Linear Circuit Analysis - Basic Concepts

Introduction to Electric Circuits and Analysis

• Electric Circuit Definition: An electric circuit is an interconnection of circuit elements connected together to achieve a specific goal (such as transferring energy or processing signals).

• Terminal Relationships: Circuit elements possess a prescribed relationship between current and voltage at their terminals (for example, Ohm's law for resistors).

• Interconnection Wires: Circuit elements are joined together via ideal conductive wires:

  • All points along an ideal wire are at the exact same electrical potential.
  • Charge is conserved across wires: all current entering one end of a wire exits at the other end.

• Circuit Analysis vs. Circuit Synthesis:

  • Circuit Analysis: The process of determining specific voltages and currents at various points within a circuit when the individual circuit elements and their exact interconnections are known.
  • Circuit Synthesis: The process of designing and choosing a specific set of circuit elements and devising their interconnections to achieve desired voltage and current characteristics.

Circuit Topology: Branches, Nodes, and Essential Nodes

• Branch Definition: A branch represents a single two-terminal circuit element within an electrical network.
  • Segments of simple connection wire are not counted as separate elements or branches.
  • Examples of two-terminal elements that form branches include independent voltage sources, independent current sources, resistors, inductors, and capacitors.

• Branch Quantities: Every branch $k$ in a circuit has an associated branch current $i_k$ and branch voltage $v_k$. These branch variables must be explicitly labeled with reference polarities ($+, -$) for voltages and reference directions (arrows) for currents.

• Example: Counting Branches in a Network:

  • In a circuit comprising a $10\, \text{V}$ voltage source, a $2\, \text{k}\Omega$ resistor, an $8\, \text{k}\Omega$ resistor, a $6\, \text{k}\Omega$ resistor, a $5\, \text{mA}$ current source, a second $2\, \text{k}\Omega$ resistor, and a $5\, \text{k}\Omega$ resistor:
  • Total number of branches = $7$ (since there are $7$ individual two-terminal elements connected together).

• Node Definition: A node is a point of connection or junction where the leads (terminals) of two or more circuit elements meet.

  • A node is not restricted to a single geometric point; it can extend across a whole continuous segment of wire connecting terminals.
  • All element leads connected to a given node are at the exact same potential, known as the node potential.

• Essential Node Definition: An essential node is a specific point of connection where three or more circuit branches come together.

• Example: Counting Nodes vs. Essential Nodes:
  • For the circuit shown above containing $7$ branches:
  • Simple Nodes (Total Nodes): There are $5$ distinct nodes in total across the network.
  • Essential Nodes: There are $3$ essential nodes where three or more element branches connect.

• Example: Node Counting in a Complex Resistor Network:
  • In a ladder-like network with a source $V_s$ and resistors $R_1$ through $R_8$:
  • Node 1: Junction between $V_s(+)$ and $R_1$.
  • Node 2: Continuous wire connecting $R_1$, $R_2$, $R_3$, and $R_4$.
  • Node 3: Bottom common wire connecting $V_s(-)$, $R_2$, $R_3$, and $R_6$
  • Node 4: Junction connecting $R_4$, $R_5$, and $R_7$
  • Node 5: Junction connecting $R_6$, $R_5$, and $R_8$
  • Node 6: Junction connecting $R_7$ and $R_8$
  • Total Nodes = $6$

Node Voltages and Datum Selection

• Absolute Potential vs. Potential Difference: In electrical circuits, physical phenomena depend strictly on potential differences (voltage measured between two points), rather than absolute potential values.

• Reference Node (Datum Node):

  • To define absolute-type node voltages, one node in the circuit is designated as the reference node or datum node (also termed ground, assigned a potential of $0\, \text{V}$).
  • The potential of all other nodes in the network is measured with respect to this datum node.
  • Selection Guidelines: While any node can serve as the reference node, it is most convenient to select:
  1. The bottom common wire node of the circuit schematic.
  2. The node that has the largest number of branch connections.

