AEE Telecom and Electronics Core · Chapter 1

Network Analysis, Filters, Laplace, Fourier and z Transforms

What to remember

Circuits are solved with KCL (sum of currents at a node is zero) and KVL (sum of voltages around a loop is zero). Thevenin, Norton and superposition simplify linear networks. Maximum power transfers when load resistance equals source (Thevenin) resistance.
Time constant of RC is τ = RC and of RL is τ = L/R. Series RLC resonance is at ω0 = 1/√(LC), with Q = ω0L/R and bandwidth = ω0/Q. Filters are classified as low pass, high pass, band pass and band stop.
Laplace transforms handle continuous-time circuits and transfer functions (stable if poles are in the left half s-plane). Fourier transforms give frequency content. The z-transform handles discrete-time signals (stable causal system has poles inside the unit circle).

1. Basic network theorems

Ohm's law: V = I R. KCL and KVL follow from conservation of charge and energy.
Mesh analysis uses loop currents and KVL. Nodal analysis uses node voltages and KCL.
Superposition (linear networks only): the response is the sum of the responses to each independent source acting alone; other voltage sources are shorted and current sources are opened. It cannot be applied to power directly.
Thevenin's theorem: a linear network behaves like a voltage source Vth in series with a resistance Rth. Vth is the open-circuit voltage; Rth is found with sources set to zero (or Vth divided by short-circuit current).
Norton's theorem: a current source IN in parallel with Rth. IN = Vth / Rth.
Maximum power transfer: RL = Rth (for AC, load impedance = complex conjugate of source impedance). Maximum power = Vth² / (4 Rth). Efficiency at that point is only 50%.
Reciprocity: in a linear bilateral network, interchanging source and response position leaves the ratio unchanged. Millman's theorem combines parallel voltage sources. Tellegen's theorem: the sum of power in all branches is zero. Compensation and substitution theorems also exist.

Worked example. A 12 V source with 3 Ω in series feeds a 6 Ω shunt resistor. Open-circuit voltage Vth = 12 × 6 / 9 = 8 V. Rth = 3 || 6 = 2 Ω. A load of 2 Ω receives the maximum power = 8² / (4 × 2) = 8 W.

2. Transients, resonance and AC circuits

Impedances: resistor R, inductor jωL, capacitor 1/(jωC).
RC circuit time constant τ = R C. RL circuit τ = L / R. A charging capacitor reaches 63.2% of its final voltage after τ. Example: R = 10 kΩ and C = 10 μF give τ = 0.1 s.
Series RLC: damping factor α = R / 2L, natural frequency ω0 = 1/√(LC). If α > ω0: overdamped; α = ω0: critically damped; α < ω0: underdamped (oscillatory).
Series resonance: ω0 = 1/√(LC); impedance minimum (equals R); current maximum. Quality factor Q = ω0 L / R = (1/R) √(L/C). Bandwidth BW = ω0 / Q = R / L (rad/s). Half-power frequencies lie at the edges of the band.
Parallel resonance: impedance maximum; Q = R √(C/L) for a parallel RLC.
Example: L = 1 H, C = 1 μF: ω0 = 1000 rad/s. With R = 10 Ω, Q = 100 and BW = 10 rad/s.
AC power: complex power S = V I* = P + jQ. Real power P in watts, reactive power Q in var, apparent power |S| in VA. Power factor = cos φ = P / |S|.
Two-port parameters:

Parameter set	Relation	Reciprocity condition	Symmetry condition
Z (open circuit)	V = Z I	Z12 = Z21	Z11 = Z22
Y (short circuit)	I = Y V	Y12 = Y21	Y11 = Y22
ABCD (transmission)	V1 = A V2 - B I2	AD - BC = 1	A = D
h (hybrid)	V1 = h11 I1 + h12 V2	h12 = -h21	h11 h22 - h12 h21 = 1

3. Filters

A filter passes some frequencies and blocks others. The cutoff frequency is where the gain falls to 1/√2 of the maximum (-3 dB, half power).

