Formula Sheet: DC Circuits

Table of Contents
1. Ohm's Law and Power
2. Resistor Networks
3. Kirchhoff's Laws
4. Circuit Analysis Methods
5. Network Theorems
6. Delta-Wye (Δ-Y) Transformations
7. Capacitors in DC Circuits
8. Inductors in DC Circuits
9. DC Steady-State Conditions
10. Resistivity and Conductivity
11. Voltage and Current Sources
12. Wheatstone Bridge
13. Node Voltage and Mesh Current Matrices
14. Millman's Theorem
15. Voltage and Current Divider Rules
16. Dependent Sources
17. Circuit Simplification Techniques
18. Average and RMS Values in DC
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Ohm's Law and Power

Ohm's Law

• Basic form: \[V = I \times R\]
  • V = voltage (volts, V)
  • I = current (amperes, A)
  • R = resistance (ohms, Ω)
• Alternative forms: \[I = \frac{V}{R}\] \[R = \frac{V}{I}\]

Power Relationships

• General power formula: \[P = V \times I\]
  • P = power (watts, W)
  • V = voltage (volts, V)
  • I = current (amperes, A)
• Power using resistance: \[P = I^2 \times R\] \[P = \frac{V^2}{R}\]
• Energy: \[W = P \times t\]
  • W = energy (joules, J or watt-hours, Wh)
  • t = time (seconds, s or hours, h)

Resistor Networks

Series Resistors

• Total resistance: \[R_{total} = R_1 + R_2 + R_3 + ... + R_n\]
• Current: Same current flows through all resistors: \[I_{total} = I_1 = I_2 = I_3 = ... = I_n\]
• Voltage division: \[V_k = V_{total} \times \frac{R_k}{R_{total}}\]
  • V_k = voltage across resistor k
  • R_k = resistance of resistor k
• Total voltage: \[V_{total} = V_1 + V_2 + V_3 + ... + V_n\]

Parallel Resistors

• Total resistance (general): \[\frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n}\]
• Two resistors in parallel: \[R_{total} = \frac{R_1 \times R_2}{R_1 + R_2}\]
• Equal resistors in parallel: \[R_{total} = \frac{R}{n}\]
  • n = number of equal resistors
• Voltage: Same voltage appears across all resistors: \[V_{total} = V_1 = V_2 = V_3 = ... = V_n\]
• Current division: \[I_k = I_{total} \times \frac{G_k}{G_{total}}\] where \[G = \frac{1}{R}\] (conductance in siemens, S)
• Current division (two resistors): \[I_1 = I_{total} \times \frac{R_2}{R_1 + R_2}\] \[I_2 = I_{total} \times \frac{R_1}{R_1 + R_2}\]
• Total current: \[I_{total} = I_1 + I_2 + I_3 + ... + I_n\]

Conductance

• Conductance definition: \[G = \frac{1}{R}\]
  • G = conductance (siemens, S or mhos, ℧)
• Parallel conductances: \[G_{total} = G_1 + G_2 + G_3 + ... + G_n\]

Kirchhoff's Laws

Kirchhoff's Current Law (KCL)

• Node equation: \[\sum I_{in} = \sum I_{out}\]
  or equivalently: \[\sum_{k=1}^{n} I_k = 0\]
  • The algebraic sum of all currents entering and leaving a node equals zero
  • Convention: currents entering are positive, currents leaving are negative (or vice versa, but be consistent)

Kirchhoff's Voltage Law (KVL)

• Loop equation: \[\sum_{k=1}^{n} V_k = 0\]
  • The algebraic sum of all voltages around any closed loop equals zero
  • Convention: voltage rises are positive, voltage drops are negative (or vice versa, but be consistent)

Circuit Analysis Methods

Nodal Analysis

• Node voltage method: Apply KCL at each node (except reference node)
• Number of equations: \(n - 1\) equations for \(n\) nodes
• Current in terms of node voltages: \[I = \frac{V_a - V_b}{R}\]
  • Current flows from node \(a\) to node \(b\) through resistance \(R\)
• For supernodes: Treat voltage source and connected nodes as a single supernode

Mesh Analysis

• Mesh current method: Apply KVL around each mesh
• Number of equations: \(m\) equations for \(m\) meshes
• Voltage drop across resistor: \[V = I_1 \times R - I_2 \times R\]
  • When two mesh currents flow through the same resistor
• For supermeshes: Treat current source and adjacent meshes as a single supermesh

Superposition Theorem

• Principle: In a linear circuit with multiple sources, the response (voltage or current) equals the sum of responses from each source acting independently
• Procedure:
  • Deactivate all sources except one (voltage sources → short circuit, current sources → open circuit)
  • Calculate response due to active source
  • Repeat for each source
  • Sum all individual responses algebraically
• Total response: \[V_{total} = V_1 + V_2 + ... + V_n\] \[I_{total} = I_1 + I_2 + ... + I_n\]

