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 & 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 & 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 & Distributed Elements Lumped elements are those elements which are very small in size & 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 & v for AC circuits & capital I, V for DC circuits. Inductor Energy of Inductor: ½ 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) & 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’ & ‘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 & 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 & B where the terminals A & 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 & current sources are open circuited) Case II: If the circuit having independent & dependent sources both. In this case, we calculate VTh & IN then the internal impedance of network N is obtained as ZTh = VTh/IN Limitations of Thevenin’s & 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 & other circuit element. Millman’s Theorem ∑ππ=1 πΈπππ πΈπ = π ∑π=1 ππ ππ = 1 ∑ππ=1 ππ πΈπ = = 1 π1 + π2 + β― + ππ πΈ1π1 + πΈ2π2 + β― + πΈπππ π1 + π2 + β― + ππ π = πΉ + ππΏ π Im = ∑ πΈπππ = πΈ1π1 + πΈ2π2 + β― + πΈπππ π=1 π Ym = ∑ ππ = π1 + π2 + β― + ππ π=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. π°= ππ¬ πΉπ + πΉπ Power Through RL = I2RL π½ππ πΉπ³ =π· (πΉπ + πΉπ³)π Now, to find the maximum value. ππ ππ πΏ = π ππ πΏ ππ 2 [(π π +π πΏ)2 . π πΏ] (π π + π πΏ)2 − (2π πΏ)(π π + π πΏ) 0 = ππ [ ] (π π + π πΏ)4 2 (π π + π πΏ)2 − (2π πΏ)(π π + π πΏ) = 0 ο° Rs = RL Mesh Analysis Super Mesh Analysis Nodal Analysis Super Node Analysis
