## Circuit Lab B/C

Test your knowledge of various Science Olympiad events.
Cathy-TJ
Member
Posts: 28
Joined: January 20th, 2019, 7:13 pm
Division: C
State: VA

### Re: Circuit Lab B/C

Nationals Placings:
2019 Circuit Lab - 1
2018 Disease Detectives - 1
2015 Picture This - 6

2019 Events:
Chemistry Lab, Circuit Lab, Disease Detectives

Competitors today, teammates tomorrow

mjcox2000
Member
Posts: 120
Joined: May 9th, 2014, 3:34 am
State: VA

### Re: Circuit Lab B/C

Diagram: https://drive.google.com/file/d/1pqa8dq ... sp=sharing
Note: this problem requires calculus. (It's possible there's a particularly clever solution that bypasses the calculus, but I can't think of one.)

You are an electrical engineer tasked with charging a capacitor $C$ , which is connected in series with resistor $R_s$ and in parallel with resistor $R_p$ . You want to charge the capacitor to a final voltage $V_f$ , and you have at your disposal a current source $I$ . (See the diagram for the circuit configuration, and note that at time $t=0$ , the capacitor is fully discharged. Also note that $I$ is constant; i.e. the current cannot vary with time.)

In the aim of efficiency, you want to dissipate as little power as possible in the resistors as you charge the capacitor to voltage $V_f$ . However, your colleague, who designed the circuit with $R_s$ , $R_p$ , and $C$ and chose the value of $V_f$ , won't let you change any of those values. The only value you can play around with is the charging current $I$ .

a) Find an equation, in terms of $C$ , $R_s$ , $R_p$ , and $V_f$ , and $I$ , for the time $t_f$ at which the capacitor is fully charged to voltage $V_f$ .

b) Find an equation, in terms of $C$ , $R_s$ , $R_p$ , and $V_f$ , and $I$ , for the amount of power dissipated in the resistors in the course of charging the capacitor to voltage $V_f$ .

c) Find an equation, in terms of $C$ , $R_s$ , $R_p$ , and $V_f$ , for the value of $I$ that offers minimum power dissipation. (I'm not convinced this part has a closed-form solution -- if not, do what you can.)
MIT ‘23
TJHSST ‘19
Longfellow MS
Nationals medals

UTF-8 U+6211 U+662F
Exalted Member
Posts: 1476
Joined: January 18th, 2015, 7:42 am
Division: C
State: PA

### Re: Circuit Lab B/C

mjcox2000 wrote:Diagram: https://drive.google.com/file/d/1pqa8dq ... sp=sharing
Note: this problem requires calculus. (It's possible there's a particularly clever solution that bypasses the calculus, but I can't think of one.)

You are an electrical engineer tasked with charging a capacitor $C$ , which is connected in series with resistor $R_s$ and in parallel with resistor $R_p$ . You want to charge the capacitor to a final voltage $V_f$ , and you have at your disposal a current source $I$ . (See the diagram for the circuit configuration, and note that at time $t=0$ , the capacitor is fully discharged. Also note that $I$ is constant; i.e. the current cannot vary with time.)

In the aim of efficiency, you want to dissipate as little power as possible in the resistors as you charge the capacitor to voltage $V_f$ . However, your colleague, who designed the circuit with $R_s$ , $R_p$ , and $C$ and chose the value of $V_f$ , won't let you change any of those values. The only value you can play around with is the charging current $I$ .

a) Find an equation, in terms of $C$ , $R_s$ , $R_p$ , and $V_f$ , and $I$ , for the time $t_f$ at which the capacitor is fully charged to voltage $V_f$ .

b) Find an equation, in terms of $C$ , $R_s$ , $R_p$ , and $V_f$ , and $I$ , for the amount of power dissipated in the resistors in the course of charging the capacitor to voltage $V_f$ .

c) Find an equation, in terms of $C$ , $R_s$ , $R_p$ , and $V_f$ , for the value of $I$ that offers minimum power dissipation. (I'm not convinced this part has a closed-form solution -- if not, do what you can.)

Not really sure if I'm on the right track but

mjcox2000
Member
Posts: 120
Joined: May 9th, 2014, 3:34 am
State: VA

### Re: Circuit Lab B/C

UTF-8 U+6211 U+662F wrote:
mjcox2000 wrote:Diagram: https://drive.google.com/file/d/1pqa8dq ... sp=sharing
Note: this problem requires calculus. (It's possible there's a particularly clever solution that bypasses the calculus, but I can't think of one.)

You are an electrical engineer tasked with charging a capacitor $C$ , which is connected in series with resistor $R_s$ and in parallel with resistor $R_p$ . You want to charge the capacitor to a final voltage $V_f$ , and you have at your disposal a current source $I$ . (See the diagram for the circuit configuration, and note that at time $t=0$ , the capacitor is fully discharged. Also note that $I$ is constant; i.e. the current cannot vary with time.)

