Unit 01: DC Resistive Circuits
Fundamentals of Circuit Analysis (2026) · Chapters 1-4
Knowledge Structure Overview
DC Resistive Circuits
├─ 1. Basic Circuit Concepts and Laws
│ ├─ 1.1 Physical Circuits and Circuit Models
│ ├─ 1.2 Fundamental Circuit Quantities
│ ├─ 1.3 Kirchhoff's Laws
│ ├─ 1.4 Passive Elements
│ └─ 1.5 Active Elements
├─ 2. Equivalent Transformations
│ ├─ 2.1 Equivalence of Two-Terminal Networks
│ ├─ 2.2 Resistor Equivalent Transformations
│ ├─ 2.3 Source Equivalent Transformations
│ ├─ 2.4 Equivalent Resistance of Passive Networks
│ └─ 2.5 Bridge Circuits
├─ 3. Systematic Analysis Methods
│ ├─ 3.1 Branch Current Method
│ ├─ 3.2 Mesh Current and Loop Current Methods
│ └─ 3.3 Node Voltage Method
└─ 4. Circuit Theorems
├─ 4.1 Superposition Theorem
├─ 4.2 Substitution Theorem
├─ 4.3 Equivalent Source Theorem
└─ 4.4 Maximum Power Transfer Theorem
1.1 Physical Circuits and Circuit Models
1.1.1 Physical Circuits
- Definition: A current path formed by electrical equipment and devices connected in a specific manner
- Components: Source (provides electrical energy), intermediate links (transmit and distribute energy), load (consumes energy)
- Function: To achieve energy transmission and conversion, or signal transmission and processing
1.1.2 Circuit Models
- Definition: A mathematical model that uses ideal circuit elements to approximate the behaviour of a physical circuit
- Ideal Circuit Elements: Idealized models with precise mathematical relationships (resistor R, capacitor C, inductor L, voltage source, current source)
- Modeling Principle: Under certain conditions, use combinations of ideal elements to reflect the main electromagnetic properties of actual devices
- Lumped-Parameter Circuits: When circuit dimensions are much smaller than signal wavelength, lumped-parameter elements can be used
1.2 Fundamental Circuit Quantities
Current
Current
The amount of charge passing through a conductor's cross-section per unit time.
$i = dq/dt$
Unit: Ampere (A)
Voltage
Voltage
The work done by the electric field force when a unit positive charge moves from point a to point b.
$u_{ab} = dw/dq$
Unit: Volt (V)
Power
Power
The work done by the electric field force per unit time (rate of energy conversion).
$p = dw/dt = ui$
Unit: Watt (W)
Potential
Electric Potential
The voltage at a point relative to a reference point.
$V_a = u_{a0}$
Reference: Ground
Reference Direction
Reference Directions for Current and Voltage
- Current Reference Direction: An assumed direction of positive charge motion. Current is positive when the actual direction agrees with the reference direction, and negative when opposite
- Voltage Reference Direction: An assumed direction of voltage drop (positive terminal to negative terminal)
- Associated Reference Direction: Current flows in at the positive terminal and out at the negative terminal
- Power Sign: $p > 0$ means the element absorbs (consumes) power; $p < 0$ means the element delivers (supplies) power
- Power Balance: In any circuit, the sum of power absorbed by all elements equals zero (conservation of energy)
1.3 Kirchhoff's Laws
KVL — Kirchhoff's Voltage Law
The algebraic sum of voltages around any loop is zero
$\sum u = 0$
KCL — Kirchhoff's Current Law
The sum of currents entering any node equals the sum of currents leaving that node
$\sum i_{in} = \sum i_{out}$
1.3.1 Kirchhoff's Voltage Law (KVL)
- Statement: The algebraic sum of all branch voltages around any loop is zero, $\sum u = 0$
- Corollary: The algebraic sum of voltage drops equals the algebraic sum of voltage rises in a loop
KVL Writing Procedure
1
Choose the loop
traversal direction
traversal direction
2
Determine reference
directions for each branch voltage
directions for each branch voltage
3
Positive if consistent,
negative if opposite
negative if opposite
4
Write the equation
$\sum u = 0$
$\sum u = 0$
1.3.2 Kirchhoff's Current Law (KCL)
- Statement: At any instant, the sum of currents entering any node equals the sum of currents leaving that node, $\sum i_{in} = \sum i_{out}$
- Equivalent Form: $\sum i = 0$ (entering positive, leaving negative)
- Generalization: KCL applies to any closed surface (generalized node)
1.4 Passive Elements
Resistor
Resistor
A two-terminal element where voltage is proportional to current. A linear resistor has a V-I characteristic that is a straight line through the origin.
