Module 1 DC Circuits bpresentation.pptx

basic electrical engineering dc circuits

1. Chapter 1 1

2. DC Circuit DC Circuit elements Voltage and Current Sources Kirchhoff’s Current and Voltage law Series resonance and Parallel resonance Superposition theorem Thevenin Theorem Norton Theorem First order RL and RC circuits 2

3. DC Circuit Thomas Edison and Alessandro Volta were pioneers in DC current. It is the constant flow of Electric Charge from high to low Potential. In DC circuits, electricity flows in constant direction with a fixed polarity that doesn’t vary with time. DC is commonly found in many low-voltage applications, especially where these are powered by Battery. 3

4. DC Circuit Elements Three types of basic elements: 1) Resistor, 2) Inductor and 3) Capacitor Resistor : It is taken as passive element since it can dissipate energy as heat as long as current flows through it. Denoted by R. 2) Inductor : An inductor is considered as passive element of circuit because it can store energy as magnetic field and can deliver that energy to the circuit. Denoted by L. 3) Capacitor: A capacitor is considered as passive element, because it can store energy in it as electric field and delivers that energy to the circuit. Denoted by C. Resistor Inductor Capacitor 4

5. Voltage Source It is an active element. Real voltage sources can be represented as ideal voltage sources in series with a resistance. The ideal voltage source having zero resistance. The magnitude of DC voltage source is represented by the uppercase symbol V S. Voltage Source 5

6. Voltage Source A voltage source can be considered as an active element. If current leaves from the positive terminal of the voltage source, the energy is delivered from the source to the circuit. As per definition of active element of electrical circuit, a battery can be considered as active element, as it continuously delivers energy to the circuit during discharging. Real voltage sources can be represented as ideal voltage sources in series with a resistance r, the ideal voltage source having zero resistance. Voltage Source and it’s I-V characteristic

7. Voltage Source An ideal voltage source has zero internal resistance so that changes in load resistance will not change the voltage supplied. The magnitude of DC voltage source is represented by the uppercase symbol V S . An ideal voltage source will supply a constant voltage at all times regardless of the value of the current being supplied producing an I-V characteristic represented by a straight line.  

8. Current Sources It is an active element. The current supplied to the circuit by an ideal current source is independent of circuit voltage. Real current sources can be represented as ideal current source in parallel with a resistance. The ideal current source having infinite resistance. Current Source 6

9. Current Source A current source is also considered as active element. The current supplied to the circuit by an ideal current source is independent of circuit voltage. Real current source can be represented as ideal current source in parallel with a resistance r, the ideal current source having infinite resistance. Because of this reason of ideal current sources, changes in load resistance will not change the current supplied. An ideal current source is called a “constant current source” as it provides a constant steady state current independent of the load connected to it producing an I-V characteristic represented by a straight line. Current Source and it’s I-V characteristic

10. Current Source The current source is if DC source then it delivers a constant DC current independent of time. Alternatively, it may be a time-dependent source. Fig above shows the symbolic representation and i -v characteristic of an independent current source, where I S is used for DC current. A transistor is an active circuit element, because, it can amplify the power of a signal. On the other hand, transformer is not an active circuit element because it does not amplify power and the power remains same both in primary and secondary side. Although, a transistor is an example of active circuit elements, but the transformer is an example of passive circuit elements.

11. Kirchhoff’s Current Law The algebraic sum of all the currents at any node point or a junction of a circuit is zero. Σ I = 0 The direction of incoming currents to a node is taken as positive while the outgoing currents is taken as negative. i 1  + i 2  – i 3  – i 4  – i 5  + i 6  = 0 Sum of incoming currents = Sum of outgoing currents i 1  + i 2  + i 6  = i 3  + i 4  + i 5 Kirchhoff's Current Law 7

