v(t) across them:
Voltage division:
i(t)
+
+
R1
v1(t)
+
v(t)
R2
-
v2(t)
v1 (t ) = v(t )
R1
R1 + R2
v2 (t ) = v(t )
R2
R1 + R2
-
12
Resistors in Series
If we wish to replace the two series
resistors with a single equivalent resistor
whose voltage-current relationship is the
same, the equivalent resistor has a value
given by
n
Req = R1 + R2
Resistors in Series
For N resistors in series, the equivalent
resistor has a value given by
n
R1
R2
Req
R3
Req = R1 + R2 + R3 + L + RN
Resistors in Parallel
n
When the terminals of two or more circuit
elements are connected to the same two
nodes, the circuit elements are said to be in
parallel.
13
Resistors in Parallel
Consider two resistors in parallel with a
voltage v(t) across them:
Current division:
i(t)
+
i1(t)
i2(t)
v(t)
R1
R2
i1 (t ) = i (t )
R2
R1 + R2
i2 (t ) = i(t )
R1
R1 + R2
-
Resistors in Parallel
If we wish to replace the two parallel
resistors with a single equivalent resistor
whose voltage-current relationship is the
same, the equivalent resistor has a value
given by
n
Req =
R1 R2
R1 + R2
Resistors in Parallel
For N resistors in parallel, the equivalent
resistor has a value given by
n
R1
Req =
R2
R3
Req
1
1
1
1
1
+
+ +L+
R1 R2 R3
RN
14
Energy Storage Elements
Capacitors store energy in an electric field.
n Inductors store energy in a magnetic field.
n Capacitors and inductors are passive
elements:
n
n Can
store energy supplied by circuit
n Can return stored energy to circuit
n Cannot supply more energy to circuit than is
stored.
Energy Storage Elements
Voltages and currents in a circuit without
energy storage elements are solutions to
algebraic equations.
n Voltages and currents in a circuit with
energy storage elements are solutions to
linear, constant coefficient differential
equations.
n
Energy Storage Elements
n
n
n
Electrical engineers (and their software tools)
usually do not solve the differential equations
directly.
Instead, they use:
n LaPlace transforms
n AC steady-state analysis
These techniques covert the solution of
differential equations into algebraic problems.
15
Energy Storage Elements
n
n
n
Energy storage elements model electrical
loads:
n Capacitors model computers and other
electronics (power supplies).
n Inductors model motors.
Capacitors and inductors are used to build
filters and amplifiers with desired frequency
responses.
Capacitors are used in A/D converters to
hold a sampled signal until it can be
converted into bits.
Capacitors
n
n
n
Capacitance occurs when two conductors are separated
by a dielectric (insulator).
Charge on the two conductors creates an electric field
that stores energy.
The voltage difference between the two conductors is
proportional to the charge.
q(t ) = C v(t )
n
n
The proportionality constant C is called capacitance.
Capacitance is measured in Farads (F).
Capacitors
The
rest i(t)
of
the
circuit
+
i(t ) = C
v(t)
dv (t )
dt
t
v (t ) =
1
i ( x )dx
C -ò¥
t
v (t ) = v(t 0 ) +
1
i ( x) dx
C tò0
16
n
Capacitance - the measure of the ability of a capacitor
to store charge
n
Voltage Equation for a Capacitor
v=q/C
C = Capacitance in Farads (F)
q = Charge on one plate in Coulombs (C)
v = Voltage across the capacitor in Volts (V)
Example: What is the charge on a 200mF capacitor with
100 Volts across its terminals?
Calculus of Capacitors
q = Cv
i=
dq
dt
i( t ) = C
dv( t )
dt
17
Capacitors
The voltage across a capacitor cannot
change instantaneously.
n The energy stored in the capacitors is given
by
1
wC (t ) = Cv 2 (t )
2
n
n
Capacitance Equation
C=
e re o A
d
eo = permittivity of air, 8.85 x 10-12 F/m
er = relative permittivity of the dielectric
A = plate area in square meters (m 2)
d = distance between plates in meters (m)
18
Capacitors
19
Capacitors
Types of Capacitors
n Electrolytic
n Oil-filled
n Ceramic
n Variable
Capacitors
n How
can you vary the capacitance of a
capacitor?