Network Theorems

Thévenin's Theorem

• Thévenin voltage (open-circuit voltage): \[V_{TH} = V_{OC}\]
  • Voltage across load terminals when load is removed (open circuit)
• Thévenin resistance: \[R_{TH} = \frac{V_{OC}}{I_{SC}}\]
  • V_OC = open-circuit voltage
  • I_SC = short-circuit current
• Alternative for R_TH: Deactivate all independent sources and calculate equivalent resistance looking into terminals
• Load current: \[I_L = \frac{V_{TH}}{R_{TH} + R_L}\]
• Load voltage: \[V_L = V_{TH} \times \frac{R_L}{R_{TH} + R_L}\]

Norton's Theorem

• Norton current (short-circuit current): \[I_N = I_{SC}\]
  • Current through load terminals when shorted
• Norton resistance: \[R_N = R_{TH} = \frac{V_{OC}}{I_{SC}}\]
• Load current: \[I_L = I_N \times \frac{R_N}{R_N + R_L}\]
• Load voltage: \[V_L = I_N \times \frac{R_N \times R_L}{R_N + R_L}\]

Thévenin-Norton Conversion

• Relationship: \[V_{TH} = I_N \times R_N\] \[I_N = \frac{V_{TH}}{R_{TH}}\] \[R_{TH} = R_N\]

Maximum Power Transfer Theorem

• Condition for maximum power transfer: \[R_L = R_{TH}\]
  • Load resistance must equal Thévenin (or Norton) resistance
• Maximum power delivered to load: \[P_{max} = \frac{V_{TH}^2}{4R_{TH}}\]
• Efficiency at maximum power transfer: \[\eta = 50\%\]

Source Transformation

• Voltage source to current source:
  Voltage source \(V_S\) in series with \(R_S\) → Current source \(I_S\) in parallel with \(R_S\) \[I_S = \frac{V_S}{R_S}\]
• Current source to voltage source:
  Current source \(I_S\) in parallel with \(R_S\) → Voltage source \(V_S\) in series with \(R_S\) \[V_S = I_S \times R_S\]
• Series resistance: Remains the same in both representations

Delta-Wye (Δ-Y) Transformations

Delta to Wye Conversion

• R₁ (connected between nodes a and n): \[R_1 = \frac{R_b \times R_c}{R_a + R_b + R_c}\]
• R₂ (connected between nodes b and n): \[R_2 = \frac{R_c \times R_a}{R_a + R_b + R_c}\]
• R₃ (connected between nodes c and n): \[R_3 = \frac{R_a \times R_b}{R_a + R_b + R_c}\]
• where:
  • R_a = delta resistance opposite to node a
  • R_b = delta resistance opposite to node b
  • R_c = delta resistance opposite to node c
• For equal resistances (R_Δ): \[R_Y = \frac{R_{\Delta}}{3}\]

Wye to Delta Conversion

• R_a (connected between nodes b and c): \[R_a = \frac{R_1 \times R_2 + R_2 \times R_3 + R_3 \times R_1}{R_1}\]
• R_b (connected between nodes a and c): \[R_b = \frac{R_1 \times R_2 + R_2 \times R_3 + R_3 \times R_1}{R_2}\]
• R_c (connected between nodes a and b): \[R_c = \frac{R_1 \times R_2 + R_2 \times R_3 + R_3 \times R_1}{R_3}\]
• For equal resistances (R_Y): \[R_{\Delta} = 3 \times R_Y\]

Capacitors in DC Circuits

Capacitance Fundamentals

• Capacitance definition: \[C = \frac{Q}{V}\]
  • C = capacitance (farads, F)
  • Q = charge (coulombs, C)
  • V = voltage (volts, V)
• Current-voltage relationship: \[i = C \frac{dv}{dt}\]
• Voltage in terms of current: \[v(t) = \frac{1}{C} \int_{0}^{t} i(\tau) d\tau + v(0)\]
  • v(0) = initial voltage across capacitor
• Energy stored: \[W = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV\]

Capacitors in Series

• Total capacitance: \[\frac{1}{C_{total}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + ... + \frac{1}{C_n}\]
• Two capacitors in series: \[C_{total} = \frac{C_1 \times C_2}{C_1 + C_2}\]
• Equal capacitors in series: \[C_{total} = \frac{C}{n}\]
• Voltage division: \[V_k = V_{total} \times \frac{C_{total}}{C_k}\]
  • Note: inverse relationship compared to resistors
• Charge: Same charge on all capacitors in series: \[Q_{total} = Q_1 = Q_2 = ... = Q_n\]