$u = Ri$
Unit: Ohm (Ω)
$p = i^2R = u^2/R$
Capacitor
Capacitor
A two-terminal element that stores electric field energy. Capacitor voltage cannot change instantaneously (memory property). Acts as an open circuit under DC steady-state conditions.
$i = C(du/dt)$
Unit: Farad (F)
$W_C = \frac{1}{2}Cu^2$
Inductor
Inductor
A two-terminal element that stores magnetic field energy. Inductor current cannot change instantaneously (memory property). Acts as a short circuit under DC steady-state conditions.
$u = L(di/dt)$
Unit: Henry (H)
$W_L = \frac{1}{2}Li^2$
1.5 Active Elements
1.5.1 Independent Sources
Ideal Voltage Source
Terminal voltage is constant, independent of external circuit. Internal resistance is zero.
V-I characteristic: a line parallel to the current axis
Ideal Current Source
Output current is constant, independent of external circuit. Internal resistance is infinite.
V-I characteristic: a line parallel to the voltage axis
1.5.2 Dependent Sources
Definition: Sources whose voltage or current is controlled by the voltage or current of another branch in the circuit.
VCVS
μ
Voltage Controlled Voltage Source
$u_2 = \mu u_1$
VCCS
$g_m$
Voltage Controlled Current Source
$i_2 = g_m u_1$
CCVS
$r_m$
Current Controlled Voltage Source
$u_2 = r_m i_1$
CCCS
β
Current Controlled Current Source
$i_2 = \beta i_1$
Note
Special Properties of Dependent Sources
- Dependent sources are not independent excitations and cannot act alone
- In the superposition theorem, they cannot be zeroed independently like independent sources
- Dependent sources must always remain in the circuit
2.1 Equivalence of Two-Terminal Networks
- Definition: Two two-terminal networks are equivalent if their port V-I relationships (VCR) are identical
- Purpose of Equivalent Transformations: To simplify circuit analysis and reduce the number of unknowns
- Note: Equivalence applies to the external circuit; internal structures may differ
2.2 Resistor Equivalent Transformations
Series
Series Equivalent of Resistors
$$R_{eq} = R_1 + R_2 + \cdots + R_n$$
- Characteristic: The current through each resistor is the same
- Voltage Divider Formula: $u_k = \dfrac{R_k}{R_1+R_2+\cdots+R_n} \cdot u$
Parallel
Parallel Equivalent of Resistors
$$G_{eq} = G_1 + G_2 + \cdots + G_n$$
- Characteristic: The voltage across each resistor is the same
- Two resistors in parallel: $R_{eq} = \dfrac{R_1 R_2}{R_1 + R_2}$ or $G_{eq} = G_1 + G_2 $
- Current Divider Formula: $i_k = \dfrac{G_k}{G_{eq}} \cdot i$
2.2.2 Resistor Y-Delta Equivalent Transformation
Δ→Y Transformation
Given three resistors in delta connection, find the equivalent star connection
$R_1 = \dfrac{R_{12}R_{31}}{R_{12}+R_{23}+R_{31}}$
$R_2 = \dfrac{R_{12}R_{23}}{R_{12}+R_{23}+R_{31}}$
$R_3 = \dfrac{R_{31}R_{23}}{R_{12}+R_{23}+R_{31}}$
$R_2 = \dfrac{R_{12}R_{23}}{R_{12}+R_{23}+R_{31}}$
$R_3 = \dfrac{R_{31}R_{23}}{R_{12}+R_{23}+R_{31}}$
Y→Δ Transformation
Given three resistors in star connection, find the equivalent delta connection
