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ELECTRICAL CIRCUITS UNIT-

INTRODUCTION TO ELECTRICAL CIRCUITS

 Concept of Network and circuit

 Types of Elements

 Types of Sources

 Source Transformation

 R-L-C Parameters

 Voltage - Current relationships for Passive Elements (For different

Input Signals : Square, Ramp, Saw tooth and Triangle)

 Kirchhoff’s Laws

INTRODUCTION:

An Electric circuit is an interconnection of various elements in which there is at least one closed path in which current can flow. An Electric circuit is used as a component for any engineering system.

The performance of any electrical device or machine is always studied by drawing its electrical equivalent circuit. By simulating an electric circuit, any type of system can be studied for e.g., mechanical, hydraulic thermal, nuclear, traffic flow, weather prediction etc.

All control systems are studied by representing them in the form of electric circuits. The analysis, of any system can be learnt by mastering the techniques of circuit theory.

The analysis of any system can be learnt by mastering the techniques of circuit theory.

Elements of an Electric circuit:

An Electric circuit consists of following types of elements.

Active elements:

Active elements are the elements of a circuit which possess energy of their own and can impart it to other element of the circuit.

Active elements are of two types

a) Voltage source b) Current source

A Voltage source has a specified voltage across its terminals, independent of current flowing through it. A current source has a specified current through it independent of the voltage appearing across it.

Passive Elements:

The passive elements of an electric circuit do not possess energy of their own. They receive energy from the sources. The passive elements are the resistance, the inductance and the capacitance. When electrical energy is supplied to a circuit element, it will respond in one and more of the following ways.

If the energy is consumed, then the circuit element is a pure resistor.

If the energy is stored in a magnetic field, the element is a pure inductor.

And if the energy is stored in an electric field, the element is a pure capacitor.

Types of Sources:

Independent & Dependent sources:

If the voltage of the voltage source is completely independent source of current and the current of the current source is completely independent of the voltage, then the sources are called as independent sources.

The special kind of sources in which the source voltage or current depends on some other quantity in the circuit which may be either a voltage or a current anywhere in the circuit are called Dependent sources or Controlled sources.

There are four possible dependent sources:

a. Voltage dependent Voltage source b. Current dependent Current source c. Voltage dependent Current source d. Current dependent Current source

The constants of proportionalities are written as B, g, a, r in which B & a has no units, r has units of ohm & g units of mhos.

Independent sources actually exist as physical entities such as battery, a dc generator & an alternator. But dependent sources are used to represent electrical properties of electronic devices such as OPAMPS & Transistors.

Ideal & Practical sources:

1. An ideal voltage source is one which delivers energy to the load at a constant terminal voltage, irrespective of the current drawn by the load.
2. An ideal current source is one, which delivers energy with a constant current to the load, irrespective of the terminal voltage across the load.
3. A Practical voltage source always possesses a very small value of internal resistance r. The internal resistance of a voltage source is always connected in series with it & for a current source; it is always connected in parallel with it. As the value of the internal resistance of a practical voltage source is very small, its terminal voltage is assumed to be almost constant within a certain limit of current flowing through the load.
4. A practical current source is also assumed to deliver a constant current, irrespective of the terminal voltage across the load connected to it.

When two ideal voltage sources of emf’s V1 & V2 are connected in parallel, what voltage appears across its terminals is ambiguous.

Hence such connections should not be made.

However if V1 = V2= V, then the equivalent voltage some is represented by V.

In that case also, such a connection is unnecessary as only one voltage source serves the purpose.

Practical voltage sources connected in parallel:

Equivalent Circuit Single Equivalent Voltage Source

Ideal current sources connected in series:

When ideal current sources are connected in series, what current flows through the line is ambiguous. Hence such a connection is not permissible.

However, it I1 = I2 = I, then the current in the line is I.

But, such a connection is not necessary as only one current source serves the purpose.

Practical current sources connected in series:

Ideal current sources connected in parallel

Two ideal current sources in parallel can be replaced by a single equivalent ideal current source.