Capacitors in Parallel

• Total capacitance: \[C_{total} = C_1 + C_2 + C_3 + ... + C_n\]
• Voltage: Same voltage across all capacitors: \[V_{total} = V_1 = V_2 = ... = V_n\]
• Charge division: \[Q_k = Q_{total} \times \frac{C_k}{C_{total}}\]
• Total charge: \[Q_{total} = Q_1 + Q_2 + ... + Q_n\]

RC Circuit Transient Response

• Time constant: \[\tau = R \times C\]
  • τ = time constant (seconds, s)
• Charging capacitor (voltage): \[v_C(t) = V_f + (V_i - V_f)e^{-t/\tau}\]
  • V_f = final voltage (steady state)
  • V_i = initial voltage at t = 0
• Simplified charging (V_i = 0): \[v_C(t) = V_f(1 - e^{-t/\tau})\]
• Discharging capacitor: \[v_C(t) = V_i e^{-t/\tau}\]
• Current during charging: \[i_C(t) = \frac{V_f - V_i}{R}e^{-t/\tau}\]
• Current during discharging: \[i_C(t) = -\frac{V_i}{R}e^{-t/\tau}\]
• Percentage of final value:
  • At t = τ: 63.2% charged
  • At t = 2τ: 86.5% charged
  • At t = 3τ: 95.0% charged
  • At t = 4τ: 98.2% charged
  • At t = 5τ: 99.3% charged (considered fully charged)

Inductors in DC Circuits

Inductance Fundamentals

• Inductance definition: \[L = \frac{\lambda}{i}\]
  • L = inductance (henries, H)
  • λ = flux linkage (weber-turns, Wb)
  • i = current (amperes, A)
• Voltage-current relationship: \[v = L \frac{di}{dt}\]
• Current in terms of voltage: \[i(t) = \frac{1}{L} \int_{0}^{t} v(\tau) d\tau + i(0)\]
  • i(0) = initial current through inductor
• Energy stored: \[W = \frac{1}{2}LI^2\]

Inductors in Series

• Total inductance (no mutual coupling): \[L_{total} = L_1 + L_2 + L_3 + ... + L_n\]
• Current: Same current flows through all inductors: \[I_{total} = I_1 = I_2 = ... = I_n\]
• Voltage division: \[V_k = V_{total} \times \frac{L_k}{L_{total}}\]

Inductors in Parallel

• Total inductance (no mutual coupling): \[\frac{1}{L_{total}} = \frac{1}{L_1} + \frac{1}{L_2} + \frac{1}{L_3} + ... + \frac{1}{L_n}\]
• Two inductors in parallel: \[L_{total} = \frac{L_1 \times L_2}{L_1 + L_2}\]
• Equal inductors in parallel: \[L_{total} = \frac{L}{n}\]
• Voltage: Same voltage across all inductors: \[V_{total} = V_1 = V_2 = ... = V_n\]

RL Circuit Transient Response

• Time constant: \[\tau = \frac{L}{R}\]
  • τ = time constant (seconds, s)
• Increasing current (energizing): \[i_L(t) = I_f + (I_i - I_f)e^{-t/\tau}\]
  • I_f = final current (steady state)
  • I_i = initial current at t = 0
• Simplified energizing (I_i = 0): \[i_L(t) = I_f(1 - e^{-t/\tau})\]
• Decreasing current (de-energizing): \[i_L(t) = I_i e^{-t/\tau}\]
• Voltage across inductor during energizing: \[v_L(t) = (V_s - I_i R)e^{-t/\tau}\]
  • V_s = source voltage
• Voltage across inductor during de-energizing: \[v_L(t) = -I_i R e^{-t/\tau}\]
• Percentage of final value:
  • At t = τ: 63.2% of final current
  • At t = 2τ: 86.5% of final current
  • At t = 3τ: 95.0% of final current
  • At t = 4τ: 98.2% of final current
  • At t = 5τ: 99.3% of final current (considered steady state)

DC Steady-State Conditions

Capacitor at Steady State

• Current: \[i_C = 0\]
  • Acts as open circuit in DC steady state
• Voltage: Constant (no change with time)

Inductor at Steady State

• Voltage: \[v_L = 0\]
  • Acts as short circuit in DC steady state
• Current: Constant (no change with time)

Resistivity and Conductivity

Material Properties

• Resistance of conductor: \[R = \rho \frac{l}{A}\]
  • ρ = resistivity (ohm-meters, Ω·m)
  • l = length (meters, m)
  • A = cross-sectional area (square meters, m²)
• Conductivity: \[\sigma = \frac{1}{\rho}\]
  • σ = conductivity (siemens per meter, S/m)
• Conductance: \[G = \sigma \frac{A}{l}\]
• Temperature coefficient of resistance: \[R_T = R_0[1 + \alpha(T - T_0)]\]
  • R_T = resistance at temperature T
  • R₀ = resistance at reference temperature T₀
  • α = temperature coefficient (per °C or per K)