• Distinction Between Node Voltages and Branch Voltages:

  • Node Voltage: The potential of a specific node relative to the chosen reference node. It is a single scalar value that applies across the entire node and all terminals connected to it.
  • Example: If node A is at $+5\, \text{V}$ relative to ground, every element terminal attached to node A is at $+5\, \text{V}$.
  • Branch Voltage: The voltage appearing directly across a single two-terminal branch element, representing the difference between the node voltages at its two ends.
  • Formula: For a branch connected between node A and node B, the branch voltage $v_{AB}$ is:       $v_{AB} = v_A - v_B$
  • Example: If one end of a resistor is attached to node A ($v_A = +5\, \text{V}$) and the other end is attached to node B ($v_B = +2\, \text{V}$), the branch voltage is $v_{AB} = 5\, \text{V} - 2\, \text{V} = 3\, \text{V}$.

• Example: Determining Node Voltages for Different Datum Choices

• A circuit contains 5 branches ($X_1$ through $X_5$) and 4 nodes (A, B, C, D). Voltmeter measurements give branch voltages as:
  • Across $X_1$: $v_{AD} = 1\, \text{V}$ ($+$ at A, $-$- at D)
  • Across $X_2$: $v_{BA} = 4\, \text{V}$ ($-$- at A, $+$ at B) $\implies v_{AB} = -4\, \text{V}$
  • Across $X_3$: $v_{BD} = 5\, \text{V}$ ($+$ at B, $-$- at D)
  • Across $X_4$: $v_{BC} = 3\, \text{V}$ ($+$ at B, $-$- at C)
  • Across $X_5$: $v_{CD} = 2\, \text{V}$ ($-$- at D, $+$ at C)

• Case (a): Datum Node is Node D ($v_D = 0\, \text{V}$):
  • $v_D = 0\, \text{V}$
  • $v_A = v_D + v_{AD} = 0 + 1 = 1\, \text{V}$
  • $v_B = v_D + v_{BD} = 0 + 5 = 5\, \text{V}$
  • $v_C = v_D + v_{CD} = 0 + 2 = 2\, \text{V}$

• Case (b): Datum Node is Node C ($v_C = 0\, \text{V}$):
  • $v_C = 0\, \text{V}$
  • $v_D = v_C - 2 = -2\, \text{V}$
  • $v_B = v_C + 3 = 3\, \text{V}$
  • $v_A = v_D + 1 = -2 + 1 = -1\, \text{V}$
• Case (c): Datum Node chosen as Node A ($v_A = 0\, \text{V}$):
  • $v_A = 0\, \text{V}$, $v_D = -1\, \text{V}$, $v_B = 4\, \text{V}$, $v_C = 1\, \text{V}$
• Case (d): Datum Node chosen as Node B ($v_B = 0\, \text{V}$):
  • $v_B = 0\, \text{V}$, $v_A = -4\, \text{V}$, $v_D = -5\, \text{V}$, $v_C = -3\, \text{V}$

Loops and Meshes

• Loop Definition: A loop is any closed path through a circuit formed by tracing through branches such that no node is encountered or traversed more than once.

• Mesh Definition: A mesh is a specific type of loop that contains no other loops within its interior boundary (also called an elementary loop or window pane).

• Example: Counting Loops and Meshes:
  • For the 3-window circuit above containing a $10\, \text{V}$ source, $2\, \text{k}\Omega$, $8\, \text{k}\Omega$, $6\, \text{k}\Omega$, $5\, \text{mA}$, $2\, \text{k}\Omega$, and $5\, \text{k}\Omega$ elements:
  • Number of Meshes: $3$ (the three individual adjacent internal windows).
  • Number of Loops: $6$ total closed paths:
    • $3$ single-mesh loops.
    • $2$ double-mesh combined loops.
    • $1$ triple-mesh outer loop enclosing the entire circuit boundary.

Fundamental Circuit Laws Overview

• Circuit analysis relies on two categories of fundamental governing physical laws:

  1. Element Laws (e.g., Ohm's Law): Define the intrinsic voltage-current terminal characteristics of specific individual elements regardless of how they are connected in a circuit.
  2. Connection Laws (Kirchhoff's Laws): Dictate the relationships between currents and voltages shared across interconnections, regardless of what types of elements make up those branches.

• Physical Foundations:

  • Kirchhoff's Current Law (KCL) stems directly from the principle of Conservation of Charge.
  • Kirchhoff's Voltage Law (KVL) stems directly from the principle of Conservation of Energy.