Practical current sources connected in parallel

Hence a voltage source Vs in series with its internal resistance R can be converted into a current source

I = , with its internal resistance R connected in parallel with it. Similarly a current source I in parallel with its internal resistance R can be converted into a voltage source V = IR in series with its internal resistance R.

Parameters:

1. Resistance:

Resistance is that property of a circuit element which opposes the flow of electric current and in doing so converts electrical energy into heat energy.

It is the proportionality factor in ohm’s law relating voltage and current.

Ohm’s law states that the voltage drop across a conductor of given length and area of cross section is directly proportional to the current flowing through it. R œ i

V=Ri

i= = GV

Where the reciprocal of resistance is called conductance G. The unit of resistance is ohm and the unit of conductance is mho or Siemens.

When current flows through any resistive material, heat is generated by the collision of electrons with other atomic particles. The power absorbed by the resistor is converted to heat and is given by the expression

P= vi= i2R where i is the resistor in amps, and v is the voltage across the resistor in volts.

Energy lost in a resistance in time t is given by

W = t

2. Inductance:

Inductance is the property of a material by virtue of which it opposes any change of magnitude and direction of electric current passing through conductor. A wire of certain length, when twisted into a coil becomes a basic conductor. A change in the magnitude of the current changes the electromagnetic field.

Increase in current expands the field & decrease in current reduces it. A change in current produces change in the electromagnetic field. This induces a voltage across the coil according to Faradays laws of Electromagnetic Induction.

Induced Voltage V = L

V = Voltage across inductor in volts

I = Current through inductor in amps

di = v dt

Integrating both sides,

Power absorbed by the inductor P = VI = Li

Energy stored by the inductor

W= = dt =

W =

Conclusions:

1) V = L

The induced voltage across an inductor is zero if the current through it is constant. That means an inductor acts as short circuit to dc.

2. For minute change in current within zero time (dt = 0) gives an infinite voltage across the inductor which is physically not at all feasible. In an inductor, the current cannot change abruptly. An inductor behaves as open circuit just after switching across dc voltage.

3. The inductor can store finite amount of energy, even if the voltage across the inductor is zero.

4. A pure inductor never dissipates energy, it only stores it. Hence it is also called as a non– dissipative passive element. However, physical inductor dissipates power due to internal resistance.

W =^ Li^2 = X 2 X (4)^2 = 16 J

3. Capacitance:

1. A capacitor consists of two metallic surfaces or conducting surfaces separated by a dielectric medium.
2. It is a circuit element which is capable of storing electrical energy in its electric field.
3. Capacitance is its capacity to store electrical energy.
4. Capacitance is the proportionality constant relating the charge on the conducting plates to the potential.

Charge on the capacitor q V

q = CV

Where C is the capacitance in farads, if q is charge in coulombs and V is the potential difference across the capacitor in volts. The current flowing in the circuit is rate of flow of charge

i = = C

The capacitance of a capacitor depends on the dielectric medium & the physical dimensions. For a parallel plate capacitor, the capacitance

C = = €0 €r

A is the surface area of plates D is the separation between plates

€ is the absolute permeability of medium€0 is the absolute permeability of free

space €r is the relative permeability of medium

i= = C

V =

The power absorbed by the capacitor P = vi = vc

Energy stored in the capacitor W = = dt

= C = Joules

This energy is stored in the electric field set up by the voltage across capacitor.

Conclusions:

1. The current in a capacitor is zero, if the voltage across it is constant, that means the capacitor acts as an open circuit to dc
2. A small change in voltage across a capacitance within zero time gives an infinite current through the capacitor, which is physically impossible. In a fixed capacitor, the voltage cannot change abruptly A capacitor behaves as short circuit just after switching across dc voltage.
3. The capacitor can store a finite amount of energy, even if the current through it is zero.
4. A pure capacitor never dissipates energy but only stores it hence it is called non-dissipative element.