Capacitors in Series
Draw three capacitors in series with a battery.
n What is the same for every capacitor in a
series?
n
n Answer:
n
Charge
This leads to the Total Capacitance Equation:
1
1
1
1
=
+
+
Ceq C1 C 2 C3
20
Capacitors in Parallel
Draw three capacitors in parallel with a battery.
n What is the same for every capacitor in
parallel?
n
n Answer:
n
Voltage
This leads to the Total Capacitance Equation:
Ceq = C1 + C2 + C3
21
Example Problems
n
What is the total capacitance for a 1mF,
a 2 mF and a 5mF capacitor is series?
n
...in parallel?
Inductors
n
n
n
n
n
Inductance occurs when current flows through a (real)
conductor.
The current flowing through the conductor sets up a
magnetic field that is proportional to the current.
The voltage difference across the conductor is
proportional to the rate of change of the magnetic flux.
The proportionality constant is called the inductance,
denoted L.
Inductance is measured in Henrys (H).
Inductance
Inductor - a circuit component that has
two terminals connected to a coil of wire
n Inductors are also called:
n
n Solenoids
n Coils
n Electromagnets
n
Circuit Symbols
22
Toroidal coils are used in a
broad range of applications
in AC electronic circuits,
such as high-frequency coils
and transformers.
23
n
Energy Stored on a Inductor
n Inductors
resistors.
n They
field.
also do not dissipate energy like
store energy in the form of a magnetic
w( t ) =
1 2
Li
2
24
Inductors in Series
n
What is the same for every inductor in a
series?
n Answer:
n
Current
This leads to the Total Inductance
Equation:
Leq = L1 + L2 + L3
Inductors in Parallel
n
What is the same for every inductor in
parallel?
n Answer:
n
Voltage
This leads to the Total Inductance
Equation:
1
1 1
1
= + +
Leq L1 L2 L3
25
Inductors
The
rest i(t)
of
the
circuit
+
v(t)
L
v (t ) = L
di(t )
dt
t
i(t ) =
1
v ( x) dx
L -ò¥
t
1
i(t ) = i(t 0 ) + ò v ( x )dx
L t0
Inductors
The current through an inductor cannot
change instantaneously.
n The energy stored in the inductor is given
by
1
wL (t ) = Li 2 (t )
2
n
26
Analysis of Circuits Containing
Energy Storage Elements
n
Need to determine:
order of the circuit.
n Forced (particular) and natural
(complementary/homogeneous) responses.
n Transient and steady state responses.
n 1st order circuits - the time constant.
n 2nd order circuits - the natural frequency
and the damping ratio.
n The
Analysis of Circuits Containing
Energy Storage Elements
n
n
n
The number and configuration of the energy
storage elements determines the order of the
circuit.
n £ # of energy storage elements
Every voltage and current is the solution to a
differential equation.
In a circuit of order n, these differential
equations are linear constant coefficient and
have order n.
Analysis of Circuits Containing
Energy Storage Elements
n
Any voltage or current in an nth order
circuit is the solution to a differential
equation of the form
d n v (t )
d n -1v(t )
+
a
+ ... + a0 v(t ) = f (t )
n -1
dt n
dt n -1
as well as initial conditions derived from
the capacitor voltages and inductor
currents at t = 0-.
27
Analysis of Circuits Containing
Energy Storage Elements
n
n
n
The solution to any differential equation consists
of two parts:
v(t) = vp(t) + vc(t)
Particular (forced) solution is vp(t)
n Response particular to the source
Complementary/homogeneous (natural)
solution is vc(t)
n Response common to all sources
Analysis of Circuits Containing
Energy Storage Elements
The particular solution vp(t) is typically a
weighted sum of f(t) and its first n
derivatives.
n If f(t) is constant, then vp(t) is constant.
n If f(t) is sinusoidal, then vp(t) is sinusoidal.
n
Analysis of Circuits Containing
Energy Storage Elements
n
The complementary solution is the
solution to
d n v (t )
d n -1v(t )
+ an-1
+ ... + a0 v(t ) = 0
n
dt
dt n -1
n
The complementary solution has the form
n
vc (t ) = å K i e si t
i =1
28
Analysis of Circuits Containing
Energy Storage Elements
n s1
through sn are the roots of the
characteristic equation
s n + an-1s n -1 + ... + a1s + a0 = 0
Analysis of Circuits Containing
Energy Storage Elements
If si is a real root, it corresponds to a
decaying exponential term K i e s t , si < 0
n If si is a complex root, there is another
complex root that is its complex conjugate,
and together they correspond to an
exponentially decaying sinusoidal term
n
i
e -s i t ( Ai cos w d t + Bi sin w d t )
Analysis of Circuits Containing
Energy Storage Elements
The steady state (SS) response of a circuit
is the waveform after a long time has
passed.