Low pass (LPF) passes frequencies below fc. High pass (HPF) passes above fc. Band pass (BPF) passes a band. Band stop (notch, BSF) blocks a band.
First-order RC low pass: H(s) = 1 / (1 + sRC); fc = 1 / (2π R C); ωc = 1/(RC). With R = 1 kΩ and C = 1 μF, ωc = 1000 rad/s. Roll-off is 20 dB per decade (6 dB per octave) per order.
Passive filters use R, L and C only. Active filters use op-amps and need a power supply; they provide gain and avoid bulky inductors.
Approximations:

Type	Feature
Butterworth	Maximally flat passband, monotonic response
Chebyshev (type I)	Ripple in passband, sharper cutoff for same order
Bessel	Maximally linear phase (constant group delay)
Elliptic (Cauer)	Ripple in passband and stopband, sharpest transition

Constant-k filter: series arm Z1 and shunt arm Z2 satisfy Z1 Z2 = k², a constant. For a constant-k low pass, cutoff fc = 1 / (π √(LC)) and nominal impedance k = √(L/C). m-derived filters give a sharper cutoff and an infinite attenuation peak.
Filter order n gives a roll-off of 20n dB/decade.

4. Laplace transform

Definition: F(s) = ∫ f(t) e^(-st) dt from 0 to ∞ (unilateral). It converts differential equations into algebra.

f(t)	F(s)
δ(t)	1
u(t)	1/s
t	1/s²
e^(-at)	1/(s + a)
sin ωt	ω/(s² + ω²)
cos ωt	s/(s² + ω²)

Properties: linearity; time shift f(t - T) u(t - T) gives e^(-sT) F(s); frequency shift e^(-at) f(t) gives F(s + a); differentiation d/dt gives sF(s) - f(0); integration gives F(s)/s; convolution in time becomes multiplication in s.
Initial value theorem: f(0+) = lim (s → ∞) s F(s). Final value theorem: f(∞) = lim (s → 0) s F(s), valid only if all poles of sF(s) are in the left half-plane.
Examples: F(s) = 10/(s + 5) gives f(0+) = 10. F(s) = 5 / (s (s + 2)) gives f(∞) = 5/2 = 2.5.
Transfer function H(s) = Y(s) / X(s) with zero initial conditions; it is the Laplace transform of the impulse response. Poles are roots of the denominator; zeros are roots of the numerator. A causal system is stable if all poles are in the left half s-plane. Poles on the imaginary axis give marginal stability.
Region of convergence (ROC) is required to define the transform uniquely.

5. Fourier series and Fourier transform

Fourier series represents a periodic signal as a sum of sinusoids at harmonics of the fundamental frequency. Dirichlet conditions ensure convergence. An even function has only cosine terms (and a DC term). An odd function has only sine terms. Half-wave symmetric signals have only odd harmonics. A square wave has odd harmonics with amplitudes decreasing as 1/n.
Parseval's theorem: power (or energy) in the time domain equals that in the frequency domain.
Fourier transform: X(jω) = ∫ x(t) e^(-jωt) dt. Pairs: δ(t) ↔ 1; 1 ↔ 2π δ(ω); e^(-at) u(t) ↔ 1 / (a + jω); rectangular pulse ↔ sinc function; Gaussian ↔ Gaussian.
Properties: time shift gives a phase factor; time scaling x(at) gives (1/|a|) X(ω/a), so compression in time spreads the spectrum; duality; convolution in time is multiplication in frequency; modulation shifts the spectrum.
Sampling theorem: a band-limited signal with highest frequency fm can be recovered if the sampling rate fs ≥ 2 fm (Nyquist rate). Sampling below this rate causes aliasing.
The Laplace transform evaluated on s = jω gives the Fourier transform if the imaginary axis lies in the ROC.

6. z-transform and discrete systems

Definition: X(z) = Σ x[n] z^(-n). It is the discrete-time counterpart of the Laplace transform; z = e^(sT).

x[n]	X(z)	ROC
δ[n]	1	All z
u[n]	z/(z - 1)		z	> 1
aⁿ u[n]	z/(z - a)		z	>	a
n aⁿ u[n]	a z / (z - a)²		z	>	a

Properties: delay by k samples multiplies X(z) by z^(-k); convolution in time becomes multiplication in z.
Stability: a causal system is stable if all poles lie inside the unit circle |z| = 1 (ROC includes the unit circle). Poles on the circle give marginal stability.
FIR filters have a finite impulse response (no feedback, always stable, can have exactly linear phase). IIR filters have feedback and need fewer coefficients for a given sharpness but can be unstable.
Bilinear transform: s = (2/T)(1 - z^(-1)) / (1 + z^(-1)) maps the left half s-plane into the inside of the unit circle and avoids aliasing, but warps frequency. Impulse invariance can cause aliasing.
DFT and FFT: direct DFT of N points needs N² complex multiplications; radix-2 FFT needs (N/2) log₂N. For N = 8: 12 multiplications compared with 64.

