EE 151: DC Circuits and Theorems

• EE 151 is Applied Electricity
• E.A Affum is the instructor.
• A. Robert and W. Fortunatus are assistants.
• Contact via email: [email protected]
• Contact via phone: 0547926139
• Location: Ceaser Building, room 316

Lecture Outline: Fundamentals of DC Circuits

• Topics include DC/AC circuits, active/passive elements, Ohm’s law, current/voltage division, V-I relationships for resistors/capacitors/inductors.
• Theorems covered: Kirchhoff's, Thevenin's, Norton's, superposition, reciprocity, delta-star transformation
• Methods like mesh/nodal analysis and ideal source equivalent circuits are also part of the curriculum.

Introduction to DC and AC Circuits

• Applied Electricity covers electricity fundamentals, generation, transmission, and distribution.
• Topics: electric circuit arrangements, transformers, motors and their applications
• AC (Alternating Current) changes magnitude and direction regularly.
• DC (Direct Current) flows in one direction and is used in low-voltage applications.

Generator Principle

• Electrical generators convert mechanical energy into electrical energy.
• The induced EMF is produced with Faraday's law of electromagnetic induction
• Generators' essential parts: magnetic field and conductors to cut the flux.
• Generators use a prime mover, such as a steam turbine, diesel engine, or electric motor.

DC Generator

• Electrical DC machines convert mechanical energy into direct current electricity (DC generator) or vice versa (DC motor) without constructional changes.
• DC machines have a field circuit and an armature circuit.
• The rotor is the rotating part, and the stator is the stationary part of the machine.

Operating Principle of DC Generator

• Essential components for converting mechanical energy into electrical energy: conductors, magnetic field, and mechanical energy
• Alternating AC current production: achieved by rotating a loop in a constant magnetic field; current direction determined by the right-hand rule
• Faraday's Law explains moving a wire through a magnetic field induces voltage.
• A potential difference is maintained with movement through the field; reversing motion reverses polarity of the potential difference across the conductor.
• Induced voltage on a loop: 2vBl under the pole face, 0 away from the pole face (v=velocity, B=magnetic field strength, l=length of conductor)

AC Generator

• AC generators, also known as alternators, convert mechanical energy into electrical energy.
• Alternating current sinusoidal output waveform is the resultant energy form.
• AC generators: stator for field, rotor, exciter field rheostat, slip rings, and brushes

Operating principle of AC Generator

• Electromagnetic force (EMF or voltage) is generated in a current-carrying conductor cutting a magnetic field.
• EMF induces in the armature
• Produces electric current through the galvanometer, slip rings, and brushes with positive and negative values.
• Fleming’s Right Hand Rule determines the direction of the induced current.

Circuit Terminologies

• Node (Junction): A point where currents split or meet (e.g., points c, d, e, f).
• Path: A connection where current flows (e.g., bc, be, fa).
• Branch: A connection (path) between two nodes (e.g., cd, cbad, df).
• Loop/Mesh: A closed path of a circuit (e.g., cghdc).
• Current (I): Directed flow of electrons (charge), measured in Amps.
• Electrical potential: Capacity to do work, measured in Volts.
• Potential difference: Difference in potentials between two charged bodies.
• Power (P): Rate at which electrical work is done, measured in Watts.
• Electrical work (W): Transfer of charge, measured in joules.
• Energy: Capacity to do work.
• Electrical Network: A combination of Resistors, Inductors, Capacitors, Voltage sources, and Current sources.

Classification of Circuit Elements

• Active source (energy sources): Includes voltage and current sources. E.g., generators, vacuum tubes, transistors, diodes
• Passive source: Either dissipates energy (heat) or stores energy. E.g., Resistors (R), inductors (L), and capacitors (C)
• Active elements amplify signal power (voltage or currents), and a transistor is a good example for radio/RF circuits.
• Resistors dissipate energy(R); inductors store energy in a magnetic field (L); capacitors store energy in electrostatic filed (C)
• Electrical network: Combination of passive and active elements using conductors
• Active network: Occurs when a connection of passive and active elements by conductors.
• Passive network: Occurs without active sources.
• Ohm's law: A simple electric network with a voltage source and resistor V = iR.

Short Circuit (SC)

• Two-terminal element where voltage must be zero
• The resistance is theoretically zero.
• Currents divert to itself from adjacent branches not having sources.
• Equations: V(t) = 0, i(t) ≠ 0, implies R -> 0 and I -> ∞.

Open Circuit (OC)

• A branch where the resistance is theoretically infinite.
• Open circuits prevent current flow.
• Equations are R -> ∞, and I = 0.