$R_{12} = \dfrac{R_1R_2+R_2R_3+R_3R_1}{R_3}$
$R_{23} = \dfrac{R_1R_2+R_2R_3+R_3R_1}{R_1}$
$R_{31} = \dfrac{R_1R_2+R_2R_3+R_3R_1}{R_2}$
$R_{23} = \dfrac{R_1R_2+R_2R_3+R_3R_1}{R_1}$
$R_{31} = \dfrac{R_1R_2+R_2R_3+R_3R_1}{R_2}$
Symmetric Case
When $R_{12}=R_{23}=R_{31}=R_\Delta$
$R_Y = R_\Delta/3$
2.3 Source Equivalent Transformations
2.3.1 Practical Source Models and Equivalent Transformations
Voltage Source Model
Ideal voltage source $U_s$ in series with internal resistance $R_0$
Current Source Model
Ideal current source $I_s$ in parallel with internal resistance $R_0$
Equivalent Transformation Relationship
Internal resistance $R_0$ remains unchanged; equivalent only for the external circuit
$U_s = I_s \cdot R_0 \qquad I_s = U_s / R_0$
Note
Equivalent Transformation Precautions
- Internal resistance $R_0$ remains unchanged during equivalent transformation
- The transformation is equivalent only for the external circuit, not internally
- Ideal voltage sources and ideal current sources cannot be directly transformed into each other
2.3.2 Series and Parallel Equivalents of Ideal Sources
Voltage Sources in Series
Σ
$U_s = U_{s1} + U_{s2}$
Add if same direction
Subtract if opposite
Voltage Sources in Parallel
=
Requires all voltage sources
to have equal voltage
Otherwise violates KVL
Current Sources in Series
=
Requires all current sources
to have equal current
Otherwise violates KCL
Current Sources in Parallel
Σ
$I_s = I_{s1} + I_{s2}$
Add if same direction
Subtract if opposite
2.4 Equivalent Resistance of Passive Two-Terminal Networks
Definition: The equivalent resistance at the port after setting all independent sources to zero (voltage sources short-circuited, current sources open-circuited).
Solution Methods
Method 1
Series-Parallel Simplification
Use resistor series-parallel formulas to simplify step by step.
Method 2
External Source Method
Apply voltage and find current, or apply current and find voltage, to calculate port equivalent resistance.
Method 3
Y-Delta Transformation
Transform Y or delta connections, then simplify.
Method 4
Symmetry Analysis
Use circuit symmetry to simplify calculations.
2.5 Bridge Circuits
Bridge
Bridge Circuits
Bridge Balance Condition: The product of opposite arm resistances are equal
$$R_1 R_3 = R_2 R_4$$
$$R_1 R_3 = R_2 R_4$$
- Balance Characteristics: No current flows in the bridge arm; it can be treated as open or short circuit
- Unbalanced bridge: Requires a Y–Δ transformation or another analysis method
3. Systematic Analysis Methods
Method 1
Branch Current Method
Use branch currents as unknowns, write KCL and KVL equations and solve simultaneously.
Equations = b
Intuitive but many equations
Method 2
Mesh Current Method
Assume circulating currents flowing around each mesh boundary as unknowns.
Equations = number of meshes
Only for planar circuits
Method 3
Node Voltage Method
Use node voltages relative to the reference node as unknowns.
