Lab 5

© N. B. Dodge 01/12

Lab 5 Worksheet Note: Lab 5 is generally the most challenging exercise in ENGR 2105. Please

watch the videos and read Lab 5 carefully at least twice and then take your time on

the exercises below to make sure that you understand the theoretical material.

1. In what quadrant of the complex plane are these numbers located?

−12+j7 __________________ −10−j50 __________________

8−j2 __________________ 1+j100 __________________

2. Use the complex conjugate to convert the expressions to a real term and an

imaginary term.

26 (6 − 𝑗4)⁄ __________________ (8 − 𝑗8) (2 + 𝑗2)⁄ __________________

3. Inductor and capacitor impedances are given as: 𝑍𝐿 = 𝑗𝜔𝐿 and 𝑍𝐶 =

1 𝑗𝜔𝐶⁄ . Assume you have a 10μF capacitor and a 10mH inductor. Calculate the impedance of these components at the following frequencies and list in

the space provided:

1 MHz (1,000,000 Hz): 𝑗𝜔𝐿 = _______________Ω 1 𝑗𝜔𝐶⁄ = _________ Ω

50KHz (50,000 Hz): 𝑗𝜔𝐿 = ______________ Ω 1 𝑗𝜔𝐶⁄ = ___________ Ω

0Hz: 𝑗𝜔𝐿 = _______________ Ω 1 𝑗𝜔𝐶⁄ = _______________ Ω

4. Different items in the time domain transform in different ways to the ω

domain:

Element Time Domain ω Domain Transform

Applied Sinusoidal

AC Voltage

𝑽𝒑 𝐜𝐨𝐬(𝝎𝒕)

(Volts)

Vp (Volts)

Series Current 𝑰𝒑 𝐜𝐨𝐬(𝝎𝒕 + 𝜽)

(Amps)

𝑰𝒑

(Amps)

Resistance R

(Ohms)

R

(Ohms)

Inductance L

(Henry’s) 𝒋𝝎𝑳

(Ohms)

Capacitance C

(Farads) 𝟏 𝒋𝝎𝑪⁄ (Ohms)

© N. B. Dodge 01/12

Given a circuit with a Time domain values:

𝑣(𝑡) = 10 cos(1000𝑡)

𝑅 = 100 Ω

L = 10 mH

C = 10 μF

Calculate the values in the ω domain for:

Applied Voltage __________Volts

Resistance __________Ω

Impedance from Inductor __________ Ω

Impedance from Capacitor __________ Ω

5. After transforming voltage and circuit to the ω domain, find the current by

dividing voltage by impedance. This usually results in a complex number. To

convert back to the time-domain, which is the answer sought, do four things:

• Rationalize using complex conjugate; the result is an X ± jY representation.

• Convert complex number to polar-coordinates:

• The radial distance is the peak current. The angle is the phase angle.

6. Based on the procedure in 5 above, convert the following ω domain currents

back into the time domain (assume ω = 1000):

I = 10+j10 _____________ I = −8+j4 _____________

7. If 𝑣(𝑡) = 10 cos(1000𝑡), R = 100 Ω, C = 100 μF , determine the expression for i(t) .