25-Nav-B6 Ocean Engineering and Offshore Structures · Undated paper
Question 7 of 8: First-Order RC Network — Step Response and Frequency Response
Nivaar worked solution (AI-drafted; not reviewed by a licensed engineer)
Notes on this paper
Paper format. PEO National Examinations,
May 2019 — 98-Mar-B6, printed for Electrical & Electronics
Engineering / Mechanical Engineering candidates. Three hours, closed
book, two approved calculators (Casio or Sharp). Eight questions of equal
value; any five constitute a complete paper, and only the first five appearing in
the answer book are marked. Constants supplied on the front page: $\pi = 3.14159$,
$1\ \text{hp} = 746\ \text{W}$. All eight questions are solved here so the solutions cover whichever five a candidate chooses.
Subject note
This paper is listed under 25-Nav-B6 “Ocean Engineering and Offshore Structures”, but the printed paper is headed 98-Mar-B6 (front page: 98-Elec-B6) and every question is Electrical & Electronics Engineering content (BJT current-mirror analysis, combinational logic, a linear dc machine, a gapped/parallel-path transformer magnetic circuit, a three-op-amp instrumentation amplifier, an induction-motor dc test and slip calculation, an RC transient/frequency-response network, and industrial power-factor correction) — zero naval-architecture or ocean-engineering content. Solved as the exam actually printed.
Mano & Ciletti, Digital Design, 6th ed. — Boolean algebra and
De Morgan’s theorems (Ch. 2), NAND/NOR universal gates (Ch. 3).
Fitzgerald, Kingsley & Umans, Electric Machinery, 7th ed. —
elementary electromechanical energy conversion / the linear dc machine (Ch. 3).
Chapman, Electric Machinery Fundamentals, 5th ed. — transformers
(Ch. 2), induction motors and the dc test (Ch. 6).
Sadiku & Alexander, Fundamentals of Electric Circuits, 7th ed.
— ac power and power-factor correction (Ch. 11), first-order transients
(Ch. 7), frequency response (Ch. 14).
Question 7: First-Order RC Network — Step Response and Frequency Response (equal value)
Given. In both configurations the network is a series resistor
$R$ feeding a shunt capacitor $C$, output taken across $C$, drawing no load
current. In (a) the source is a dc supply $V_s$ switched on at $t=0$ with
$v_o(0^-)=0$; in (c) the same network is driven by a sinusoidal source $v_i$ of
variable frequency and the steady-state response is required. No numeric $R,C$ are
given — the answer is a symbolic transfer function and its sketch.
The working writes the dc supply as $V_s$ ($\equiv V_I$ on the paper) and the output as $v_o$ ($\equiv V_O$).
Find. (a) $v_o(t)/V_s$; (b) its sketch over $5\tau$; (c)
$H(j\omega)=v_o/v_i$; (d) its magnitude sketch over 4 decades about the corner
frequency.
Figure 7 — (a) dc-switched RC low-pass, (b) the same
network driven by a variable-frequency ac source.
Approach. (a)/(b): solve the first-order charging ODE for the
capacitor voltage. (c)/(d): replace $C$ by its impedance $1/j\omega C$ and write
the voltage-divider transfer function.
(a) Time-domain step response. For $t\ge0$, KVL gives
$V_s=iR+v_o$ with $i=C\,dv_o/dt$, so
$$RC\frac{dv_o}{dt}+v_o=V_s,\qquad v_o(0)=0
\;\Longrightarrow\;\boxed{\frac{v_o(t)}{V_s}=1-e^{-t/\tau}},\quad \tau=RC.$$
(b) Sketch over $5\tau$. $v_o/V_s$ rises monotonically from 0,
reaching $1-e^{-1}=63.2\%$ at $t=\tau$, $86.5\%$ at $2\tau$, $95.0\%$ at $3\tau$,
$98.2\%$ at $4\tau$ and $99.3\%$ at $5\tau$ — visually indistinguishable from
fully charged.
(c) Frequency-domain transfer function. $C$ has impedance
$1/(j\omega C)$, and with no load current the network is an unloaded divider:
$$H(j\omega)=\frac{v_o}{v_i}=\frac{1/(j\omega C)}{R+1/(j\omega C)}
=\boxed{\frac{1}{1+j\omega RC}}=\frac{1}{1+j(\omega/\omega_c)},\quad
\omega_c=\frac{1}{RC}.$$
(d) Magnitude sketch, 4 decades about $f_c=\omega_c/2\pi$.
$|H|=1/\sqrt{1+(\omega/\omega_c)^2}$: flat at 0 dB for $f\ll f_c$, $-3$ dB
($1/\sqrt2$) exactly at $f_c$, then falling at $-20$ dB/decade
($-20$ dB two decades above $f_c$, $-40$ dB four decades above) — the
standard single-pole low-pass Bode magnitude.
