# W03 Analysis of DC Circuits. Yrd. Doç. Dr. Aytaç Gören

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1 W03 Analysis of DC Circuits Yrd. Doç. Dr. Aytaç Gören

2 ELK Contents W01 Basic Concepts in Electronics W02 AC to DC Conversion W03 Analysis of DC Circuits (self and condenser) W04 Transistors and Applications (H-Bridge) W05 Op Amps and Applications W06 Sensors and Measurement (1/2) W07 Sensors and Measurement (2/2) W08 Midterm W09 Basic Concepts in Digital Electronics (Boolean Algebra, Decimal to binary, gates) W10 Digital Logic Circuits (Gates and Flip Flops) W11 PLC s W12 Microprocessors W13 Data Acquisition, D/A and A/D Converters. 2 Yrd. Doç. Dr. Aytaç Gören

3 ELK 2018 W01 Contents 1. Kirchoffs Circuit Law 2. Basic definitions for circuit analysis 3. Circuit Analysis 1. Mesh Current Analysis 2. Nodal Voltage Analysis 4. Thevenins Theorem 5. Nortons Theorem 6. Transient Analysis 7. Transient Analysis Capacitor 8. Transient Analysis - Inductor Extra Reference for this week: 3 Yrd. Doç. Dr. Aytaç Gören

4 Reminder Yrd. Doç. Dr. Aytaç Gören

5 Reminder Parameter Symbol Measuring Unit Description Voltage Volt V or E Current Ampere I or i Resistance Ohm R or Ω Conductance Siemen G or Capacitance Farad C Charge Coulomb Q Inductance Henry L or H Power Watts W Impedance Ohm Z Frequency Hertz Hz Unit of Electrical Potential V = I R Unit of Electrical Current I = V R Unit of DC Resistance R = V I Reciprocal of Resistance G = 1 R Unit of Capacitance C = Q V Unit of Electrical Charge Q = C V Unit of Inductance V L = -L(di/dt) Unit of Power P = V I or I 2 R Unit of AC Resistance Z 2 = R 2 + X 2 Unit of Frequency ƒ = 1 T Yrd. Doç. Dr. Aytaç Gören

6 Reminder Prefix Symbol Multiplier Power of Ten Terra T 1,000,000,000, Giga G 1,000,000, Mega M 1,000, kilo k 1, none none centi c 1/ milli m 1/1, micro µ 1/1,000, nano n 1/1,000,000, pico p 1/1,000,000,000, Yrd. Doç. Dr. Aytaç Gören

7 Yrd. Doç. Dr. Aytaç Gören Kirchoffs Circuit Law In complex circuits such as bridge or T networks, we can not simply use Ohm's Law alone to find the voltages or currents circulating within the circuit. Kirchoff developed a pair or set of rules which deal with the conservation of current and energy within electrical circuits. The rules are commonly known as: Kirchoffs Circuit Laws with one of these laws dealing with current flow around a closed circuit, Kirchoffs Current Law, (KCL) and the other which deals with the voltage around a closed circuit, Kirchoffs Voltage Law, (KVL).

8 Kirchoffs Circuit Law Kirchoffs Current Law or KCL, states that the "total current or charge entering a junction or node is exactly equal to the charge leaving the node as it has no other place to go except to leave, as no charge is lost within the node". (Conservation of Charge) The term Node in an electrical circuit generally refers to a connection or junction of two or more current carrying paths or elements such as cables and components

9 Kirchoffs Circuit Law Kirchoffs Voltage Law or KVL, states that "in any closed loop network, the total voltage around the loop is equal to the sum of all the voltage drops within the same loop" which is also equal to zero. In other words the algebraic sum of all voltages within the loop must be equal to zero.

10 Basic definitions for circuit analysis The terms, used frequently in circuit analysis: Path - a line of connecting elements or sources with no elements or sources included more than once. Node - a node is a junction, connection or terminal within a circuit were two or more circuit elements are connected or joined together giving a connection point between two or more branches. A node is indicated by a dot. Branch - a branch is a single or group of components such as resistors or a source which are connected between two nodes. Loop - a loop is a simple closed path in a circuit in which no circuit element or node is encountered more than once. Mesh - a mesh is a single open loop that does not have a closed path.

11 Example Using Kirchoffs Current Law (KCL) the equations are given as: At node A : I 1 + I 2 = I 3 At node B : I 3 = I 1 + I 2 Using Kirchoffs Voltage Law, KVL the equations are given as; Loop 1 is given as : 10 = R 1 x I 1 + R 3 x I 3 = 10I I 3 Loop 2 is given as : 20 = R 2 x I 2 + R3 x I3 = 20I I 3 Loop 3 is given as : = 10I 1-20I 2 As I 3 is the sum of I1 + I2 we can rewrite the equations as; 10 = 10I (I 1 + I 2 ) = 50I I 2 20 = 20I (I 1 + I 2 ) = 40I I 2

12 Example I 1 = Amps (Wrong Direction) I 2 = Amps I 3 = I 1 + I 2 At node B : I 3 = I 1 + I = Amps Using Kirchoff's Circuit Laws is as follows: 1. Assume all voltage sources and resistances are given. 2. Label each branch with a branch current. 3. Find Kirchoff's first law equations for each node. 4. Find Kirchoff's second law equations for each of the independent loops of the circuit. 5. Use Linear simultaneous equations as required to find the unknown currents.

