# 1. DC Circuits ```Electrical Science
Chapter 1: DC Circuits
1. Passive Circuit Components
A Passive element is defined as one which cannot supply average power i.e.,
greater than zero over an infinite time interval
2. Unilateral &amp; Bilateral
In the Bilateral element the voltage current relations remain same for current
flowing in either direction e.g., Resisters,
A Unilateral element has different relation b/w voltage and current for the two
possible directions of current e.g., Diodes
3. Linear &amp; Non-Linear Elements
An element is said to be linear, if its voltage current characteristic is at all times
or straight through the origin.
An element which doesn’t follow ohm’s law, those elements are said to be
Non-Linear elements.
4. Lumped &amp; Distributed Elements
Lumped elements are those elements which are very small in size &amp; in which
simultaneous action takes place for any given cause at the same instance of
time.
Distribute elements are those elements which are not electrically separable for
analytical purposes.
NOTE: We Use small i &amp; v for AC circuits &amp; capital I, V for DC circuits.
Inductor
Energy of Inductor: &frac12; LI2
Power of Inductor: (Li di)/dt watts
The Induced voltage across an inductor is zero if the current flowing through it
is constant that mean an inductor acts as short circuit to DC.
A small change in current within zero time through an inductor gives an infinite
voltage across the inductor, which is physically impossible. In a fixed inductor
the current cannot change abruptly
A pure inductor never dissipate energy, only stores. That is why it is also called
a non-dissipated passive element. However physical inductors dissipate power
due to internal resistance.
Capacitor
Formulas:
Q = CV
i = dQ/dt = cdv/dt
p = Vi = vcdv/dt
1. Current in a capacitor is 0 if voltage across it is constant, that means
capacitor acts as open circuit to DC
2. A small change in voltage across a capacitance within zero time gives an
infinity current through the capacitance, which is physically impossible.
That mean in a fixed capacitance the voltage cannot change abruptly.
3. The capacitor can a finite amount of energy even if the current through
it is zero.
4. A pure capacitor never dissipates energy but only stores that is why it is
called non dissipated passive element however physical capacitors
dissipate power due to internal resistance.
Representations
Voltage source can be divided as:
1. Independent source
2. Dependent source
Dependent Source
Short Forms
(i)
(ii)
(iii)
(iv)
VCVC – Voltage Controlled Voltage Source
CCVS – Current Controlled Voltage Source
VCCS – Voltage Controlled Current Source
CCCS – Current Controlled Current Source
Taking the sign convention – The entry terminal point of current through a
resistor or source is the sign of the voltage across the resistor or source.
Kirchhoff Voltage Law
Kirchhoff Voltage law (KVL) states that the algebraic sum of all branch voltages
around any closed path in a circuit is always zero at all instance of time
OR
In any closed electrical circuits or mesh, the algebraic sum of all electromotive
forces (EMF) &amp; Voltage drops in resistors is equal to zero i.e., in any closed
circuit or mesh.
(Algebraic sum of emfs) + (Algebraic sum of voltage drops) = 0
Kirchhoff Current Law
Kirchhoff Current Law (KCL) states that the sum of the current entering into
any node is equal to the sum of current leaving that node. Node is a junction of
two or more branches.
Ques. Find i1, i2, i3, i4
Node (Definition)
Node is a junction of two or more branches.
Superposition Theorem
The superposition theorem states that in any linear network containing two or
more sources, the response in any element is equal to the algebraic sum of the
responses caused by individual sources acting alone while the other sources
are non-operative i.e.,
other current sources are open circuited and other voltage sources are shortcircuited.
Thevenin’s Theorem
Thevenin’s states that w.r.t terminal pair AB, the network N maybe replaced by
a voltage source Vth in series with an internal impedance Zth.
The voltage source Vth called the Thevenin’s Voltage is a potential difference
b/w terminals ‘a’ &amp; ‘b’ and Zth is the internal impedance of the network N, with
all the sources set to zero i.e., voltage sources are short circuited &amp; current
sources are open circuited.