Exam traps

Superposition cannot be used for power, and sources are turned off by shorting voltage sources and opening current sources.
Maximum power transfer gives only 50% efficiency.
Bandwidth of series RLC is R/L; Q rises when R falls.
Cutoff is the -3 dB point, where power is half, not where gain is zero.
Butterworth is flat passband; Chebyshev has ripple; Bessel has linear phase.
Final value theorem fails when poles are on the imaginary axis or in the right half-plane.
Stability: left half s-plane is the same as inside the unit circle in z.
Nyquist rate is twice the highest frequency, not equal to it.

One-liners

1. KCL: algebraic sum of currents at a node is zero.
2. KVL: algebraic sum of voltages in a loop is zero.
3. RC time constant is RC; RL time constant is L/R.
4. Series resonance frequency ω0 = 1/√(LC).
5. Q of series RLC = ω0 L / R.
6. A first-order filter rolls off at 20 dB per decade.
7. Laplace transform of a unit step is 1/s.
8. Laplace transform of δ(t) is 1.
9. Fourier transform of δ(t) is 1.
10. z-transform of aⁿ u[n] is z/(z - a).
11. Nyquist rate = 2 × highest signal frequency.
12. A causal stable discrete system has poles inside the unit circle.

Practice questions

At a node, currents of 5 A enter, and 2 A and another current I leave. The value of I is
1. 7 A
2. 10 A
3. 3 A
4. 2 A
Answer
C. 3 A
KCL: currents in = currents out, so I = 5 - 2 = 3 A.
Maximum power is delivered to a load when the load resistance equals
1. Twice the Thevenin resistance
2. Infinity
3. Zero
4. The Thevenin resistance of the source network
Answer
D. The Thevenin resistance of the source network
Maximum power transfer theorem: RL = Rth.
At maximum power transfer, the efficiency is
1. 100%
2. 50%
3. 75%
4. 25%
Answer
B. 50%
Half of the power is dissipated in the source resistance.
A Thevenin source has Vth = 10 V and Rth = 5 Ω. The Norton current is
1. 2 A
2. 0.5 A
3. 50 A
4. 15 A
Answer
A. 2 A
IN = Vth / Rth = 10/5 = 2 A.
A 12 V source with 3 Ω series resistance feeds a 6 Ω resistor across the terminals. The open-circuit voltage at the terminals across the 6 Ω resistor is
1. 4 V
2. 8 V
3. 6 V
4. 12 V
Answer
B. 8 V
Voltage divider: 12 × 6/(3 + 6) = 8 V.
A 12 V source with 3 Ω series resistance feeds a 6 Ω shunt resistor. The Thevenin resistance seen at the terminals across the 6 Ω resistor is
1. 2 Ω
2. 4.5 Ω
3. 3 Ω
4. 9 Ω
Answer
A. 2 Ω
3 Ω in parallel with 6 Ω gives 18/9 = 2 Ω.
With Vth = 8 V and Rth = 2 Ω, the maximum power to a matched load is
1. 4 W
2. 16 W
3. 32 W
4. 8 W
Answer
D. 8 W
P = Vth² / (4 Rth) = 64 / 8 = 8 W.
When applying superposition, a voltage source that is not acting is
1. Left in place
2. Replaced by an open circuit
3. Replaced by a short circuit
4. Replaced by a 1 Ω resistor
Answer
C. Replaced by a short circuit
Current sources not acting are opened.
Superposition theorem cannot be directly applied to the calculation of
1. Voltage
2. Current
3. Power
4. Both voltage and current in linear circuits
Answer
C. Power
Power is a non-linear (squared) quantity.
The time constant of a series RC circuit with R = 10 kΩ and C = 10 μF is
1. 0.1 s
2. 1 s
3. 100 s
4. 0.01 s
Answer
A. 0.1 s
τ = RC = 10⁴ × 10⁻⁵ = 0.1 s.
The time constant of an RL circuit is
1. L R²
2. L / R
3. R L
4. R / L
Answer
B. L / R