12. Kirchhoff’s Voltage Law The algebraic sum of the voltages (or voltage drops) in any closed path of network that is transverse in a single direction is zero. Σ E + Σ V = 0 So, equation can be written, -V 1 + (-V 2 ) + iR 1 +iR 2 = 0 For the dependent sources in the circuit, KVL can also be applied.   Kirchhoff’s Voltage Law 8

13. Series Resistances When two or more resistors are connected together end-to-end in a single branch, the resistors are said to be connected together in series. Series circuits are voltage dividers . Resistors in Series carry the same current. I R1 = I R2 = I R3 =I AB R T = R 1 +R 2 +R 3 V Total  = V R1  + V R2  + V R3  + ……. + V N Total resistance is generally known as the  Equivalent Resistance  R S  = R 1  + R 2  + R 3 + ……. + R N Series Resistances Equivalent Circuit 9

14. Voltage Dividers A voltage divider   is a simple circuit which turns a large voltage into a smaller one. Using just two series resistors and an input voltage, we can create an output voltage that is a fraction of the input. Voltage dividers are one of the most fundamental circuits in electronics. Different voltages, or voltage drops, appear across each resistor within the series network. Connecting resistors in series like this across a single DC supply has one major advantage, different voltages appear across each resistor producing a very handy circuit called a Voltage Divider Network. This simple circuit splits the supply voltage proportionally across each resistor in the series chain with the amount of voltage drop being determined by the resistors value and the current through a series resistor circuit is common to all resistors. So, a larger resistance will have a larger voltage drop across it, while a smaller resistance will have a smaller voltage drop across it.

15. Voltage Dividers A voltage divider involves applying a voltage source across a series of two resistors. You may see it in fig below. The circuit shown consists of just two resistors, R 1  and R 2  connected together in series across the supply voltage V in . One side of the power supply voltage is connected to resistor R 1 , and the voltage output  V out  is taken from across resistor R 2 . The value of this output voltage is given by the corresponding formula. Voltage Divider Circuit

16. Voltage Divider If more resistors are connected in series to the circuit then different voltages will appear across each resistor in turn with regards to their individual resistance R values providing different but smaller voltage points from one single supply. The voltage divider equation assumes that you know three values of the above circuit: the input voltage (V in ), and both resistor values (R 1  and R 2 ). Given those values, we can use this equation to find the output voltage ( V out ): This equation states that the output voltage is directly proportional to the input voltage and the ratio of R 1  and R 2 .  

17. Parallel Resistances When two or more resistors are connected so that both of their terminals are respectively connected to each terminal of the other resistor or resistors, they are said to be connected in parallel. Parallel circuits are Current dividers . The voltage across each resistor is exactly the same. V R1 = V R2 = V R3 = V AB The currents flowing through them are not the same. Generalize the equation for N resistors in parallel with the equation:   Equivalent Circuit Parallel Resistances 10

18. Current Dividers Resistors in parallel divide up the current. When we have a current flowing through resistors in parallel, we can express the current flowing through a single resistor as ratio of currents and resistances, without ever knowing the voltage. The electrical current entering the node of a parallel circuit is divided into the branches. Current divider formula is employed to calculate the magnitude of divided current in the circuits. Current Divider

19. Current Dividers The current I has been divided into I 1  and I 2  in two parallel branches with the resistance R 1  and R 2  and V is the voltage drop across the resistance R 1  and R 2 . From the ohm’s law, we know that V= IR ……. (1) So, we can write equation of the current as, and …………. (2) Let the total resistance of the circuit be R and is given by the equation shown below, ………… (3) So, we can write that, ………. (4) But, V=I 1 R 1 =I 2 R 2 ………..(5)  

20. Current Dividers Thus, from equation (4), So, and Thus, in the current division rule, it is said that the current in any of the parallel branches is equal to the ratio of opposite branch resistance to the total resistance, multiplied by the total current. We can generalize the equation for N resistors in parallel with the equation: Where, I K  is the current flowing through resistor k and I is the current flowing through all the resistors.    

