Network Analysis: Units and Concepts

• Network analysis involves various methods to find voltages and currents in a circuit.
• Thoroughly understanding associated terms is essential before analysis.
• Simplify a network to make analysis easier.
• Discuss techniques for combining series and parallel connections of R, L, and C elements.

Systems of Units

• Engineers use measurable quantities, and communication must be in a standard language.
• The International System of Units (SI) serves as a standard measurement language.
• SI has six principal units from which other physical quantities are derived.
• Length: Meter (m)
• Mass: Kilogram (kg)
• Time: Second (s)
• Electric Current: Ampere (A)
• Temperature: Kelvin (K)
• Luminous Intensity: Candela (Cd)
• The SI unit uses prefixes based on powers of 10 to relate larger and smaller units to the base unit.
• Tera (T): 10^12
• Giga (G): 10^9
• Mega (M): 10^6
• Kilo (K): 10^3
• Milli (m): 10^-3
• Micro (µ): 10^-6
• Nano (n): 10^-9
• Pico (p): 10^-12

Basic Concepts and Definitions

• Electric charge is the most basic quantity in an electric circuit
• Charge is an electrical property of atomic particles, measured in coulombs (C)
• Denoted by the letter q or Q.
• Atoms consist of electrons, protons, and neutrons
• An electron has a negative charge equal to 1.602x10^-19 C, while a proton carries a positive charge of the same magnitude
• A neutron has no charge
• Equal numbers of protons and electrons leave an atom neutrally charged.
• Current is defined as the motion of charge through a conducting material, measured in Ampere (A)
• Denoted by the letter i or I.
• The ampere (A) represents the quantity of total charge passing through a cross-section of a conducting material per unit second.
• Formula: I = Q/t (Q is charge in Coulombs, I is current in amperes, t is time in seconds)
• Current is the rate of charge passing through a point in an electric circuit.
• Formula: i = dq/dt
• The charge transferred between times t1 and t2 is q = integral from t1 to t2 of i dt
• Constant current (DC) is denoted by I, while time-varying current (AC) is represented by i or i(t)
• Direct Current (DC): A current that remains constant with time.
• Alternating Current (AC): A current that varies with time.
• Voltage (or potential difference) is the energy required to move charge from one point to another, measured in volts (V)
• Denoted by the letter v or V.
• Mathematically, Vab = dw/dq
• Where w is energy in joules (J) and q is charge in coulombs (C).
• 1 volt = 1 joule/coulomb = 1 newton-meter/coulomb
• Voltage is always measured across a circuit element.
• Power is the time rate of expending or absorbing energy, measured in watts (W)
• Denoted by the letter p or P.
• Formula: p = dw/dt
• Where p is power in watts (W), w is energy in joules (J), and t is time in seconds (s).
• From voltage and current equations, p = v*I

Sign of Power

• Plus sign: Power is absorbed by the element. (Resistor, Inductor)
• Minus sign: Power is supplied by the element. (Battery, Generator)

Passive Sign Convention

• If the current enters through the positive polarity of the voltage, p = +vi
• If the current enters through the negative polarity of the voltage, p = -vi
• Energy is the capacity to do work, measured in joules (J)
• The energy absorbed or supplied by an element from time 0 to t is given by, w = integral from 0 to t of pdt = integral from 0 to t of vidt
• Electric power utility companies measure energy in watt-hours (WH) or kilowatt-hours (KWH).
• 1 WH = 3600 J

Ohm's Law

• Georg Simon Ohm (1787–1854) found the relationship between current and voltage for a resistor.
• Ohm's law: at constant temperature, voltage (V) across a conducting material is directly proportional to the current (I) flowing through the material
• V=RI
• R is resistance of the material.
• Ohm's law is not applicable to non-linear elements like diodes or transistors
• Ohm's law is not applicable for non-metallic conductors like silicon carbide.
• An element is the basic building block of a circuit and an electric circuit is an interconnection of the elements.
• Circuit analysis is the process of determining voltages across (or the currents through) the elements of the circuit

Types of Circuit Elements

• Active elements (Energy sources): capable of generating or delivering energy (e.g., generators, batteries)
• Passive elements (Loads): capable of receiving energy (e.g., resistors, capacitors, inductors)
• Active elements generate energy & the most important are voltage or current sources delivering power/energy to the circuit.
  • Independent Sources: provide a specified voltage or current completely independent of other circuit elements.
    • Ideal Independent Voltage Source: gives a constant voltage across its terminals irrespective of the current drawn
    • Ideal Independent Current Source: gives a constant current through its terminals irrespective of the voltage appearing
  • Dependent (Controlled) Sources: an active element where the source quantity is controlled by another voltage or current

Passive Elements (Loads)

• Passive elements receive energy and some (inductors and capacitors) store a finite amount that is later returned
• Passive element can't supply average power greater than zero over an infinite time interval.
  • Resistor: Material's characteristic of resisting the flow of electric charge and resistance is measured in ohms (Ω).