Equations = n-1
Widely applicable
3.1 Branch Current Method
3.1.1 Two Types of Constraints and Circuit Equations
KCL Constraints
For a circuit with n nodes
Independent KCL equations = n-1
KVL Constraints
For a planar circuit with b branches and n nodes
Independent KVL equations = b-(n-1)
Element Constraints (VCR)
The voltage-current relationship for each branch
3.1.2 Writing Branch Current Equations
1
Assign reference
directions for each branch current
directions for each branch current
2
Write KCL equations
for n-1 nodes
for n-1 nodes
3
Write KVL equations
for independent loops
for independent loops
4
Express branch voltages
using branch currents (VCR)
using branch currents (VCR)
5
Solve simultaneously
b equations total
b equations total
3.1.3 Circuit Topology (Graph Theory Basics)
Graph Theory
Circuit Topology Fundamentals
- Graph: A collection of nodes and branches
- Connected Graph: A path exists between any two nodes
- Tree: A connected subgraph containing all nodes but no loops
- Tree Branch: A branch belonging to the tree; count = n-1
- Link: A branch not belonging to the tree; count = b-(n-1)
- Fundamental Loop: A loop formed by one link and several tree branches
3.2 Mesh Current and Loop Current Methods
3.2.1 Standard Form of Mesh Current Equations
For planar circuits, using mesh currents as unknowns. Standard form:
$$\begin{bmatrix}
R_{11} & R_{12} & \cdots & R_{1m} \\
R_{21} & R_{22} & \cdots & R_{2m} \\
\vdots & \vdots & \ddots & \vdots \\
R_{m1} & R_{m2} & \cdots & R_{mm}
\end{bmatrix}
\begin{bmatrix} i_{m1} \\ i_{m2} \\ \vdots \\ i_{mm} \end{bmatrix} =
\begin{bmatrix} u_{s11} \\ u_{s22} \\ \vdots \\ u_{smm} \end{bmatrix}$$
- Self-resistance $R_{ii}$: The sum of all resistances in mesh $i$; always positive
- Mutual resistance $R_{ij}$ $(i \neq j)$: The sum of resistances on the common branch of mesh $i$ and mesh $j$. Positive when the two mesh currents flow in the same direction on the common branch, negative when opposite (if all mesh currents are assigned clockwise, mutual resistance is always negative)
- $u_{sii}$: The algebraic sum of all voltage source rises in mesh $i$
3.2.2 Mesh Current Equations with Current Sources
Current Source on a Boundary Branch
That mesh current is known; write the equation directly
Current Source on a Common Branch
Introduce the voltage across the current source as an additional unknown; add one equation
3.2.3 Mesh Current Equations with Dependent Sources
1
Treat the dependent source
as an independent source first
as an independent source first
2
Express the controlling quantity
in terms of mesh currents
in terms of mesh currents
3
Substitute into
the equations and simplify
the equations and simplify
3.3 Node Voltage Method
3.3.1 Standard Form of Node Voltage Equations
Using node voltages (potential differences relative to the reference node) as unknowns. Standard form:
$$\begin{bmatrix}
G_{11} & G_{12} & \cdots & G_{1,n-1} \\
G_{21} & G_{22} & \cdots & G_{2,n-1} \\
\vdots & \vdots & \ddots & \vdots \\
G_{n-1,1} & G_{n-1,2} & \cdots & G_{n-1,n-1}
\end{bmatrix}
\begin{bmatrix} u_{n1} \\ u_{n2} \\ \vdots \\ u_{n,n-1} \end{bmatrix} =
\begin{bmatrix} i_{s11} \\ i_{s22} \\ \vdots \\ i_{s,n-1,n-1} \end{bmatrix}$$
- Self-conductance $G_{ii}$: The sum of all conductances connected to node $i$; always positive
- Mutual conductance $G_{ij}$ $(i \neq j)$: The sum of conductances directly connecting node $i$ and node $j$; always negative
- $i_{sii}$: The algebraic sum of all current source currents flowing into node $i$ (positive for inflow)
3.3.2 Node Voltage Equations with Ideal Voltage Source Branches
Voltage Source Between a Node and the Reference
That node voltage is known; one less equation to write
Voltage Source Between Two Non-Reference Nodes
Introduce the current through the voltage source as an additional unknown; add one equation
3.3.3 Node Voltage Equations with Dependent Sources
Approach: Similar to the mesh method, first treat the dependent source as an independent source, then express the controlling quantity in terms of node voltages and substitute.