Voltage and Current Sources

Ideal Sources

• Ideal voltage source:
  • Maintains constant voltage regardless of load current
  • Internal resistance = 0 Ω
• Ideal current source:
  • Maintains constant current regardless of load voltage
  • Internal resistance = ∞ Ω

Practical Sources

• Practical voltage source:
  Ideal voltage source \(V_s\) in series with internal resistance \(R_s\)
  Terminal voltage: \[V_t = V_s - I \times R_s\]
• Practical current source:
  Ideal current source \(I_s\) in parallel with internal resistance \(R_s\)
  Terminal current: \[I_t = I_s - \frac{V_t}{R_s}\]

Wheatstone Bridge

Bridge Configuration

• Balanced condition: \[\frac{R_1}{R_2} = \frac{R_3}{R_4}\]
  or equivalently: \[R_1 \times R_4 = R_2 \times R_3\]
  • When balanced, voltage across bridge detector = 0
  • No current flows through detector
• Unknown resistance: \[R_x = R_3 \times \frac{R_2}{R_1}\]
  • If \(R_x\) = \(R_4\), and bridge is balanced

Node Voltage and Mesh Current Matrices

Matrix Formulation for Nodal Analysis

• General form: \[[G][V] = [I]\]
  • [G] = conductance matrix (siemens, S)
  • [V] = node voltage vector (volts, V)
  • [I] = current source vector (amperes, A)
• Diagonal elements: Sum of all conductances connected to node k
• Off-diagonal elements: Negative sum of conductances between nodes

Matrix Formulation for Mesh Analysis

• General form: \[[R][I] = [V]\]
  • [R] = resistance matrix (ohms, Ω)
  • [I] = mesh current vector (amperes, A)
  • [V] = voltage source vector (volts, V)
• Diagonal elements: Sum of all resistances in mesh k
• Off-diagonal elements: Negative sum of resistances shared between meshes

Millman's Theorem

Parallel Voltage Sources

• Common voltage (for sources with series resistances): \[V = \frac{\frac{V_1}{R_1} + \frac{V_2}{R_2} + ... + \frac{V_n}{R_n}}{\frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}}\]
  or using conductances: \[V = \frac{V_1 G_1 + V_2 G_2 + ... + V_n G_n}{G_1 + G_2 + ... + G_n}\]
• Equivalent resistance: \[\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}\]

Voltage and Current Divider Rules

Voltage Divider (General)

• Output voltage: \[V_{out} = V_{in} \times \frac{R_{out}}{R_{total}}\]
  • Valid for series resistors only

Current Divider (General)

• Branch current: \[I_{branch} = I_{total} \times \frac{R_{other}}{R_{branch} + R_{other}}\]
  • Valid for two parallel resistors
• Using conductances: \[I_k = I_{total} \times \frac{G_k}{G_{total}}\]

Dependent Sources

Types of Dependent Sources

• Voltage-Controlled Voltage Source (VCVS):
  \(V_{out} = \mu \times V_{in}\)
  • μ = voltage gain (dimensionless)
• Current-Controlled Current Source (CCCS):
  \(I_{out} = \beta \times I_{in}\)
  • β = current gain (dimensionless)
• Voltage-Controlled Current Source (VCCS):
  \(I_{out} = g_m \times V_{in}\)
  • g_m = transconductance (siemens, S)
• Current-Controlled Voltage Source (CCVS):
  \(V_{out} = r_m \times I_{in}\)
  • r_m = transresistance (ohms, Ω)

Circuit Simplification Techniques

Series-Parallel Reduction

• Procedure:
  • Identify series and parallel combinations
  • Replace with equivalent resistance
  • Repeat until single equivalent resistance remains
• Check for validity: Series requires same current; parallel requires same voltage

Combining Identical Sources

• Identical voltage sources in series: \[V_{total} = n \times V\]
  • n = number of sources
  • All sources must have same polarity
• Identical current sources in parallel: \[I_{total} = n \times I\]
  • All sources must have same direction

Average and RMS Values in DC

Average Value

• Average (DC) value: \[V_{avg} = \frac{1}{T} \int_{0}^{T} v(t) dt\]
  • T = period (for periodic signals)
  • For pure DC: \(V_{avg}\) = DC value

RMS Value

• Root Mean Square (effective) value: \[V_{rms} = \sqrt{\frac{1}{T} \int_{0}^{T} v^2(t) dt}\]
• For DC: \[V_{rms} = V_{DC}\]
• Power using RMS: \[P = V_{rms} \times I_{rms}\]