Kirchhoff's Current Law (KCL)

• Statement of KCL: At any instant in time, the algebraic sum of all currents entering a node must equal the sum of all currents leaving that node.   $\sum i_{\text{entering}} = \sum i_{\text{leaving}}$   Alternatively, the algebraic sum of all currents at any node is zero:   $\sum_{k=1}^{n} i_k = 0$

• Applying KCL across Circuit Nodes:
  • Node A: $i_1 = i_2$
  • Node B: $i_2 = i_3 + i_4$
  • Node C: $i_4 = i_5 + i_6$
  • Node D: $i_3 + i_5 + i_6 = i_1$

• Example: Calculating Unknown Branch Currents:
  • Given KCL at node B: $i_2 = i_3 + i_4 \implies i_4 = i_2 - i_3$
  • Condition (a): At $t_1$, $i_2 = 5\, \text{A}$ and $i_3 = 2\, \text{A}$.     $i_4 = 5 - 2 = 3\, \text{A}$     Since $i_4$ is positive, current flows in the assigned reference direction (to the right):     $i_{X_4} = 3\, \text{A}\ (\rightarrow)$
  • Condition (b): At $t_2$, $i_2 = 6\, \text{A}$ and $i_3 = 7\, \text{A}$.     $i_4 = 6 - 7 = -1\, \text{A}$     Since the result is negative, current actually flows opposite to the reference direction (to the left):     $i_{X_4} = 1\, \text{A}\ (\leftarrow)$

Double-Subscript Notation for Currents and Voltages

• Current Notation: Reference directions for current variables can be defined using double subscripts corresponding to labeled element terminals $a$ and $b$:
  • $i_{ab}$ denotes current flowing in the direction from terminal $a$ to terminal $b$
  • $i_{ba}$ denotes current flowing in the direction from terminal $b$ to terminal $a$
  • Relationship: $i_{ba} = -i_{ab}$

• Voltage Notation: Reference polarities for voltage variables use double subscripts as well:
  • $v_{ab}$ represents the potential at terminal $a$ relative to terminal $b$ (positive reference polarity at $a$, negative at $b$).
  • $v_{ba}$ represents the potential at terminal $b$ relative to terminal $a$ (positive reference polarity at $b$, negative at $a$).
  • Relationship: $v_{ba} = -v_{ab}$

Kirchhoff's Voltage Law (KVL)

• Statement of KVL: At any instant in time, the sum of all voltage rises along any closed loop must equal the sum of all voltage drops around that loop.   $\sum v_{\text{rises}} = \sum v_{\text{drops}}$   Alternatively, the algebraic sum of all branch voltages around any closed path (loop) in an electrical circuit equals zero:   $\sum_{k=1}^{m} v_k = 0$

• Formulating KVL Equations for Circuit Loops:
  • Loop $X_1 X_2 X_3$: $v_1 + v_2 = v_3$
  • Loop $X_3 X_4 X_5$: $v_3 = v_4 + v_5$
  • Loop $X_5 X_6$: $v_5 = v_6$
  • Loop $X_1 X_2 X_4 X_5$: $v_1 + v_2 = v_4 + v_5$
  • Loop $X_1 X_2 X_4 X_6$: $v_1 + v_2 = v_4 + v_6$
  • Loop $X_3 X_4 X_6$: $v_3 = v_4 + v_6$

• Sign Conventions for Traversing Loops:
  • When traversing a branch along a loop in a chosen direction:
  • Moving from $+$ to $-$- terminal across an element is a potential drop: add $+v_a$ to the KVL equation.
  • Moving from $-$- to $+$ terminal across an element is a potential rise: subtract $-v_a$ from the KVL equation.

• KVL Loop Analysis Example:
  • Loop 1 (Clockwise): $-v_a + v_b + v_c = 0 \implies v_a = v_b + v_c$
  • Loop 2 (Clockwise): $-v_c - v_d + v_e = 0 \implies v_e = v_c + v_d$
  • Loop 3 (Outer Loop Clockwise): $-v_a + v_b - v_d + v_e = 0$

Conservation of Power

• Principle of Power Conservation: In any electrical circuit, the total power absorbed by all elements at any instant must equal the total power delivered/supplied by all sources.   $\sum P_{\text{absorbed}} = \sum P_{\text{supplied}}$   Alternatively, the algebraic sum of power for all branches in a circuit equals zero:   $\sum_{k=1}^{b} P_k = 0$

Independent and Dependent Sources

• Independent Voltage Sources: Maintains a prescribed voltage $v = v_S$ across its terminals regardless of the current passing through it.