The current builds up to 63.2% in one time constant.
A charging capacitor reaches what fraction of its final voltage after one time constant?
1. 99%
2. 50%
3. 36.8%
4. 63.2%
Answer
D. 63.2%
v = V(1 - e⁻¹) = 0.632 V.
At steady state under DC, an ideal capacitor behaves as
1. A short circuit
2. A voltage source
3. An open circuit
4. A resistor
Answer
C. An open circuit
No current flows through a capacitor in DC steady state.
A series RLC circuit has L = 1 H and C = 1 μF. The resonant frequency is
1. 1000 rad/s
2. 1 rad/s
3. 100 rad/s
4. 10⁶ rad/s
Answer
A. 1000 rad/s
ω0 = 1/√(LC) = 1/√10⁻⁶ = 1000 rad/s.
With L = 1 H, C = 1 μF and R = 10 Ω in series, the quality factor is
1. 0.01
2. 10
3. 1000
4. 100
Answer
D. 100
Q = ω0 L / R = 1000 × 1 / 10 = 100.
A series RLC circuit has L = 1 H, C = 1 μF and R = 10 Ω. Its bandwidth in rad/s is
1. 100
2. 10
3. 0.1
4. 1000
Answer
B. 10
BW = ω0 / Q = 1000/100 = 10 (also R/L = 10).
A series RLC circuit has R = 4 Ω, L = 1 H and C = 1 F. The response is
1. Overdamped
2. Critically damped
3. Underdamped
4. Undamped
Answer
A. Overdamped
α = R/2L = 2 and ω0 = 1, so α > ω0.
A series RLC circuit has R = 2 Ω, L = 1 H and C = 1 F. The response is
1. Underdamped
2. Unstable
3. Overdamped
4. Critically damped
Answer
D. Critically damped
α = 1 and ω0 = 1, so α = ω0.
At series resonance the impedance of an RLC circuit is
1. Infinite
2. Maximum
3. Minimum, equal to R
4. Purely capacitive
Answer
C. Minimum, equal to R
X_L and X_C cancel, so only R remains and current is maximum.
A load draws P = 800 W and Q = 600 var. The power factor is
1. 0.75
2. 0.8
3. 1.0
4. 0.6
Answer
B. 0.8
|S| = √(800² + 600²) = 1000 VA; pf = 800/1000.
A two-port network has A = 2, B = 3, C = 1, D = 2 (ABCD parameters). The network is
1. Non-reciprocal, since AD - BC ≠ 1
2. Symmetric, since B = C
3. Active, since A = D
4. Reciprocal, since AD - BC = 1
Answer
D. Reciprocal, since AD - BC = 1
AD - BC = 4 - 3 = 1.
The cutoff frequency of a first-order RC low pass filter is
1. 2π R C
2. R C
3. 1 / (2π R C)
4. 1 / (R C²)
Answer
C. 1 / (2π R C)
At this frequency gain falls to 1/√2 (-3 dB).
An RC low pass filter has R = 1 kΩ and C = 1 μF. The cutoff in rad/s is
1. 1
2. 10⁶
3. 159
4. 1000
Answer
D. 1000
ωc = 1/(RC) = 1/(10³ × 10⁻⁶) = 1000.
At the cutoff frequency, the gain magnitude of a filter is
1. 0.707 of the maximum
2. 0.5 of the maximum
3. Zero
4. 1.414 of the maximum
Answer
A. 0.707 of the maximum
-3 dB corresponds to half power.
The roll-off of a second-order filter beyond cutoff is about
1. 10 dB per decade
2. 40 dB per decade
3. 60 dB per decade
4. 20 dB per decade
Answer
B. 40 dB per decade
Each order gives 20 dB per decade.
A filter with a maximally flat passband and no ripple is
1. Bessel
2. Butterworth
3. Chebyshev
4. Elliptic
Answer
B. Butterworth
Chebyshev has ripple; Bessel is optimised for linear phase.
Equiripple response in the passband with a sharper cutoff than Butterworth of the same order is a feature of
1. RC filter
2. Butterworth filter
3. Chebyshev filter
4. Bessel filter
Answer
C. Chebyshev filter
Type I Chebyshev trades passband ripple for steeper roll-off.
Which filter has the most linear phase response (constant group delay)?
1. Bessel
2. Chebyshev
3. Elliptic
4. Butterworth
Answer
A. Bessel
Bessel filters preserve pulse shape.
