RC Circuit with Current Source: Is This Solution Correct?

[goonking]

iC1(t) = io (e^-t/RC) = (0.003A)(e^-t/0.005)

U1(t) = [Is - iC1(t)] 20k

are the answers then.

Suppose the question also asks for the I_R1(t)

and U_C1(t)

how would the equations look like? The voltage on the capacitor will be exponentially increasing and the current through R1 will be exponentially increasing too.

UC1(t) = Uo (1 - e^-t/RC) = IR(1 - e^-t/RC) = (0.003A)(30000ohms)(1 - e^-t/(0.005)

I_R1(t) = Io ( 1 - e^-t/RC) = (I V) 1 - e^-t/RC) = [0.0045A * 90V](1 - e^-t/(0.005)
?

 

[cnh1995]

UC1(t) = Uo (1 - e^-t/RC) = IR(1 - e^-t/RC) = (0.003A)(30000ohms)(1 - e^-t/(0.005)

IR1(t) = Io ( 1 - e^-t/RC) = (I V) 1 - e^-t/RC) = [0.0045A * 90V](1 - e^-t/(0.005)

Those equations give zero current at t=0.
Your earlier expression for Ic(t) was right, except for the time constant. Use it to find U1(t).

U1(t) = [Is - iC1(t)] 20

 

[goonking]

Those equations give zero current at t=0.
Your earlier expression for Ic(t) was right, except for the time constant. Use it to find U1(t).

I_R1(t) = [Is - iC1(t)] = [7.5x10^-3 A] - [(0.003A)(e^-t/0.005)]

so when t is a high number, all the current flows through R1.

 

[cnh1995]

I_R1(t) = [Is - iC1(t)] = [7.5x10^-3 A] - [(0.003A)(e^-t/0.005)]

so when t is a high number, all the current flows through R1.

Right! Multiplying it with 20k will give you U1(t)..

 

[goonking]

Those equations give zero current at t=0.
Your earlier expression for Ic(t) was right, except for the time constant. Use it to find U1(t).

oops, made an error

IR1(t) = Io ( 1 - e^-t/RC) = ( 7.5x10^-3 A) ( 1 - e^-t/RC) = ( 7.5x10^-3 A) ( 1-e^t/0.005)

I got confused, didn't know if Io was the current at t=0 from the current source or the current going through the R1 resistor at t=0.

only way the equation makes sense is if Io is the current source which is 7.5x10^-3

 

[goonking]

anywhoo, after reading some problems online, many of them like to have a "chart" or write down what exactly is happening when t→ -∞ , t = 0 , and t →∞
it probably helps organize things a bit so I will try that approach for this problem too, and hopefully refine my understanding a bit more.

For t → -∞
The switch is open, therefore no current flows through the capacitor and R2:
I_C1(-∞) = 0 A

I_R1(-∞) = 7.5 x 10^-3 A ∴ U_R1(-∞) = I_source * R1 = (0.0075 A) (20,000Ω) = 150 V

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For t = 0

The switch is now closed.

Voltage across the capacitor is 0 ∴ Q = VC = (0)C = 0, there is no charge on the capacitor at t = 0
Capacitor is closed, current is flowing through: I_C1(0) = [20 / (20+30)]* 7.5 x 10^-3 A = 0.003 A

Current through R1 is 0.0045A: I_R1(0) = 0.0045 A ∴ U_R1(0) = ( 0.0045 A) (20,000 Ω) = 90 V

(but capacitor is parallel to R1, therefore voltage across it must be 90 too, but we stated before voltage across capacitor is 0! (unless this just means the MAX possible voltage across the capacitor is 90 V, can someone clarify this?)

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For t→ ∞
The switch is STILL closed.
The capacitor is fully charged, there is no current flowing through it so it becomes open.
I_C1(∞) = 0 A
and voltage across it should be max? aka 90V? U_C1(∞) = 90 V ?