13 Circuit Analysis While Kirchoff s Laws give us the basic method for analysing any complex electrical circuit, there are different ways of improving upon this method by using Mesh Current Analysis Nodal Voltage Analysis that results in a lessening of the math's involved and when large networks are involved this reduction in maths can be a big advantage.

14 Circuit Analysis One simple method of reducing the amount of math's involved is to analyse the circuit using Kirchoff's Current Law equations to determine the currents, I 1 and I 2 flowing in the two resistors. Then there is no need to calculate the current I 3 as its just the sum of I 1 and I 2. So Kirchoff's second voltage law simply becomes: Equation No 1 : 10 = 50I I 2 Equation No 2 : 20 = 40I I 2 therefore, one line of math's calculation have been saved.

15 Mesh Current Analysis A more easier method of solving the above circuit is by using Mesh Current Analysis or Loop Analysis which is also sometimes called Maxwell s Circulating Currents method. Instead of labelling the branch currents we need to label each "closed loop" with a circulating current. As a general rule of thumb, only label inside loops in a clockwise direction with circulating currents as the aim is to cover all the elements of the circuit at least once

16 Mesh Current Analysis Kirchoff's voltage law equation can be applied in the same way as before to solve them but the advantage of this method is that it ensures that the information obtained from the circuit equations is the minimum required to solve the circuit as the information is more general and can easily be put into a matrix form.

17 Mesh Current Analysis [ V ] gives the total battery voltage for loop 1 and then loop 2. [ I ] states the names of the loop currents [ R ] is called the resistance matrix.

18 Mesh Current Analysis The basic procedure for solving Mesh Current Analysis equations is as follows: 1. Label all the internal loops with circulating currents. (I 1, I 2,...I L etc) 2. Write the [ L x 1 ] column matrix [ V ] giving the sum of all voltage sources in each loop. 3. Write the [ L x L ] matrix, [ R ] for all the resistances in the circuit as follows; o R 11 = the total resistance in the first loop. o R nn = the total resistance in the nth loop.

19 Nodal Voltage Analysis Nodal Voltage Analysis uses the "Nodal" equations of Kirchoff's first law to find the voltage potentials around the circuit. By adding together all these nodal voltages the net result will be equal to zero. For each node we apply Kirchoff's first law equation, that is: "the currents entering a node are exactly equal in value to the currents leaving the node" then express each current in terms of the voltage across the branch.

20 As V a = 10v and V c = 20v Nodal Voltage Analysis

21 Thevenins Theorem Thevenins Theorem states that "Any linear circuit containing several voltages and resistances can be replaced by just a Single Voltage in series with a Single Resistor". it is possible to simplify any "Linear" circuit, to an equivalent circuit with just a single voltage source in series with a resistance connected to a load as shown below. Thevenins Theorem is especially useful in analyzing power or battery systems and other interconnected circuits where it will have an effect on the adjoining part of the circuit.

22 Thevenins Theorem Example the value of the voltage required Vs is the total voltage across terminals A and B with an open circuit and no load resistor Rs connected. The value of resistor Rs is found by calculating the total resistance at the terminals A and B with all the emf s removed

23 Thevenins Theorem Example The Equivalent Resistance (Rs) The Equivalent Voltage (Vs)

24 Thevenins Theorem Example Thevenins Equivalent circuit is shown below with the 40Ω resistor connected. The basic procedure for solving a circuit using Thevenins Theorem is as follows: 1. Remove the load resistor R L or component concerned. 2. Find R S by shorting all voltage sources or by open circuiting all the current sources. 3. Find V S by the usual circuit analysis methods. 4. Find the current flowing through the load resistor R L.

25 Nortons Theorem Nortons Theorem states that "Any linear circuit containing several energy sources and resistances can be replaced by a single Constant Current generator in parallel with a Single Resistor". The value of this "constant current" is one which would flow if the two output terminals where shorted together while the source resistance would be measured looking back into the terminals, (the same as Thevenin).

26 Nortons Theorem To find the Nortons equivalent of the above circuit we firstly have to remove the centre 40Ω load resistor and short out the terminals A and B to give us the following circuit.

27 Nortons Theorem When the terminals A and B are shorted together the two resistors are connected in parallel across their two respective voltage sources and the currents flowing through each resistor as well as the total short circuit current can now be calculated as:

28 Nortons Theorem The value of the internal resistor Rs is found by calculating the total resistance at the terminals A and B giving us the following circuit.