21. Superposition Theorem In any linear, active, bilateral network having more than one source, the response across any element is the sum of the responses obtained from each source considered separately and all other sources are replaced by their internal resistance. The superposition theorem is used to solve the network where two or more sources are present and connected. In this method, we will consider only one independent source at a time. So, we have to eliminate the remaining independent sources from the circuit. We can eliminate the voltage sources by shorting their two terminals and similarly, the current sources by opening their two terminals. 11

22. Superposition Theorem Follow these steps in order to find the response in a particular branch using superposition theorem . Find the response in a particular branch by considering one independent source and eliminating the remaining independent sources present in the network. Repeat above Step for all independent sources present in the network. Add all the responses in order to get the overall response in a particular branch when all independent sources are present in the network. 12

23. Explanation of Superposition Theorem Let us understand the superposition theorem with the help of an example. The circuit diagram shown below consists of a two voltage sources V 1  and V 2 . Superposition theorem Superposition theorem taking source V 1 First, take the source V 1  alone and short circuit the V 2  source as shown in the circuit.

24. Explanation of Superposition Theorem Here, the value of current flowing in each branch, i.e. I 1 ’, I 2 ’ and I 3 ’ is calculated by the following equations. The difference between the above two equations gives the value of the current, Now, activating the voltage source V 2  and deactivating the voltage source V 1 by short circuiting it, find the various currents, i.e. I 1 ’’, I 2 ’’, I 3 ’’ flowing in the circuit diagram. Superposition theorem taking source V 2  

25. Explanation of Superposition Theorem Here, And the value of the current I 3 ’’ will be calculated by the equation, For finding the actual current, add or subtract the current due to individual source. If the current is in same direction add them. If current is in opposite direction, subtract them.  

26. Explanation of Superposition Theorem As per the superposition theorem the value of current I 1 , I 2 , I 3  is now calculated as below equation, Direction of current should be taken care while finding the current in the various branches. The application of superposition theorem is used only for linear circuits as well as the circuit which has more supplies. The Superposition theorem cannot be useful for power calculations but this theorem works on the principle of linearity. As the power equation is not linear.  

27. Thevenin Theorem Any linear active network consisting of independent or dependent voltage and current source and the network elements can be replaced by an equivalent circuit having a voltage source in series with a resistance. The voltage source being the open circuited voltage across the open circuited load terminals. The resistance being the internal resistance of the source. The voltage source present in the Thevenin’s equivalent circuit is called as Thevenin’s equivalent voltage or simply Thevenin’s voltage, V Th . The resistor present in the Thevenin’s equivalent circuit is called as Thevenin’s equivalent resistor or simply Thevenin’s resistor, R Th . 13

28. Thevenin Theorem The basic procedure for solving a circuit using  Thevenin’s Theorem  is as follows: Remove the load resistor R L  or component concerned. Find source resistance by shorting all voltage sources or by open circuiting all the current sources. Find source voltage by the usual circuit analysis methods. Find the current flowing through the load resistor. Equivalent Circuit 14

29. Explanation of Thevenin’s Theorem The current flowing through a resistor connected across any two terminals of a network by an equivalent circuit having a voltage source V th in series with a resistor R th . Where V th  is the open circuit voltage between the required two terminals called the Thevenin voltage and the R th  is the equivalent resistance of the network as seen from the two terminals with all other sources replaced by their internal resistances called Thevenin resistance. The Thevenin’s statement is explained with the help of a circuit shown below.