Norton Theorem
Norton’s Theorem states that w.r.t terminal pair AB, the network N maybe
replaced by a current source IN in parallel with an internal impedance Zn.
The current source IN called the Norton’s current that is the current that would
have flown from A &amp; B where the terminals A &amp; B are shorted together and Zn
is the Norton’s impedance or equivalent impedance with all sources set to
zero.
Procedure to obtain Vth or Zth and IN or Zn
(i)
(ii)
(iii)
Remove the portion of the network across which the Thevenin or
Norton’s equivalent circuit is to be found.
Mark the terminals AB of the remaining two terminal circuit.
(a) Thevenin’s Vn: Obtain the open circuit voltage at terminals AB that
is Vth keeping all the sources at their normal values.
(b) Norton’s IN: Short the terminals AB, obtain the short circuit
current IN following from A to B keeping all the sources of the normal values.
(iv) Calculate ZTh = ZN
Case I: If the circuit having only independent sources. In this case, set all the
sources at zero values (i.e., voltage sources are short circuited &amp; current
sources are open circuited)
Case II: If the circuit having independent &amp; dependent sources both. In this
case, we calculate VTh &amp; IN then the internal impedance of network N is
obtained as ZTh = VTh/IN
Limitations of Thevenin’s &amp; Norton Theorem
(i)
(ii)
(iii)
Not Applicable to circuits consisting of unilateral elements, non-linear
elements like diodes, etc.
Not Applicable to the circuit consisting of load series or parallel with
controlled or dependent source.
Not applicable to the circuit consisting of magnetic coupling b/w any
node &amp; other circuit element.
Millman’s Theorem
∑𝑛𝑖=1 𝐸𝑖𝑌𝑖
𝐸𝑚 = 𝑛
∑𝑖=1 𝑌𝑖
𝑍𝑚 =
1
∑𝑛𝑖=1 𝑌𝑖
𝐸𝑚 =
=
1
𝑌1 + 𝑌2 + ⋯ + 𝑌𝑛
𝐸1𝑌1 + 𝐸2𝑌2 + ⋯ + 𝐸𝑛𝑌𝑛
𝑌1 + 𝑌2 + ⋯ + 𝑌𝑛
𝒁 = 𝑹 + 𝒋𝑿
𝑛
Im = ∑ 𝐸𝑖𝑌𝑖 = 𝐸1𝑌1 + 𝐸2𝑌2 + ⋯ + 𝐸𝑛𝑌𝑛
𝑖=1
𝑛
Ym = ∑ 𝑌𝑖 = 𝑌1 + 𝑌2 + ⋯ + 𝑌𝑛
𝑖=1
Maximum Power Transfer Theorem
The Maximum Power Transfer Theorem states that maximum power is
delivered from a source to a load when the load resistance is equal to source
resistance.
𝑰=
𝐕𝐬
𝑹𝒔 + 𝑹𝒍
Power Through RL = I2RL
𝑽𝒔𝟐 𝑹𝑳
=𝑷
(𝑹𝒔 + 𝑹𝑳)𝟐
Now, to find the maximum value.
𝑑𝑝
𝑑𝑅𝐿
=
𝑑
𝑑𝑅𝐿
𝑉𝑠 2
[(𝑅𝑠+𝑅𝐿)2 . 𝑅𝐿]
(𝑅𝑠 + 𝑅𝐿)2 − (2𝑅𝐿)(𝑅𝑠 + 𝑅𝐿)
0 = 𝑉𝑠 [
]
(𝑅𝑠 + 𝑅𝐿)4
2
(𝑅𝑠 + 𝑅𝐿)2 − (2𝑅𝐿)(𝑅𝑠 + 𝑅𝐿) = 0
 Rs = RL
Mesh Analysis
Super Mesh Analysis
Nodal Analysis
Super Node Analysis
```