EE-201: Network Analysis and Synthesis – Units 1
& 2 Notes
Unit 1: Introduction
Signals: A signal conveys information by variation of a physical quantity. A continuous-time signal is
defined for all real time instants (e.g. analog voltage) 1 , whereas a discrete-time signal exists only at
discrete instants (e.g. sampled data) 1 . Signals can be deterministic (completely specified by a formula,
no uncertainty) or random (stochastic, modeled probabilistically) 2 . Key properties include even signals
satisfying x(t) = x(−t) and odd signals satisfying x(t) = −x(−t) 3 4 . A signal is periodic if it
repeats every period T , i.e. x(t) = x(t + T )
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; otherwise it is aperiodic. Energy and power
∞
characterize signal magnitude: an energy signal has finite energy E = ∫−∞ ∣x(t)∣2 dt , while a power
T
1
∫−T ∣x(t)∣2 dt
signal has finite average power P = limT →∞ 2T
6
. (A signal cannot be both energy-
and power-type simultaneously.)
Periodic signals can be expanded in harmonics. Any real periodic waveform x(t) with period T can be
synthesized as a Fourier series:
∞
x(t) = ∑ Cn ejnω0 t ,
ω0 = 2π
T ,
n=−∞
T
where the complex coefficients are Cn = T1 ∫0 x(t)e−jnω0 t dt . This representation expresses x(t) as a
sum of complex sinusoids (harmonics) with frequencies nω0
7
8
. For example, a square wave’s
Fourier series consists of odd harmonics (Fig. shown above). The Fourier coefficients encode how much
of each sinusoidal component is present, enabling synthesis and analysis of periodic signals 7 8 .
For aperiodic signals, the analogous transform is the Continuous-Time Fourier Transform (CTFT). Any
finite-energy continuous signal f (t) can be represented by
F (ω) = ∫
∞
f (t) e−jωt dt,
f (t) =
−∞
where F (ω) is the spectrum of f (t)
9
∞
1
∫ F (ω) ejωt dω,
2π −∞
. In effect, the FT extends the Fourier series to a continuum of
frequencies for nonperiodic signals.
LTI Systems: A Linear Time-Invariant (LTI) system satisfies superposition (linearity) and time-shift
invariance 10 . Linearity means a x1 (t) + b x2 (t) produces a y1 (t) + b y2 (t) if y1 , y2 are outputs for
inputs x1 , x2
11 .
Time invariance means shifting the input in time shifts the output by the same
amount. An LTI system is fully characterized by its impulse response h(t) : for any input x(t) , the output
is the convolution y(t) = x(t) ∗ h(t)
12 .
Because convolution is commutative, associative and
distributive, these properties carry over to LTI systems 13 14 . Key LTI system properties include: Causality: A causal LTI has h(t) = 0 for t < 0 , so output depends only on present/past input 15 .
- Stability: An LTI is BIBO-stable if its impulse response is absolutely integrable (∫ ∣h(t)∣dt < ∞ ) 16 ;
then bounded inputs yield bounded outputs.
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- Memory: A memoryless LTI has output y(t) = k x(t) and impulse response h(t) = kδ(t) (no
memory); otherwise the system has memory (dynamic) 17 .
LTI systems enjoy convenient analysis via eigenfunctions: complex exponentials est pass through
unchanged except for a scale factor H(s) . In particular, a sinusoidal input ejωt yields output
H(jω)ejωt , where H(jω) is the frequency response. Because of this, representing signals via Fourier
series or transform greatly simplifies LTI analysis.
Modeling by Differential Equations: Many circuits and systems are governed by linear ODEs. For
example, a series RLC circuit obeys Kirchhoff’s voltage law VL + VR + VC = E(t) . Substituting VR =
RI , VL = L dI/dt , VC = C1 q (where I = dq/dt ) yields
L
dI
1
+ R I + q = E(t).
dt
C
2
1
dE
Differentiating again and using I = dq/dt gives a second-order ODE L ddtI2 + R dI
dt + C I = dt
18 .
Such equations, along with initial conditions (e.g. initial current and capacitor voltage), define the
system’s response to arbitrary inputs. Solving these ODEs (homogeneous + particular) yields the circuit’s
transient and steady-state behavior.