30. Explanation of Thevenin’s Theorem Let us consider a simple DC circuit as shown in the figure above, where we have to find the load current I L  by the Thevenin’s theorem. In order to find the equivalent voltage source,   R L  is removed from the circuit as shown in the figure below and V OC  or V TH  is calculated. Thevenin Theorem  

31. Explanation of Thevenin’s Theorem Now, to find the internal resistance of the network (Thevenin’s resistance or equivalent resistance) in series with the open circuit voltage V OC , also known as Thevenin’s voltage V TH , the voltage source is removed or we can say it is deactivated by a short circuit (as the source does not have any internal resistance) as shown in the figure below, Thevenin Theorem  

32. Explanation of Thevenin’s Theorem Equivalent Circuit of Thevenin’s Theorem As per Thevenin’s Statement, the load current is determined by the circuit shown above and the equivalent Thevenin’s circuit is obtained. The load current I L is given by: Where, V TH is the Thevenin’s equivalent voltage. It is an open circuit across the terminal AB known as load terminal. R TH is the Thevenin’s equivalent resistance, as seen from the load terminals where all the sources are replaced by their internal impedance R I is the load resistance.  

33. Norton Theorem A linear active network consisting of independent or dependent voltage source and current sources and the various circuit elements can be substituted by an equivalent circuit consisting of a current source in parallel with a resistance. The current source being the short-circuited current across the load terminal and the resistance being the internal resistance of the source network. The Norton’s theorems reduce the networks equivalent to the circuit having one current source, parallel resistance and load . Norton’s theorem is the converse of Thevenin’s Theorem. It consists of the equivalent current source instead of an equivalent voltage source as in Thevenin’s theorem. 15

34. Norton Theorem The basic procedure for solving a circuit using  Norton’s Theorem  is as follows: Remove the load resistor R L  or component concerned. Find Source resistor by shorting all voltage sources or by open circuiting all the current sources. Find Source current  by placing a shorting link on the output terminals. Find the current flowing through the load resistor R L . In a circuit, power supplied to the load is at its maximum when the load resistance is equal to the source resistance. Equivalent Circuit 16

35. Explanation of Norton’s Theorem Let us consider a circuit diagram as shown in fig below. Circuit diagram consists of voltage source, resistors and one load resistors. Norton’s Theorem In order to find the current through the load resistance I L  as shown in the circuit diagram above, the load resistance has to be short- circuited as shown in the figure as shown.

36. Explanation of Norton’s Theorem Now, the value of current I flowing in the circuit is found out by the equation, And the short-circuit current I SC  is given by the equation shown below, Now the short circuit is removed, and the independent source is deactivated and the value of the internal resistance is calculated.  

37. Explanation of Norton’s Theorem As per the Norton’s Theorem, the equivalent source circuit would contain a current source in parallel to the internal resistance, the current source being the short-circuited current across the shorted terminals of the load resistor. The Norton’s Equivalent circuit is represented as below. Finally, the load current I L  calculated by the equation shown below, Where, I L  is the load current I sc  is the short circuit current R int  is the internal resistance of the circuit R L  is the load resistance of the circuit Equivalent Circuit  

38. First Order RL Circuit A  RL Series Circuit   consists basically of an inductor of inductance, L connected in series with a resistor of resistance, R. The resistance “R” is the DC resistive value of the wire turns or loops that goes into making up the inductors coil. Kirchhoff’s voltage law (KVL), The value of the current at any instant of time as:   17

39. First Order RL Circuit An inductor in an electrical circuit opposes the flow of current, ( i ) through it. While this is perfectly correct, we made the assumption that it was an ideal inductor which had no resistance or capacitance associated with its coil windings. For real world purposes we can consider our simple coil as being an “Inductance”, L in series with a “Resistance”, R. In other words, forming an RL Series Circuit. A RL Series Circuit consists basically of an inductor of inductance, L connected in series with a resistor of resistance, R. The resistance “R” is the DC resistive value of the wire turns or loops that goes into making up the inductors coil.