Resistance Factors

• Material
• Geometrical Shape.
• Resistance is proportional to its length and inversely proportional to its cross-sectional area: R = ρL/A
  • The proportionality constant is resistivity of the conductor

Conductance

• Inverse of resistance known, inverse of resistivity is conductivity, its symbol is G & conductivity is σ, units are Siemens per meter

• Power dissipated in a resistor: P = VI = I²R = V²/R

• Power may also be expressed in terms of G as: P = VI = V²G = I²/G

• Energy lost in the resistor from time 0 to t: W = integral from 0 to t of Pdt = integral from 0 to t of I²R dt = I²Rt

• Where V is in volts, I is in amperes, R is in ohms, and energy W is in joules

• Inductor: A wire of certain length, when twisted into a coil becomes a basic inductor

• If current passing through inductor, an electromagnetic field is formed

• Change in current produces change in electromagnetic field + induces voltage across coil

• Voltage across the inductor is directly proportional to the time rate of change of current

• v = L(di/dt)

• L = inductance

• Unit of inductance is Henry (H)

• The current in an inductor depends on the integral of voltage across its terminal and the initial current in the coil

• Power absorbed by inductor: P = vi = L * i * (di/dt)

• The energy stored by the inductor is W = 1/2 Li²

• The induced voltage across an inductor is zero if the current through it is constant (inductor acts as short circuit to DC)

• A small change in current in zero time through an inductor gives an infinite voltage (physically impossible + inductor opposes the sudden changes in currents)

• The inductor can store finite amount of energy and a pure inductor never dissipates energy

• called a non-dissipative passive element but physical inductors dissipate power due to internal resistance.

• Capacitor: Any two conducting surfaces separated by an insulating medium

• Stores energy in form of electric field by opposite charges on the two electrodes (+) - (-).

• When a voltage source (v) is connected to the capacitor a + charge q on one plate and a – charge q on the other = stores the electric charge

• The amount of charge stored (q) is directly proportional to the applied voltage v so that q = C*v

  • C is the constant of proportionality, is known as the capacitance = unit is the farad (F).

• C=ϵA/d

  • A is the surface area of each plate, d is the distance between the plates, and is the permittivity of the dielectric material

• The current flowing through the capacitor is i = dq/dt = C (dv/dt)

     Equation as: v (t  = 1/C integration of i dt + v(0)
     The voltage across the terminals of a capacitor depends upon the integral of the current through it and the initial voltage.
     The power absorbed by the capacitor is P = vi = vC (dv/dt)
    Energy stored by the capacitor is W = 1/2 Cv²
  

• The current in a capacitor is zero if the voltage across it is constant (capacitor acts as an open circuit to DC).

• A small change in voltage in zero time gives an infinite current (physically impossible + A capacitor will oppose the sudden changes in voltages)

• The capacitor can store a finite amount of energy and a pure capacitor never dissipates energy = a non-dissipative passive element BUT physical capacitors dissipate power due to internal resistance

Network/Circuit Terminology

• Network Elements: individual components (resistor, inductor, capacitor, diode, voltage source, current source, etc.) -Network: interconnection of network elements. -Circuit: network with at least one closed path -Branch: an element of a network having only two terminals. -Node: point of connection between two or more branches. -Loop: any closed path in a circuit -Mesh/Independent Loop: a loop without any other loops in it.

Kirchhoff's Laws

• Kirchhoff's voltage and current laws are the most common -Several useful relationships can be derived based on these laws -Formally known as Kirchhoff's current law (KCL) and Kirchhoff's voltage law (KVL).

Kirchhoff's Current Law (KCL)

-Also called Kirchhoff's first law/nodal law. -KCL is based on conservation of charge, demanding the algebraic sum of charges in a system to remain constant. -Statement: algebraic sum of the currents meeting at any junction/node is zero = algebraic means the quantity along with its sign (pos or neg) . -Mathematically, KCL dictates Σin=0 where N is the # of branches connected- the node By this law, currents entering a node may be regarded as positive, while currents leaving the node may be taken as negative/vice versa. -Alternate Statement: the currents flowing towards a junction = sum of the currents flowing away from the junction.

Kirchhoff’s Voltage Law (KVL)

• Also = Kirchhoff's second law/ loop/mesh law. Based on conservation of energy. -Statement: algebraic sum of all=voltages around a closed path /closed loop @ any instant is zero AND Algebraic sum @value + polarity -KVL implies to take care assigning proper signs/polarities for voltages in different sections of the circuit.
• The polarity of the voltages across active elements is fixed on terminals -The polarity of the voltage drop across the passive elements should be assigned with reference to the direction of the current WITH higher potential> to lower potential.
  • Entry point of current = + polarity of voltage drop across the element+ the exit point= - polarity.
  • The direction of currents in branches initially known either = the set/ assumed direction. After assigning, algebraic sum is accounted around a closed loop, +assign a+ sign for ascending/ -sign for-potentials.

Resistive Networks

• series resistors & voltage division as two/more resistor =series if same current flows -Combine resistors by combining =2 @ time
• If applying KVL v-v1-v2= 0
• Combining equations v=v1+v2=(Resistor1+Resistor2)

Equivalent Resistance & Voltage Drop

• Req= R1+R2

• To determine across each resistor = divide by the resistors V1 by (Resistor1/Resistor2) = V2 by (Resistor1+ Resistor2)*Resistor2 Notice =direct proportion> the larger resistance, the larger voltage drop= principle of voltage division+ a voltage divider, if divider has N (R1,R2,… RN), the nth resistor (RN) will have a drop of Vn =RN/(R1+R2+…+RN).

• parallel>2/ more at =voltage (circuit in Fig. 1.20, -Equivalent Resistance= total= R1 R2

• parallel+ equal product /sum BUT applies only 2 =Eq. (6), if+ theReq = R1/2+=Eq. (5)+the result+the N resistors Req is the reciprocal -Current Division= to derive currents (total= at +enters node -This =that =shared inversely= this = +the circuit known=

Inductors

• Two/more =series= if at =current (connection+ shown (a), (b)). Have the = (loop) -The equivalent inductance= -The (parallel+ =if+ @element. Consider+ parallel connection (a),+ @, with+circuit+ -By connecting Req parallel<smallest= -Note= combine @=as-=

Capacitive Networks

• (circuits)= series-parallel< reducing<. This, with, and.. + to-Replace< = series if The
• With N + With=

• = With (b).
• Applying>= (loop (1))

• That (v)
• Where