Ohm's Law

• With physical conditions of the conductor stable, the ratio between potential difference and current is constant.
• Formula: V/I = constant, V = IR
• Power Equations: P = VI = I²R = V²/R.
• Current flows in either direction if terminals are interchanged.
• Diodes when forward-biased have higher current, When reversed-biased, exhibit very low current.
• A graph where current is on the X axis represents a straight line representing the resistance.
• Limitations: It cannot be applied to non-metallic conductors/ non-linear devices (Zener diode)

Voltage-Current Relations

• Resistance R: V = Ri, i = V/R
• Inductance L: V = L(di/dt), i = (1/L)∫V dt
• Capacitance C: V = (1/C)∫i dt, i = C(dv/dt)
• Voltage across each element in a circuit for i = Im * sin(ωt)
• For a resistor: VR = Vm * sin(ωt), where Vm = R * Im
• For an inductor: VL = Vm * sin(ωt + π/2), where `Vm = ωL * Im
• For a capacitor: Vc VC = Vm * sin(ωt - π/2), where Vm=Im / ωC

Resistors in Series

• Resistors get arranged in series when the current flows through them is the same
• Resistors in Series: NO JUNCTION
• Total resistance: 𝑅𝑇 = 𝑅1 + 𝑅2 +...+𝑅𝑁
• For series circuits: the current is the same, voltage depends on resistance, and voltage drops/resistances/powers are additive
• Applied voltage equals the sum of individual voltages.

Resistors in Parallel

• Resistors in parallel circuits have the same voltage across them.
• Colloquially, two resistors are in parallel if you can move from one to the other without passing through another element.
• In circuits, components must be able to be circled without passing through to be in parrallel

Effective Resistance of a Circuit

• Formulas: RT = (2//3) + 1 = (2*3)/(2+3) + 1 = 11/5 or RT = (2//2) + 1 = (2*2)/(2+2) + 1 = 2

Internal Resistance of Sources

• Voltage sources have internal resistance (r) that is very small in value and is a series resistor connected externally to the source.
• Current sources have internal resistance (r) that is very high in value and is connected externally across the source.

Capacitors

• Capacitors: devices storing electrical energy in an electrostatic field, measured in Farads
• Capacitance (C): Q/V (Q=charge, V=potential difference) and ε₀εrA/d (ε₀=absolute permitivity, εr=relative permitivity, A=electrode area, d=spacing)
• E = 1/2 QV = Q²/2C = 1/2CV² is the formula for energy stored
• V-I relationship: i = C dv/dt or v =(1/C)∫idt + v(0)
• When voltage is V(Constant), dv/dt = 0, and i = 0, it acts as an open circuit.

Capacitors in Series

• The total capacitance is less than individual capacitances.
• Capacitances diminish in series
• Series capacitance is (1/Ctotal) = (1/C1) + (1/C2) + ... + (1/Cn)

Capacitors in Parallel

• Behaving as a capacitor where the area is the sum of surface areas of each capacitors
• Capacitances add in parallel.
• Effective parallel capacitance: Ctotal= C1+ C2+ ... + Cn

Inductors

• Store electrical energy in a field using conducting wire wrapped around a core of ferromagnetic material.
• V-I relationship: v = L(di/dt) or == (1/L)∫vdt +i(0)
• When the current is DC (constant), di/dt = 0, and v = 0; acts as a short circuit to direct current.

Inductors in Series

• The inductance is the amount of voltage dropped across an inductor for an amount of current change through it.
• When inductors are sharing current, then the voltage change will be additive.
• Effective series inductance: Ltotal = L1 + L2 + ... + Ln.

Current Division Rule

• Used to distribute current in parallel branches.
• With Two Resistance R1 and R2 in a parallel:
• The current flowing thru R1 is I/R1 and through R2 is I2
• Formulas Include: R1(R2/ R1+R2) I
• Where: -Ι1 = − (R2/ R1+R2 ),and Ι2 = − (R1/ R1+R2) - V is the total Voltage - Is is the Current at branch 1 -I2 is the Current at branch 2
• When three resistance are in parallel
• With Three Resistance R1,R2, and R3 in a parallel -Formulas include: = ((R2 * R3 )(R1R2 + R2R3 + R3R1 ))I

Voltage drop

• When voltage runs through a resistor, there results a voltage drop
• The magnitude of the drop results in the product of resistance
• V2 = V1-V2 where V2 is the Voltage Drop

Voltage Division Rule

• Given ‘n’ resistances connected in series across a voltage source (V), the equivalent resistance
• 𝑇 is the sum of all individual resistances.
• Formula: Vx= V(Rx÷ RT) where Vx is any resistance