All the current from the current source should be flowing through R1: I_R1(∞) = 7.5 x 10^-3 A
∴ U_R1(∞) = I_source * R1 = (0.0075 A) (20,000Ω) = 150 V
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the question asks to find expressions for I_C1(t) and U_R1(t) for t ≥ 0+ (does this mean I would need the time constant , RC, with R being the equivalent resistance in the circuit WHEN the switch is JUST closed? (but that isn't what we established earlier, Req needs to be 50k ohms)

Looking at Req when t <0 , switch is open, Req is just R1, Req = 20k ohm
Req when t =0, switch is closed, current is flowing through R1 and R2, Req = 20k*30k / 50k = 12k ohms
Req when t > 0, switch is still closed, but capacitor is fully charged, therefore it becomes open, now we are back to all current flowing through R1, Req = 20k ohm

The only way Req = 50k ohms is when current source is open (which is when the capacitor is in series with R1 and R2, and supplying them two resistors with energy), but I don't understand why it would just open (unless it requires me to know about thevenin's theorem, but I'm trying to understand this problem with elementary principles) would greatly appreciate if someone can point out the flaws in my thought process/logic.

 

[goonking]

Going through the textbook, I found an example problem is similar to the one in OP.

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So in this problem, once the switch is closed, the voltage source becomes shorted. Then Req is solved.

I can see it now, in my problem, once the switch is closed, the source, since it was a current source, becomes open, then C1 , R1, and R2 are in series.
Then Req = 50k ohms.

So basically, the only time to solve for time constant is when the switch is closed, all voltage sources become shorted,and all current sources become open, then we solve for Req accordingly.

 

[cnh1995]

IR1(t) = Io ( 1 - e^-t/RC) = ( 7.5x10^-3 A) ( 1 - e^-t/RC) = ( 7.5x10^-3 A) ( 1-e^t/0.005)

It still gives zero current at t=0.
This expression for Ic is right.

iC1(t) = io (e^-t/RC)

I1=Is-Ic...

 

[cnh1995]

but I don't understand why it would just open (unless it requires me to know about thevenin's theorem, but I'm trying to understand this problem with elementary principles)

Equivalent resistance of a network between a and b is nothing but the Thevenin resistance of the network viewed from the terminals a and b. One way to convince yourself that R1 and R2 are in series is to find the "discharging" time constant, since charging time constant=discharging time constant(numerically, although the definitions differ). For the capacitor to discharge, current source should be opened i.e. existing sources should be removed.

 

[goonking]

After learning a bit about thevenin and norton circuits , this problem has become easier to visual.

For the capacitor to discharge, current source should be opened i.e. existing sources should be removed.

I understand this now but in this case (after a long period and the capacitor is full charged) , the switch is never opened again after it was closed, so the capacitor will never discharge and the voltage across it will remain 90 V.

Correct?

 

[cnh1995]

so the capacitor will never discharge and the voltage across it will remain 90 V.

It will remain 150V.

 

[NascentOxygen]

anywhoo, after reading some problems online, many of them like to have a "chart" or write down what exactly is happening when t→ -∞ , t = 0 , and t →∞

Usually the times consided are t=0^- (just before the switch changes), t=0⁺ (immediately after the switching occurs), and t→∞ (the final equilibrium).

Current through R1 is 0.0045A: I_R1(0) = 0.0045 A ∴ U_R1(0) = ( 0.0045 A) (20,000 Ω) = 90 V

(but capacitor is parallel to R1, therefore voltage across it must be 90 too, but we stated before voltage across capacitor is 0!

Not correct. It is the combination R2-C which is parallel to R1, so at t=0⁺ the 90V is across that series pair R2-C, not across C only. With the voltage across C being zero, it leaves all 90V across R2 at this moment.

The only way Req = 50k ohms is when current source is open (which is when the capacitor is in series with R1 and R2, and supplying them two resistors with energy), but I don't understand why it would just open (unless it requires me to know about thevenin's theorem, but I'm trying to understand this problem with elementary principles) would greatly appreciate if someone can point out the flaws in my thought process/logic.

The current source doesn't open.

It is undoubtedly easiest to explain using Thévenin's theorem why $\tau$=(R1+R2)C for the capacitor charging. But you can still get a feel for why it must be if you consider the influence of R1 on the charging rate of C. The effect of lowering resistance R1 is to divert more of the current away from the capacitor, thereby slowing down the charging rate of the capacitor. In the calculations a slowing in the charging rate can be accounted for by a larger effective R, and mathematically you arrive at this using R1+R2 but not R1 || R2.

(The picture is further complicated because R1 also changes the voltage towards which the capacitor is charging.)