29 Nortons Theorem Nortons equivalent circuit. The basic procedure for solving a circuit using Nortons Theorem is as follows: 1. Remove the load resistor R L or component concerned. 2. Find R S by shorting all voltage sources or by open circuiting all the current sources. 3. Find I S by placing a shorting link on the output terminals A and B. 4. Find the current flowing through the load resistor R L.

30 Transient Analysis The time constant Electrical or Electronic circuits or systems suffer from some form of "time-delay" between its input and output, when a signal or voltage, either continuous, (DC) or alternating (AC) is firstly applied to it. This delay is generally known as the Time Constant of the circuit and it is the time response of the circuit when a step voltage or signal is firstly applied. The resultant time constant of any circuit or system will mainly depend upon the reactive components either capacitive or inductive.

31 Transient Analysis - Capacitor When an increasing DC voltage is applied to a discharged Capacitor the capacitor draws a charging current and "charges up", and when the voltage is reduced, the capacitor discharges in the opposite direction. Because capacitors are able to store electrical energy they act like small batteries and can store or release the energy as required. The charge on the plates of the capacitor is given as: Q = CV. This charging (storage) and discharging (release) of a capacitors energy is never instant but takes a certain amount of time to occur with the time taken for the capacitor to charge or discharge to within a certain percentage of its maximum supply value being known as its Time Constant (τ).

32 Transient Analysis - Capacitor RC Charging Circuit The figure below shows a capacitor, (C) in series with a resistor, (R) forming a RC Charging Circuitconnected across a DC battery supply (Vs) via a mechanical switch. When the switch is closed, the capacitor will gradually charge up through the resistor until the voltage across it reaches the supply voltage of the battery.

33 Transient Analysis - Capacitor RC Charging Circuit If C is fully "discharged" and the switch (S) is fully open, these are the initial conditions of the circuit, then t = 0, i = 0 and q = 0. When the switch is closed the time begins at t = 0 and current begins to flow into the capacitor via the resistor. Since the initial voltage across the capacitor is zero, (V c = 0) the capacitor appears to be a short circuit and the maximum current flows through the circuit restricted only by the resistor R.

34 Transient Analysis - Capacitor RC Charging Circuit The current no flowing around the circuit is called the Charging Current and is found by using Ohms law as:i = V R /R.

35 Transient Analysis - Capacitor RC Charging Circuit The capacitor now starts to charge up as shown, with the rise in the RC charging curve steeper at the beginning because the charging rate is fastest at the start and then slow down as the capacitor takes on additional charge at a slower rate. As the capacitor charges up, the voltage difference between Vs and Vc reduces, so to does the circuit current, i. Then at the final condition, t =, i = 0, q = Q = CV. Then at infinity the current diminishes to zero, the capacitor acts like an open circuit condition therefore, the voltage drop is entirely across the capacitor.

36 Transient Analysis - Capacitor RC Charging Circuit As the capacitor charges the potential difference across its plates increases with the actual time taken for the charge on the capacitor to reach 63% of its maximum possible voltage, in our curve 0.63Vs being known as the Time Constant, (T) of the circuit. V is related to charge on a capacitor given by the equation, Vc = Q/C, the voltage across the value of the voltage across the capacitor, (Vc) at any instant in time during the charging period is given as:

37 Transient Analysis - Capacitor RC Charging Circuit Vc is the voltage across the capacitor Vs is the supply voltage RC is the time constant of the RC charging circuit After a period equivalent to 4 time constants, (4T) the capacitor in this RC charging circuit is virtually fully charged and the voltage across the capacitor is approx 0.99Vs. The time period taken for the capacitor to reach this 4T point is known as the Transient Period. As the capacitor is fully charged no more current flows in the circuit. The time period after this 5T point is known as the Steady State Period.

38 Transient Analysis - Inductor An inductor could not change instantaneously, but would increase at a constant rate determined by the self-induced emf in the inductor. In other words, an inductor in a circuit opposes the flow of current, ( i ) through it. An LR 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.

39 Transient Analysis - Inductor The above LR 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, i begins to flow through the circuit but does not rise rapidly to its maximum value of I max as determined by the ratio of V / R (Ohms Law). This limiting factor is due to the presence of the self induced emf within the inductor as a result of the growth of magnetic flux, (Lenz's Law).

40 Transient Analysis - Inductor After a time the voltage source neutralizes the effect of the self induced emf, the current flow becomes constant and the induced current and field are reduced to zero. Kirchoffs Voltage Law, (KVL) to define the individual voltage drops:

41 Transient Analysis - Inductor Expression for the Current in an LR Series Circuit Since the voltage drop across the resistor, V R is equal to IxR (Ohms Law), it will have the same exponential growth and shape as the current. However, the voltage drop across the inductor, V L will have a value equal to: Ve (-Rt/L).

42 Ref for this week: 42 Yrd. Doç. Dr. Aytaç Gören

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