n DC SS if response approaches a
constant.
n AC SS if response approaches a sinusoid.
n The transient response is the circuit
response minus the steady state response.
n
29
Analysis of Circuits Containing
Energy Storage Elements
Transients usually are associated with the
complementary solution.
n The actual form of transients usually
depends on initial capacitor voltages and
inductor currents.
n Steady state responses usually are
associated with the particular solution.
n
First-Order Circuits
Any circuit with a single energy storage
element, an arbitrary number of sources,
and an arbitrary number of resistors is a
circuit of 1st order.
n Any voltage or current in such a circuit is
the solution to a 1st order differential
equation.
n
First-Order Circuits
n
Examples of 1st order circuits:
n Computer
RAM
dynamic RAM stores ones as charge on a
capacitor.
n The charge leaks out through transistors
modeled by large resistances.
n The charge must be periodically refreshed.
nA
30
First-Order Circuits
n
Examples of 1st order circuits (Cont’d):
n The RC low-pass filter for an envelope detector in a
superheterodyne AM receiver.
n Sample-and-hold circuit:
n The capacitor is charged to the voltage of a
waveform to be sampled.
n The capacitor holds this voltage until an A/D
converter can convert it to bits.
n The windings in an electric motor or generator can
be modeled as an RL 1st order circuit.
First-Order Circuits
+
vS(t)
n
+
vR(t)
-
R
+
vC(t)
C
-
-
1st Order Circuit:
n
n
One capacitor and one resistor
The source and resistor may be equivalent to a
circuit with many resistors and sources.
First-Order Circuits
+ vR(t) -
vS(t)
n
+
-
R
+
vC(t)
C
i(t)
-
Let’s derive the (1st order) differential
equation for the mesh current i(t).
31
First-Order Circuits
KVL around the loop:
vS (t ) = v R (t ) + vC (t )
We have
vR (t ) = Ri (t )
vC (t ) = vC (0 ) +
t
1
i (x )dx
C ò0
First-Order Circuits
The KVL equation becomes:
Ri (t ) + vC (0 ) +
t
1
i (x )dx = vS (t )
C ò0
Differentiating both sides w.r.t. t, we have
R
or
di (t ) 1
dv (t )
+ i(t ) = S
dt
C
dt
di (t ) 1
1 dvS (t )
+
i (t ) =
dt
RC
R dt
First-Order Circuits
iL(t)
iR(t)
iS(t)
R
+
L
v(t)
-
n
1st Order Circuit:
n
n
One inductor and one resistor
The source and resistor may be equivalent to a
circuit with many resistors and sources.
32
First-Order Circuits
iL(t)
iR(t)
R
iS(t)
+
v(t)
L
-
n
Let’s derive the (1st order) differential
equation for the node voltage v(t).
First-Order Circuits
KCL at the top node:
We have
iS (t ) = iR (t ) + iL (t )
iR (t ) =
v(t )
R
iL (t ) = iL (0) +
t
1
v( x )dx
L ò0
First-Order Circuits
The KVL equation becomes:
v(t )
1
+ iL (0) + ò v (x )dx = iS (t )
R
L0
t
Differentiating both sides w.r.t. t, we have
or
1 dv(t ) 1
di (t )
+ v(t ) = S
R dt
L
dt
dv (t ) R
di (t )
+ v (t ) = R S
dt
L
dt
33
First-Order Circuits
n
n
For all 1st order circuits, the diff. eq. can be
written as
dv(t ) 1
+ v(t ) = f (t )
dt
t
The complementary solution is given by
vC (t ) = Ke
-t
t
where K is evaluated from the initial conditions.
First-Order Circuits
n
The time constant of the complementary
response is t.
an RC circuit, t = RC
n For an RL circuit, t = L/R
n For
n
t is the amount of time necessary for an
exponential to decay to 36.7% of its initial
value.
First-Order Circuits
n
The particular solution vp(t) is usually a
weighted sum of f(t) and its first derivative.
n If
n If
f(t) is constant, then vp(t) is constant.
f(t) is sinusoidal, then vp(t) is sinusoidal.
34