• Constant/DC Voltage Source: Provides a constant fixed terminal voltage (e.g., $12\, \text{V}$).
• AC Voltage Source: Provides a time-varying sinusoidal terminal voltage (e.g., $5 \cos(2\pi t)\, \text{V}$).

• Independent Current Sources: Maintains a prescribed current $i = i_S$ through its terminals regardless of the voltage appearing across it.

• DC Current Source: Supplies a constant DC current (e.g., $2\, \text{A}$).
• AC Current Source: Supplies a time-varying AC current (e.g., $3 \sin(100\pi t)\, \text{A}$).

• Dependent (Controlled) Sources: A source whose output value (voltage or current) is controlled by a voltage or current existing elsewhere in the circuit network. Controlled sources are represented visually by diamond-shaped symbols.

• Four Classifications of Dependent Sources:
  1. Voltage-Controlled Voltage Source (VCVS):
  • Output voltage: $v = k_v v_X$
  • Control parameter $k_v$ is dimensionless ($\text{V/V}$).
  1. Current-Controlled Current Source (CCCS):
  • Output current: $i = k_i i_X$
  • Control parameter $k_i$ is dimensionless ($\text{A/A}$).
  1. Current-Controlled Voltage Source (CCVS):
  • Output voltage: $v = k_r i_X$
  • Control parameter $k_r$ has dimensions of Resistance ($\text{V/A} = \Omega$).
  1. Voltage-Controlled Current Source (VCCS):
  • Output current: $i = k_g v_X$
  • Control parameter $k_g$ has dimensions of Conductance ($\text{A/V} = \text{S}$ or $\Omega^{-1}$).

Interconnection Rules for Sources

• Voltage Source Interconnections:

  • Voltage sources can be connected in series to yield an equivalent total voltage:     $v_S = v_{S1} + v_{S2}$
  • Forbidden Connection: Independent voltage sources of different values must never be connected in parallel, as this directly violates Kirchhoff's Voltage Law (KVL).

• Current Source Interconnections:

  • Current sources can be connected in parallel to yield an equivalent total current:     $i_S = i_{S1} + i_{S2}$
  • Forbidden Connection: Independent current sources of different values must never be connected in series, as this directly violates Kirchhoff's Current Law (KCL).

• Parallel Current Source Examples:
  • Two parallel $5\, \text{A}$ current sources pointing in the same direction yield an equivalent current:     $i = 5\, \text{A} + 5\, \text{A} = 10\, \text{A}$
  • Two parallel current sources of $5\, \text{A}$ (up) and $3\, \text{A}$ (down) yield an equivalent current:     $i = 5\, \text{A} - 3\, \text{A} = 2\, \text{A}$

• Series Voltage Source Examples:
  • Series Aiding Voltages (Addition): A $10\, \text{V}$ source in series aiding a $5\, \text{V}$ source yields:     $v_{AB} = 10\, \text{V} + 5\, \text{V} = 15\, \text{V}$
  • Series Opposing Voltages (Subtraction): A $10\, \text{V}$ source opposing a $5\, \text{V}$ source yields:     $v_{AB} = 10\, \text{V} + (-5\, \text{V}) = 5\, \text{V}$

Worked Examples and Comprehensive Exercises

• Exercise 1.7: Finding Unknown Currents via KCL

• Configuration (a): Single node with $1\, \text{A}$ entering from left, $3\, \text{A}$ entering from right, and $i_a$ leaving downwards.     $1 + 3 = i_a \implies i_a = 4\, \text{A}$

• Configuration (b): Two interconnected nodes forming a subnetwork:

  • Currents entering: $3\, \text{A}$ (left), $1\, \text{A}$ (top-left), $i_b$ (bottom-right).
  • Current leaving: $2\, \text{A}$ (right).     $3 + 1 + i_b = 2 \implies 4 + i_b = 2 \implies i_b = -2\, \text{A}$