The cutoff frequency of a constant-k low pass filter (L, C) is
1. 1 / (2π √(LC))
2. 1 / (π √(LC))
3. √(LC)
4. 1 / (LC)
Answer
B. 1 / (π √(LC))
fc = 1/(π√(LC)) for T or π sections.
The Laplace transform of the unit step u(t) is
1. 1
2. 1/s²
3. 1/s
4. s
Answer
C. 1/s
∫ e^(-st) dt from 0 to ∞ = 1/s.
The Laplace transform of e^(-3t) is
1. 1 / (s + 3)
2. 3 / s
3. 1 / (s - 3)
4. s / (s + 3)
Answer
A. 1 / (s + 3)
L{e^(-at)} = 1/(s + a).
F(s) = 10 / (s + 5). The initial value f(0+) is
1. 10
2. 2
3. 0
4. 5
Answer
A. 10
f(0+) = lim s F(s) = lim 10 s/(s + 5) = 10.
F(s) = 5 / (s (s + 2)). The final value f(∞) is
1. 0
2. 5
3. 10
4. 2.5
Answer
D. 2.5
f(∞) = lim s F(s) = 5/2 = 2.5 (poles at 0 and -2 allow the theorem).
A system has poles at s = -2 and s = -3. It is
1. Conditionally stable
2. Stable
3. Unstable
4. Marginally stable
Answer
B. Stable
All poles are in the left half-plane.
Consider: 1. A causal LTI system is stable if all poles are in the left half s-plane. 2. Poles on the imaginary axis give an unstable system with exponential growth. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
Imaginary-axis poles give marginal stability (sustained oscillation), not exponential growth.
The Fourier series of an even periodic function contains
1. Only sine terms
2. Only cosine terms (and DC)
3. Only odd harmonics
4. Only DC
Answer
B. Only cosine terms (and DC)
Even symmetry removes the sine terms.
A half-wave symmetric periodic signal contains
1. Only even harmonics
2. Only odd harmonics
3. Only DC
4. Only the fundamental
Answer
B. Only odd harmonics
Half-wave symmetry makes even harmonics zero.
The Fourier transform of the impulse δ(t) is
1. 1
2. δ(ω)
3. 1/jω
4. 2π δ(ω)
Answer
A. 1
∫ δ(t) e^(-jωt) dt = 1.
A signal limited to 5 kHz must be sampled at not less than
1. 2.5 kHz
2. 20 kHz
3. 5 kHz
4. 10 kHz
Answer
D. 10 kHz
Nyquist rate = 2 fm = 10 kHz.
Compressing a signal in time (x(2t)) causes its spectrum to
1. Shrink in frequency
2. Remain unchanged
3. Spread out in frequency
4. Shift to higher frequency only
Answer
C. Spread out in frequency
Time scaling x(at) gives (1/|a|) X(ω/a).
The z-transform of aⁿ u[n] is
1. z / (z + a)
2. 1 / (z - a)
3. z / (z - a)
4. a z / (z - 1)
Answer
C. z / (z - a)
It is Σ (a/z)ⁿ = 1/(1 - a z⁻¹), ROC |z| > |a|.
A causal discrete system with a pole at z = 1.2 is
1. Non-causal
2. Stable
3. Marginally stable
4. Unstable
Answer
D. Unstable
The pole is outside the unit circle.
Number of complex multiplications in a 16-point radix-2 FFT is (N/2) log₂N =
1. 16
2. 64
3. 32
4. 256
Answer
C. 32
(16/2) × 4 = 32 (direct DFT would need 256).
Consider: 1. FIR filters can have exactly linear phase. 2. IIR filters are always stable. Which is/are correct?
1. 1 only
2. 2 only
3. Both 1 and 2
4. Neither 1 nor 2
Answer
A. 1 only
IIR filters have feedback and can be unstable, so 2 is wrong.
Match the filter with its main feature: 1 Butterworth, 2 Chebyshev, 3 Bessel, 4 Elliptic. P Linear phase, Q Flat passband, R Passband ripple with monotonic stopband, S Ripple in both bands.
1. 1-S, 2-P, 3-R, 4-Q
2. 1-R, 2-Q, 3-S, 4-P
3. 1-P, 2-S, 3-Q, 4-R
4. 1-Q, 2-R, 3-P, 4-S
Answer
D. 1-Q, 2-R, 3-P, 4-S
Butterworth is flat, Chebyshev I has passband ripple, Bessel has linear phase, Elliptic has ripple in both bands.

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