ECE 115 Intro. to Electrical & Computer Engineering

Exam #1University of Illinois at Chicago

Spring 2016

Name: Lab: M2 T8 W2 Th11 Th2

Trigonometric Identities

cos(−θ) = cos(θ) sin(−θ) = − sin(θ) sin(θ) = cos(

θ − π

2

)

Cosines and Sines of common angles

θ (radians) θ (degrees) cos θ sin θ

0 0 1 0

π/6 30√3/2 1/2

π/4 45√2/2

√2/2

π/3 60 1/2√3/2

π/2 90 0 1

Euler’s Formula

ejθ = cos θ + j sin θ e−jθ = cos θ − j sin θ

Complex Numbers

rectangular form z = a+ j b a is the real part of z, b is the imaginary part of z

polar form z = Ae jθ A is the magnitude of z, θ is the angle of z

rectangular → polar z = a+ j b →√

a2 + b2 e j tan−1(b/a) = Ae jθ

polar → rectangular z = Ae jθ → A cos θ + j A sin θ = a+ j b

complex conjugate z = a+ j b → z∗ = a− j b

complex conjugate z = Ae jθ → z∗ = Ae−jθ

Complex Number Properties

j =√−1 = ejπ/2 = e−j3π/2

j2 = −1 = ejπ = e−jπ

j3 = −j = ej3π/2 = e−jπ/2

j4 = 1 = ej0 = ej2π = e−j2π

ImpedancesR → ZR = R, L → ZL = jωL, C → ZC = 1/(jωC) = −j/(ωC)

Series: R1 +R2 or Z1 + Z2, Parallel: R1||R2 =R1R2

R1 +R2or Z1||Z2 =

Z1Z2

Z1 + Z2

Ohm’s Law, Voltage Division, Current DivisionOhm’s Law — Assuming passive sign convention: V = IR or V = IZ

Voltage Division — For R1 and R2 in series with Vs: V1 =R1

R1 +R2Vs, V2 =

R2

R1 +R2Vs

Current Division — For R1 and R2 in parallel with Is: I1 =R2

R1 +R2Is, I2 =

R1

R1 +R2Is

Voltage Division — For Z1 and Z2 in series with Vs: V1 =Z1

Z1 + Z2Vs, V2 =

Z2

Z1 + Z2Vs

Current Division — For Z1 and Z2 in parallel with Is: I1 =Z2

Z1 + Z2Is, I2 =

Z1

Z1 + Z2Is

Prof. Vahe Caliskan 1 of 6 Posted: February 19, 2016

ECE 115 Intro. to Electrical & Computer Engineering

Exam #1University of Illinois at Chicago

Spring 2016

Problem 1 (4 pts) In the circuit shown below, Vs = 24V, R1 = 3Ω and R2 = 9Ω.Determine I1, I2, V1 and V2. Show all work.

+ ++

−

−−

R1

R2

I1

I2V1

V2Vs

Problem 2 (2 pts) Determine the power dissipated for R1 and R2 (Pdiss,R1, Pdiss,R2) in Problem 1.

Problem 3 (4 pts) In the circuit shown below, Is = 9A, R1 = 2Ω and R2 = 4Ω.Determine I1, I2, V1 and V2. Show all work.

++

−−R1 R2

I1 I2

V1 V2Is

Problem 4 (2 pts) Determine the power dissipated for R1 and R2 (Pdiss,R1, Pdiss,R2) in Problem 3.

Prof. Vahe Caliskan 2 of 6 Posted: February 19, 2016

ECE 115 Intro. to Electrical & Computer Engineering

Exam #1University of Illinois at Chicago

Spring 2016

Problem 5 (4 pts) Find the equivalent resistance Req for the following circuit. Show all work.

Req

10Ω20Ω

30Ω40Ω 50Ω

Problem 6 (4 pts) Given that vin(t) = 40 sin(377t)V and RL = 10Ω, find vp(t), vs(t), ip(t) and is(t).Show all work.

+

+ +

−− −

vin(t)

2:3

RL

ip(t)

vp(t)

is(t)

vs(t)

Problem 7 (4 pts) Given that vs(t) = 60 sin(377t)V and RL = 12Ω, find vp(t), vin(t), ip(t) and is(t).Show all work.

+

+

+

−−

−vin(t)

5:3

RL

ip(t)

vp(t)

is(t)

vs(t)

Prof. Vahe Caliskan 3 of 6 Posted: February 19, 2016

ECE 115 Intro. to Electrical & Computer Engineering

Exam #1University of Illinois at Chicago

Spring 2016

Problem 8 (8 pts)Show all work for the following problems. It may help you to sketch the complex number first.

(a) Express −4− j3 in polar form.

(b) Express −6 + j6 in polar form.

(c) Express 4ej π/6 in rectangular form.

(d) Express 4e−j 45 in rectangular form.

(e) Express the product (−4− j3)(−6 + j6) in polar form.

(f) Express the quotient−4− j3

−6 + j6in polar form.

(g) Express the sum 4ej π/6 + 4e−j 45 in rectangular form.

(h) Express the difference 4ej π/6 − 4e−j 45 in rectangular form.

Prof. Vahe Caliskan 4 of 6 Posted: February 19, 2016

ECE 115 Intro. to Electrical & Computer Engineering

Exam #1University of Illinois at Chicago

Spring 2016

Problem 9 (2 pts)Find the total complex impedance Ztotal (in Ω) for the following circuit combinations at an angular frequencyω = 5000 rad/s. Make sure that your final result for Ztotal is in rectangular form.

(a) 50Ω resistor in series with a 50µF capacitor.

(b) 2mH inductor in parallel with a 10µF capacitor.

Problem 10 (3 points)Given the following circuit with complex impedances, find the complex voltage V . Make sure that your finalresult for V is in rectangular form. Show all work.

++

−− V

4Ω

j4Ω5∠60 V

Problem 11 (3 points)Find the complex current I using current division. Make sure that your final result for I is in rectangularform. Show all work.

I

−j5Ω j10Ω20∠30 A

Prof. Vahe Caliskan 5 of 6 Posted: February 19, 2016

ECE 115 Intro. to Electrical & Computer Engineering

Exam #1University of Illinois at Chicago

Spring 2016

Extra Workspace

Prof. Vahe Caliskan 6 of 6 Posted: February 19, 2016