In the aim of efficiency, you want to dissipate as little power as possible in the resistors as you charge the capacitor to voltage $V_f$ . However, your colleague, who designed the circuit with $R_s$ , $R_p$ , and $C$ and chose the value of $V_f$ , won't let you change any of those values. The only value you can play around with is the charging current $I$ .

a) Find an equation, in terms of $C$ , $R_s$ , $R_p$ , and $V_f$ , and $I$ , for the time $t_f$ at which the capacitor is fully charged to voltage $V_f$ .

b) Find an equation, in terms of $C$ , $R_s$ , $R_p$ , and $V_f$ , and $I$ , for the amount of power dissipated in the resistors in the course of charging the capacitor to voltage $V_f$ .

c) Find an equation, in terms of $C$ , $R_s$ , $R_p$ , and $V_f$ , for the value of $I$ that offers minimum power dissipation. (I'm not convinced this part has a closed-form solution -- if not, do what you can.)

Not really sure if I'm on the right track but

You're on the right track, but
MIT ‘23
TJHSST ‘19
Longfellow MS
Nationals medals

UTF-8 U+6211 U+662F
Exalted Member
Posts: 1476
Joined: January 18th, 2015, 7:42 am
Division: C
State: PA

### Re: Circuit Lab B/C

mjcox2000 wrote:
You're on the right track, but

Agh! Well that sucks.
Take Two

mjcox2000
Member
Posts: 120
Joined: May 9th, 2014, 3:34 am
State: VA

### Re: Circuit Lab B/C

UTF-8 U+6211 U+662F wrote:
mjcox2000 wrote:
You're on the right track, but

Agh! Well that sucks.
Take Two

That's right!
A few notes

MIT ‘23
TJHSST ‘19
Longfellow MS
Nationals medals

UTF-8 U+6211 U+662F
Exalted Member
Posts: 1476
Joined: January 18th, 2015, 7:42 am
Division: C
State: PA

### Re: Circuit Lab B/C

Explain how you could solve for the resistance between two ends of a bridge circuit using Kirchoff's rules.

wec01
Member
Posts: 185
Joined: February 22nd, 2019, 4:02 pm
State: VA

### Re: Circuit Lab B/C

UTF-8 U+6211 U+662F wrote:Explain how you could solve for the resistance between two ends of a bridge circuit using Kirchoff's rules.

2019 Nationals Medals:
4th place Fossils
5th place Sounds of Music
2nd place Thermodynamics

UTF-8 U+6211 U+662F
Exalted Member
Posts: 1476
Joined: January 18th, 2015, 7:42 am
Division: C
State: PA

### Re: Circuit Lab B/C

wec01 wrote:
UTF-8 U+6211 U+662F wrote:Explain how you could solve for the resistance between two ends of a bridge circuit using Kirchoff's rules.

Yep, your turn. Also mention source injection , as applying Kirchoff's laws is rather difficult without it.

wec01
Member
Posts: 185
Joined: February 22nd, 2019, 4:02 pm
State: VA

### Re: Circuit Lab B/C

Calculate the values of I1, I2, and I3 given:

2019 Nationals Medals:
4th place Fossils
5th place Sounds of Music
2nd place Thermodynamics

Cathy-TJ
Member
Posts: 28
Joined: January 20th, 2019, 7:13 pm
Division: C
State: VA

### Re: Circuit Lab B/C

wec01 wrote:

Calculate the values of I1, I2, and I3 given:

Nationals Placings:
2019 Circuit Lab - 1
2018 Disease Detectives - 1
2015 Picture This - 6

2019 Events:
Chemistry Lab, Circuit Lab, Disease Detectives

Competitors today, teammates tomorrow

mjcox2000
Member
Posts: 120
Joined: May 9th, 2014, 3:34 am
State: VA

### Re: Circuit Lab B/C

Edit: Cathy-TJ beat me to an answer.
MIT ‘23
TJHSST ‘19
Longfellow MS
Nationals medals

wec01
Member
Posts: 185
Joined: February 22nd, 2019, 4:02 pm
State: VA

### Re: Circuit Lab B/C

Cathy-TJ wrote:
wec01 wrote:

Calculate the values of I1, I2, and I3 given:

2019 Nationals Medals:
4th place Fossils
5th place Sounds of Music
2nd place Thermodynamics

Cathy-TJ
Member
Posts: 28
Joined: January 20th, 2019, 7:13 pm
Division: C
State: VA

### Re: Circuit Lab B/C

Calculate the resistance between terminals A and B in the infinite chain of resistors where all resistors are 1 ohm.
Nationals Placings:
2019 Circuit Lab - 1
2018 Disease Detectives - 1
2015 Picture This - 6

2019 Events:
Chemistry Lab, Circuit Lab, Disease Detectives

Competitors today, teammates tomorrow

mjcox2000
Member
Posts: 120
Joined: May 9th, 2014, 3:34 am
State: VA

### Re: Circuit Lab B/C

Cathy-TJ wrote:Calculate the resistance between terminals A and B in the infinite chain of resistors where all resistors are 1 ohm.

MIT ‘23
TJHSST ‘19
Longfellow MS
Nationals medals

### Who is online

Users browsing this forum: No registered users and 2 guests