• Configuration (c): Interconnected nodes with entering currents $1\, \text{A}$ (top-left), $3\, \text{A}$ (top-right), $i_c$ (bottom-left), $4\, \text{A}$ (bottom-right):     $1 + 3 + i_c + 4 = 0 \implies 8 + i_c = 0 \implies i_c = -8\, \text{A}$

  • Exercise 1.9: Determining Voltages $v_c$ and $v_e$ via KVL

• A multi-loop network has branch voltages given as: Element A = $3\, \text{V}$ ($+$ top), Element B = $5\, \text{V}$ ($-$- left, $+$ right), Element C = $v_c$ ($+$ top), Element D = $-10\, \text{V}$ ($-$- left, $+$ right), Element E = $v_e$ ($+$ top).

• Applying KVL to Left Loop (Clockwise):$-3 - 5 + v_c = 0 \implies v_c = 8\, \text{V}$

• Applying KVL to Middle Loop (Clockwise):$-v_c - (-10) + v_e = 0 \implies -8 + 10 + v_e = 0 \implies 2 + v_e = 0 \implies v_e = -2\, \text{V}$

  • Example: Circuit Analysis with Independent Sources

• Find $V_2$, $V_6$, and $V_I$ in a single-loop circuit containing a $10\, \text{V}$ voltage source, a $2\, \text{k}\Omega$ resistor (labeled $V_2$ with $-$- left, $+$ right), a $6\, \text{k}\Omega$ resistor (labeled $V_6$ with $-$- left, $+$ right), and a $5\, \text{mA}$ independent current source (pointing up, labeled $V_I$ with $+$ top, $-$- bottom).

• Current Analysis: Since it is a single continuous loop, the current everywhere in the loop is fixed by the $5\, \text{mA}$ source: $I = 5\, \text{mA}$ flowing clockwise.

• Calculating Resistor Voltages:

  • Current $5\, \text{mA}$ enters the $2\, \text{k}\Omega$ resistor from the left, producing a left-to-right voltage drop of:       $v_{\text{drop}} = 5\, \text{mA} \times 2\, \text{k}\Omega = 10\, \text{V}$
  • Since $V_2$ is defined as $-$- on left and $+$ on right ($V_2 = v_{\text{right}} - v_{\text{left}}$):       $V_2 = -10\, \text{V}$
  • Current $5\, \text{mA}$ enters the $6\, \text{k}\Omega$ resistor from the left, producing a left-to-right voltage drop of:       $v_{\text{drop}} = 5\, \text{mA} \times 6\, \text{k}\Omega = 30\, \text{V}$
  • Since $V_6$ is defined as $-$- on left and $+$ on right ($V_6 = v_{\text{right}} - v_{\text{left}}$):       $V_6 = -30\, \text{V}$

• Calculating Voltage $V_I$ across Current Source via KVL:

  • Traversing clockwise starting at the bottom-left corner:       $-10 + (-V_2) + (-V_6) + V_I = 0$$-10 + 10 + 30 + V_I = 0 \implies 30 + V_I = 0 \implies V_I = -30\, \text{V}$

  • Problem 1.29: Branch Current Calculations using KCL

• In the ladder grid network shown, KCL is applied systematically to every node to determine all unknown branch current magnitudes and directions based on given branch currents ($2\, \text{A}$, $5\, \text{A}$, $10\, \text{A}$, $13\, \text{A}$).

  • Problem 1.33: Branch Voltage Calculations using KVL

• Using the given branch voltages ($+4\, \text{V}$, $+2\, \text{V}$, $3\, \text{V}$), KVL equations are written around fundamental loops to determine the magnitude and polarity of every unlabelled branch voltage.

  • Problem 1.37: Circuit Analysis and Power Balance Verification

• Part (a): KCL and KVL are applied simultaneously to solve for all unknown branch voltages and currents across the bridge network.
• Part (b): Conservation of power is verified by calculating $P_k = v_k i_k$ for each branch $k$ and verifying that:     $\sum P_{\text{absorbed}} = \sum P_{\text{supplied}} \implies \sum P = 0$