4. Circuit Theorems
Theorem 1
Superposition Theorem
In a linear circuit, the current (or voltage) in any branch equals the algebraic sum of the contributions from each independent source acting alone.
Only for linear circuits
Theorem 2
Substitution Theorem
If the voltage u or current i of a branch is known, that branch can be replaced by an ideal voltage or current source.
Linear or nonlinear
Theorem 3
Equivalent Source Theorem
A source-containing linear two-terminal network can be replaced by a Thevenin or Norton equivalent circuit.
Thevenin + Norton
Theorem 4
Maximum Power Transfer
When $R_L = R_{eq}$, the load receives maximum power.
Efficiency 50%
4.1 Superposition Theorem
Statement: In a linear circuit, the current (or voltage) in any branch equals the algebraic sum of the currents (or voltages) produced in that branch by each independent source acting alone.
Application Steps
1
Retain only one
independent source at a time
independent source at a time
2
Set other independent sources to zero
(voltage sources shorted, current sources opened)
(voltage sources shorted, current sources opened)
3
Calculate the response
for each independent source separately
for each independent source separately
4
Algebraically add
all responses
all responses
Note
Superposition Theorem Limitations
- Applicable only to linear circuits
- Power cannot be directly calculated using the superposition theorem (power is a quadratic function of voltage or current)
- Dependent sources cannot be zeroed independently; they must always remain in the circuit
4.2 Substitution Theorem
Theorem
Substitution Theorem
- Statement: In a linear or nonlinear circuit, if the voltage u (or current i) of a branch is known, that branch can be replaced by an ideal voltage source with voltage equal to u (or an ideal current source with current equal to i), without changing the voltages and currents in the rest of the circuit
- Significance: Allows complex problems to be decomposed into simpler ones
4.3 Equivalent Source Theorem
4.3.1 Thevenin's Theorem
Thevenin
Thevenin's Theorem
$$\text{Source-containing linear two-terminal network} \equiv U_{oc} \text{ (open-circuit voltage) in series with } R_{eq}$$
Solution Steps:
1
Find open-circuit voltage $U_{oc}$
Disconnect the branch of interest
Disconnect the branch of interest
2
Find equivalent resistance $R_{eq}$
Set independent sources to zero
Set independent sources to zero
3
Draw the equivalent circuit
Connect the branch of interest
Connect the branch of interest
Methods for finding $R_{eq}$:
- Method 1: Direct calculation after setting independent sources to zero
- Method 2: $R_{eq} = U_{oc} / I_{sc}$ (open-circuit voltage divided by short-circuit current)
- Method 3: External source method
4.3.2 Norton's Theorem
Norton
Norton's Theorem
$$\text{Source-containing linear two-terminal network} \equiv I_{sc} \text{ (short-circuit current) in parallel with } R_{eq}$$
Relationship with Thevenin's Theorem:
$$U_{oc} = I_{sc} \cdot R_{eq}$$
Norton and Thevenin circuits can be equivalently transformed into each other
4.4 Maximum Power Transfer Theorem
Theorem
Maximum Power Transfer Theorem
Condition: When the load resistance $R_L$ equals the Thevenin equivalent resistance $R_{eq}$ of the source-containing two-terminal network, the load receives maximum power.
$$P_{max} = \frac{U_{oc}^2}{4R_{eq}}$$
$$P_{max} = \frac{U_{oc}^2}{4R_{eq}}$$
Matching condition: $R_L = R_{eq}$
Transfer efficiency: 50%