DC Circuit Analysis: KCL and Ohm's Law

DC Circuit Analysis

• Electrical sources include current and voltage sources.
• There are three types of loads: inductive (L), resistive (R), and capacitive (C).
• Sources can be current or voltage controlled.

Circuit Elements

• Includes voltage sources (Vs, Vs2) and loads (R1, R2, R3, R4).
• Has 2 sources and 4 Loads
• Complete circuit represented as ABCDEFA (loop).
• Meshes included are ABEFA and BCDEB.
• Example circuit consists of 6 elements.
• Example circuit includes nodes A, B, C, E, and F.
• Branches included are AB, BC, CD, BE, EF, and FA.

Open and Short Circuits

• In an open circuit, current (I) is 0, and resistance (R) is infinite.
• In a short circuit, resistance (R) is at a minimum, current (I) is at a maximum, and voltage (V) exists.

Basic Formulas

• V = IR (Ohm's Law)
• I = V/R
• I ∝ V (Current is proportional to Voltage)
• V ∝ I (Voltage is proportional to Current)

Kirchhoff's Current Law (KCL)

• Current entering a node/junction equals the current leaving it.

Kirchhoff's Voltage Law (KVL)

• The overall algebraic sum of all voltages in a closed circuit is 0.

Current Divider Law

• Formulas to calculate current division in a parallel circuit:
  • I1 = I * R2 / (R1 + R2)
  • I2 = I * R1 / (R1 + R2)
  • I = I1+I2

Source Conversion/Source Transformation

• A voltage source (Vs) in series with a resistor (R) can be converted to a current source (Is) in parallel with the same resistor, where Is = Vs/R.

Mesh Analysis

• Considers planar networks that can be drawn on a plane
• Directions of current assumed in clockwise direction for all meshes.
• Frame the network using KVL
• Regions can convert source to voltage if not
• ABEFA = I1 -> R1 & R3
• BCDEB = I2-> R2 & R3

Active and Passive Elements

• Active elements activate without external power sources.
• Passive elements require external sources to activate.

Mesh Analysis Steps

• Identify types of sources.
• Have U sources and will thus need sourace conversions
• Identify meshes & loops
  • 2 meshes: ABEFA, BCDEB
  • 1 loop: ABCDEFA
• Designate currents to meshes and mark the directions.
• Frame KVL equations.

Nodal Analysis

• Involves current sources.
• Identify all valid nodes in the circuit.
• Designate one node as the reference node (ground).
• Assign unknown voltages to the remaining nodes.
• Frame the KCL equations for every node to identify the unknown's.

Nodal Analysis Example

• B = Is1 = i1 + i2
• C = IS2 = i2 - i3
• i1 = (VB - VX)/R1
• i2 = (-VC + VB)/R2
• i3 = (VC - VX)/R3

Superposition Theorem

• Useful for linear DC networks containing more than one EMF source.
• The overall response equals the algebraic sum of responses due to each independent source acting alone. The other sources are set to zero.

Thévenin's Theorem

• Definition
  • Any linear two-terminal circuit can be replaced by a voltage source (VTH) in series with a resistor (RTH).

Steps for Applying Thévenin's Theorem

• Identify the load resistor across which to find the current.
• Mark the terminals of the load resistor as points A and B.
• Open circuit the terminals A and B.
• Use circuit analysis techniques (KVL, KCL, superposition, etc.) to find the Thevenin voltage (VTH) across the open-circuited terminals.
• Short circuit the terminals A and B.
• Find the current through the short circuit.
• Use the formula RTH = VTH / ISC to calculate the Thévenin resistance.
• Redraw the circuit with the load resistor connected to the Thévenin equivalent circuit (VTH and RTH).

AC Circuits

• Involves AC voltage and current continuously changing amplitude.
• Varies sinusoudal
• Definitions
  • Effective (RMS) value: The value of an AC quantity equivalent to a DC quantity in terms of power dissipation.
  • Instantaneous value: The value of an alternating quantity at a specific instant.
  • Maximum (peak) value: The maximum value attained by an alternating quantity in a cycle.
  • Average value (mean value): The average of an alternating quantity over a given interval.

AC Waveforms

• Sinusoidal
• Casinusoidal

AC Waveform

• The flux is determined where 'B' the coil Length for coil 'I' and the breath for coil 'b', Omega 'w' is angular velocity.
• A is the cross section area
• The formula can be calculated -W = 8 / t so 8 = wt -Linear of the coil is measured and the sin component at right angles to Magnetic field. -V1 = Blasiin8 -Vac =Vm sin8

Effective Root Mean Square Value

• Equivalent to the direct current that produces the same heat in the same resistor.

Average or Mean value

• the Sum of all the instantaneous values divided by the number of values, taken over the interval

Full Cycle Average of Sinusodial

• equal to zero since it is Symmetrical
• Half Cycle Average of sinusodial = (2/ π)*Im = 0.637Im

Root Mean Square of sinusodial quantity

Puley Resistive Cicuit

• there is no phase difference in circuit and same path

Series RL cicuit

• Vin = VR +VL

Serries RC

• Vin = VR+VL