Transient Responses (R, L, C circuits): Under inputs such as impulses, steps, ramps or sinusoids, RLC
circuits exhibit characteristic transient responses. First-order circuits (one energy storage element) have
simple exponentials. For example, a series RL circuit with a DC step E at t = 0 has current
I(t) =
E
(1 − e−t/τ ),
R
with I(0) = 0 , approaching E/R as t → ∞
τ=
L
,
R
19 . Similarly, a series RC circuit with step
E has
capacitor voltage
VC (t) = E(1 − e−t/τ ),
starting from VC (0) = 0 and asymptotically reaching E
τ = R C,
20 . In both cases the time constant
τ
determines the exponential rate of change.
More generally, different input waveforms produce different forced responses: an impulse input
(mathematically a δ-function) produces a system’s impulse response (derivative of step response); a
ramp input yields a rising exponential plus particular (often leading to a quadratic term in RLC); and a
sinusoidal input results in a steady-state oscillation (plus transient). For second-order circuits, the
homogeneous solution yields under-, critically-, or over-damped exponentials depending on R/2L vs.
1/ LC
21
22 . (For brevity we omit full derivations, but the method is to solve the ODE for each input
type.) In all cases, the total response is the sum of the natural (homogeneous) and forced (particular)
solutions, matched to initial conditions.
T
Key Formulas: Fourier series: x(t) = ∑ Cn ejnω0 t with Cn = T1 ∫0 x(t)e−jnω0 t dt
∞
∞
1
∫−∞ F(ω)ejωt dω
transform: F(ω) = ∫−∞ x(t)e−jωt dt , x(t) = 2π
E
−tL/R
) 19 . RC step response: VC (t) = E(1 − e−t/RC )
R (1 − e
∞
∫−∞ x(τ )h(t − τ ) dτ (commutative/distributive) 13 .
2
9
7
8
. CT Fourier
. RL step response: I(t) =
20 .
LTI convolution:
y(t) =
Unit 2: Network Topology and Graph Theory
Network Graphs: In network topology, an electric circuit is represented by a graph of nodes and
branches. A node is a junction where two or more elements (branches) meet 23 . Each branch in the
graph represents a single two-terminal element (resistor, source, etc.) connecting two nodes 23 . To
construct a graph, ideal voltage sources are replaced by shorts and ideal current sources by opens (to
reflect their internal impedances), then elements are drawn as lines between node points 24 . An
oriented (directed) graph assigns a reference direction (arrow) to each branch; otherwise it is unoriented
25 . Only the connectivity matters in topology, not the element values.
A spanning tree (or simply tree) of the graph is a set of branches connecting all nodes without forming
any closed loop 26 . A tree on n nodes has exactly n − 1 branches (called twigs) 27 . Any branch not
included in a chosen tree is called a link (or chord) 28 . The set of all non-tree branches is the cotree, the
complement of the tree. By construction, adding any single link to the tree will form exactly one loop,
and removing any twig will partition the network into two isolated parts.
Fundamental Loops (Tie-Sets): A fundamental loop (also called a tie-set or independent loop) is a
cycle formed by adding one link to the spanning tree 29 . Each link defines one fundamental loop that
contains that link plus the unique tree branches closing the path back to itself 29 . Thus there are L =
b − (n − 1) fundamental loops if b is total branches and n nodes. In matrix form, the fundamental loop
matrix B has one row per loop and one column per branch (size (b − n + 1) × b ). Its entries are +1 or
–1 for branches in the loop (depending on orientation) and 0 otherwise 29
30 . Tie-set matrices relate
branch currents to independent loop currents. For analysis, one typically assigns an arbitrary current
around each fundamental loop; branch currents are then linear combinations of these loop currents.
Fundamental Cut-Sets: A cut-set is a set of branches whose removal disconnects the graph into two
parts. A fundamental cut-set (f-cutset) is formed by choosing one twig of the tree: removing that twig
(and the necessary links to sever connectivity) partitions the network 31 . Each twig defines one
fundamental cut-set which includes that twig and some links; there are exactly n − 1 fundamental cutsets (one per twig). The fundamental cut-set matrix C has one row per cut-set and one column per
branch (size (n − 1) × b ). Matrix entries are +1 or –1 for branches in the cut-set (matching chosen
orientations) and 0 for others 32 33 . Cut-set equations express Kirchhoff’s node-voltage laws: the sum
of voltages in each cut-set is zero in steady state.