40. First Order RL Circuit The  RL series circuit is connected across a constant voltage source, (the battery) and a switch. Assume that the switch, S is open until it is closed at a time t = 0, and then remains permanently closed producing a “step response” type voltage input. The current begins to flow through the circuit but does not rise rapidly to its maximum value of Imax as determined by the ratio of V / R (Ohms Law). We can use Kirchhoff’s Voltage Law, (KVL) to define the individual voltage drops that exist around the circuit and then use it to as an expression for the flow of current. Kirchhoff’s voltage law (KVL) gives us: The voltage drops across the resistor, R is I*R (Ohms Law), The voltage drops across the inductor, L is by now our familiar expression L(di/dt),  

41. First Order RL Circuit Then the final expression for the individual voltage drops around the LR series circuit can be given as: We can see that the voltage drop across the resistor depends upon the current, while the voltage drop across the inductor depends upon the rate of change of the current, di/dt. When the current is equal to zero at time t = 0 the above expression, which is also a first order differential equation, can be rewritten to give the value of the current at any instant of time as: Where:      V is in Volts       is in Ohms      L is in Henry     t is in Seconds    e is the base of the Natural Logarithm = 2.71828  

42. First Order RL Circuit The Time Constant of the LR series circuit is given as L/R and in which V/R represents the final steady state current value after five times constant values. Once the current reaches this maximum steady state value at 5Շ, the inductance of the coil has reduced to zero acting more like a short circuit and effectively removing it from the circuit. Therefore, the current flowing through the coil is limited only by the resistive element in Ohms of the coil’s windings. A graphical representation of the current growth representing the voltage/time characteristics of the circuit can be presented as below figure.

43. First Order RL Circuit Since the voltage drop across the resistor, V R  is equal to I*R (Ohms Law), it will have the same exponential growth and shape as the current. However, the voltage drops across the inductor, V L  will have a value equal to:   Ve (-Rt/L) . Then the voltage across the inductor, V L will have an initial value equal to the battery voltage at time t = 0 or when the switch is first closed and then decays exponentially to zero as represented in the above curves. The time required for the current flowing in the LR series circuit to reach its maximum steady state value is equivalent to about 5 times the constants. This time constant τ, is measured by Շ = L/R, in seconds. The transient time of any inductive circuit is determined by the relationship between the inductance and the resistance. For example, for a fixed value resistance the larger the inductance the slower will be the transient time and therefore a longer time constant for the LR series circuit. However, for a fixed value inductance, by increasing the resistance value the transient time and therefore the time constant of the circuit becomes shorter. This is because as the resistance increases the circuit becomes more and more resistive as the value of the inductance becomes negligible compared to the resistance.

44. First Order RC Circuit In an RC circuit , the capacitor   stores energy between a pair of plates. When voltage is applied to the capacitor, the charge builds up in the capacitor and the current drops off to zero. We are assuming that the circuit has a constant voltage source V. This equation does not apply if the voltage source is variable. The time constant in the case of an RC circuit is:   18

45. First Order RC Circuit In this theory, we see how to solve the differential equation arising from a circuit consisting of a resistor and a capacitor. In an RC circuit, the capacitor   stores energy between a pair of plates. When voltage is applied to the capacitor, the charge builds up in the capacitor and the current drops off to zero. RC series circuit

46. First Order RC Circuit At Constant Voltage condition, the voltage across the resistor and capacitor are as follows: Kirchhoff's voltage law says the total voltages must be zero. So, applying this law to a series RC circuit results in the equation: One way to solve this equation is to turn it into a differential equation, by differentiating throughout with respect to  t : Solving for  i  gives us the required expression:  

47. First Order RC Circuit We are assuming that the circuit has a constant voltage source, V. This equation does not apply if the voltage source is variable. The time constant in the case of an RC circuit is:   The function of i has an   exponential decay shape as shown in the graph. The current stops flowing as the capacitor becomes fully charged. Applying our expressions from above, we have the following expressions for the voltage across the resistor and the capacitor:  

48. First Order RC Circuit While the voltage over the resistor drops, the voltage over the capacitor rises as it is charged: Curve of V R and V c The curve for voltage across resistor and voltage across capacitor are as shown in Fig above. If the voltage across capacitance changes instantaneously then the current requires infinite power. Hence voltage across capacitor cannot change instantaneously.

49. Thank You 19