Kirchhoff`s Laws:

Kirchhoff’s laws are more comprehensive than Ohm's law and are used for solving electrical networks which may not be readily solved by the latter.

Kirchhoff`s laws, two in number, are particularly useful in determining the equivalent resistance of a complicated network of conductors and for calculating the currents flowing in the various conductors.

1. Kirchhoff`s Current Law (KCL)

In any electrical network, the algebraic sum of the currents meeting at a point (or junction) is Zero.

That is the total current entering a junction is equal to the total current leaving that junction.

Consider the case of a network shown in Fig (a ).

I 1 +(-I 2 )+(I 3 )+(+I 4 )+(-I 5 ) = 0

I 1 +I 4 -I 2 -I 3 -I 5 = 0

Or

I 1 +I 4 = I 2 +I 3 +I 5

Or

Incoming currents =Outgoing currents

of the current through that branch.

(b) Sign of IR Drop

Now, take the case of a resistor (Fig. 2.4). If we go through a resistor in the same direction as the current, then there is a fall in potential because current flows from a higher to a lower potential.. Hence, this voltage fall should be taken -ve. However, if we go in a direction opposite to that of the current, then there is a rise in voltage. Hence, this voltage rise should be given a positive sign.

Consider the closed path ABCDA in Fig.

As we travel around the mesh in the clockwise direction, different voltage drops will have the following signs :

I 1 R 1 is - ve (fall in potential) I 2 R 2 is - ve (fall in potential) I 3 R 3 is + ve (rise in potential) I 4 R 4 is - ve (fall in potential) E 2 is - ve (fall in potential) E 1 is + ve (rise in potential)

Using Kirchhoff's voltage law, we get

-I 1 R 1 – I 2 R 2 – I 3 R 3 – I 4 R 4 – E 2 + E 1 = 0

Or I 1 R 1 + I 2 R 2 – I 3 R 3 + I 4 R 4 = E 1 – E 2

Assumed Direction of Current:

In applying Kirchhoff's laws to electrical networks, the direction of current flow may be assumed either clockwise or anticlockwise. If the assumed direction of current is not the actual direction, then on solving the question, the current will be found to have a minus sign. If the answer is positive, then assumed direction is the same as actual direction. However, the important point is that once a particular direction has been assumed, the same should be used throughout the solution of the question.

Kirchhoff's laws are applicable both to d.c. and a.c. voltages and currents. However, in the case of alternating currents and voltages, any e.m.f. of self-inductance or that existing across a capacitor should be also taken into account.

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Now, VL(t) =L 𝑑𝑖 𝑑𝑡(𝑡)

=110-350=0.05V 0<t< =110-30=0V 2<t<

=110-3(-50) =-0.05V 4<t<

The voltage waveform is shown in following figure.

2. A 0.5uF capacitor has voltage waveform v(t) as shown in following figure, plot i(t) as function of t?

Solution:

From the given waveform,

For 0<t<2, v(t) is a ramp of slope =(40/2)=

Therefore v(t)=20t

Therefore (^) i(t)=C𝑑𝑣 (𝑡)=0.5*10 20=110 A=10uA 𝑑𝑡

For 2<t<4, v(t) is constant

Therefore v(t)=40V

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Therefore (^) i(t)=C𝑑𝑣 (^) 𝑑𝑡(𝑡)=0.5*10 *0=0A

For 4<t<8, v(t) is a ramp with slope = 0−40 8−4 = −

Therefore v(t)=-10t+80 (According to straight line equation i.e. y=mx+c)

Therefore i(t)=C𝑑𝑣 (^) 𝑑𝑡(𝑡)=0.5*10 *(-10)=-5uA

The current waveform is shown in following figure

3. A Pure Inductance Of 3mh Carries A Current Of The Waveform Shown In Fig. Sketch The Waveform Of V (t) And P(t).Determine The Average Value Of Power

Fig

Solution:

i (t)=5t for 0<t< i (t)=10 for 2<t<

i (t)=-10t+50 for 4<t< i (t)=-10 for 6<t<