Incidence Matrix: The incidence matrix A of the graph (size n × b ) encodes node-branch connectivity
34 . Each row corresponds to a node, each column to a branch. The entry
Aij = +1 if branch j leaves
node i , −1 if it enters node i , and 0 if branch j is not connected to node i
34 . By Kirchhoff’s current
law, each column of A has one +1 and one –1 (a branch connects two nodes). The incidence matrix can
be used to write node equations and plays a central role in matrix methods (e.g. writing A i = 0
enforces KCL, where i is the vector of branch currents).
Tree, Loops, and Cut-Sets in Analysis: In network analysis, choosing a tree effectively reduces
unknowns. Kirchhoff’s voltage law is applied to each fundamental loop, yielding loop equations in terms
of loop currents (tie-set method). Kirchhoff’s current law is applied to each fundamental cut-set, yielding
node voltage relations (cut-set or node method). Loop matrices and cut-set matrices provide systematic
equations: for loops, B v = 0 (sum of branch voltages around each loop is zero); for cuts, C i = 0 (sum
of branch currents in each cut is zero). Together with element constitutive relations (Ohm’s law, etc.),
these yield solvable linear equations for circuit analysis.
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Key Definitions: A tree is a subgraph connecting all n nodes with no loops 26 . A twig is a branch of the
tree; a link (chord) is a branch outside the tree 35 28 . A fundamental loop is a loop containing exactly
one link 29 . A fundamental cut-set is the minimal set of branches (one twig + links) whose removal
disconnects the network 31 . In summary, network topology uses graph theory (trees, loops, cut-sets,
incidence) to organize Kirchhoff’s laws into a systematic framework for circuit analysis 23 29 .
Important Formulas: Incidence matrix A (n × b ): one row per node, one column per branch; entries
+1/–1/0 as above 34 . Fundamental loop (tie-set) matrix B ((b − n + 1) × b ): one row per loop, branch
entries ±1/0 29
30 . Fundamental cut-set matrix
C ((n − 1) × b ): one row per twig, entries ±1/0 31
33 . These matrices greatly facilitate writing the nodal/loop equations of any linear network in compact
form.
References: Concepts and formulas are drawn from standard signals-and-systems and networkanalysis texts 7 9 10 16 18 19 20 23 26 29 31 , ensuring definitions and derivations are
precise and up-to-date.
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Signals Classification
https://www.tutorialspoint.com/signals_and_systems/signals_classification.htm
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8.2: Continuous Time Fourier Transform (CTFT) - Engineering LibreTexts
https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Signal_Processing_and_Modeling/
Signals_and_Systems_(Baraniuk_et_al.)/08%3A_Continuous_Time_Fourier_Transform_(CTFT)/
8.02%3A_Continuous_Time_Fourier_Transform_(CTFT)
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Signals and Systems – Properties of Linear Time-Invariant (LTI) Systems
https://www.tutorialspoint.com/signals-and-systems-properties-of-linear-time-invariant-lti-systems
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Linear time-invariant system - Wikipedia
https://en.wikipedia.org/wiki/Linear_time-invariant_system
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3.9 Application: RLC Electrical Circuits – Differential Equations
https://ecampusontario.pressbooks.pub/diffeq/chapter/3-9-application-rlc-electrical-circuits/
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9.5: Transient Response of RL Circuits - Engineering LibreTexts
https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/DC_Electrical_Circuit_Analysis__A_Practical_Approach_(Fiore)/09%3A_Inductors/9.5%3A_Transient_Response_of_RL_Circuits
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8.4: Transient Response of RC Circuits - Engineering LibreTexts
https://eng.libretexts.org/Bookshelves/Electrical_Engineering/Electronics/DC_Electrical_Circuit_Analysis__A_Practical_Approach_(Fiore)/08%3A_Capacitors/8.4%3A_Transient_Response_of_RC_Circuits
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Transient Response of RLC Circuit
https://www.eeeguide.com/transient-response-of-rlc-circuit/
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27
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Network Topology | Types of Graph | Tree | Loop Properties
https://www.eeeguide.com/network-topology/
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Network Theory Topology Matrices
https://www.tutorialspoint.com/network_theory/network_theory_topology_matrices.htm
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