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#### henryparrot

##### New Member
Hi i have been asked to analyse the attached circuit. I need to find the voltage across the resistor Rc as a function of V and time. I dont know what method to use, any help would be much appreciated.

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#### user_88

##### Member
Just look at the voltage across Rc as a Voltage Divider :
Vc= Rc*V
......ΣR

The actual transient solution does not appear to be tractable, significant, or useful .... for several trial values of R1, R, and C in a circuit simulation.

... I did not get Mathcad to successfully produce any other response than the steady state .... a good exercise though.

#### henryparrot

##### New Member
Hi i managed to get the following solution, but I am not sure whether it is right.
Vc(t) = VoRc/R1 + Ke^(-t/RcCc)
where K is a constant. It doesn't seem correct to me, since if you choose the resistors in a certain way then you can get Vc(t) > V0.

#### user_88

##### Member
Hi i managed to get the following solution, but I am not sure whether it is right.
Vc(t) = VoRc/R1 + Ke^(-t/RcCc)
where K is a constant. It doesn't seem correct to me, since if you choose the resistors in a certain way then you can get Vc(t) > V0.

I looked at the problem in a manner such that the required voltage across Rc could be found with an impedance divider.... Zc/(ΣZ).

In the attachment below, each of the R and C groupings is the respective impedance, found by adding the two element admittances, and then taking the reciprocal value of the sum.

For convenience, I assumed that the excitation voltage was a step input.

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#### MrAl

##### Well-Known Member
Hi there,

The actual form of the transient response will look like this:

Vo(t)/V=Ae^at+Be^bt+Ce^ct+D

where

a,b,c are negative, and
D is positive and
A+B+C+D=0

These constants are a bit hard to find when all the values are in
algebraic variable form rather than actual constants like 1,2,3, etc.

The state vector differential equation solution would look like this
however:

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#### MrAl

##### Well-Known Member
Hi again,

(Note: Here R1=Rp,R2=Rc,R3=Rk, and Rs=R1 from the original circuit.
In other words, here R1,R2,R3 are the resistors in parallel with C1,C2,C3
and Rs is the resistor in series with V in the original circuit.
The solution is the same, with different component names.)

Here is a slightly different form for the time solution for the
voltage across R2:

v2(t,V)=V*(A*(e^(a*t)-1)+B*(e^(b*t)-1)+C*(e^(c*t)-1))

Note that this isnt a particularly complicated solution, but we do
need to find all the constants A,B,C,a,b, and c, so what we do
here is first find these constants A,B,C, and a,b,c and then we have the
solution for all time t.

First we define some sundry constants, all of which depend on the circuit components
and the voltage of the battery:

Code:
B3=C1*C2*C3*R1*R2*R3*Rs
B2=C1*C2*R1*R2*R3+C1*C2*R1*R2*Rs+C1*C3*R1*R2*R3+C1*C3*R1*R3*Rs+C2*C3*R1*R2*R3+C2*C3*R2*R3*Rs
B1=C1*R1*R2+C1*R1*R3+C1*R1*Rs+C2*R1*R2+C2*R2*R3+C2*R2*Rs+C3*R1*R3+C3*R2*R3+C3*R3*Rs
B0=R1+R2+R3+Rs

A2=V*R2*C1*C3*R1*R3
A1=V*R2*(C1*R1+C3*R3)
A0=V*R2
Next we find the three real roots of this equation:
B3*x^3+B2*x^2+B1*x+B0=0

using whatever method we have available to find numerical roots such as:

(a,b,c)=roots("B3*x^3+B2*x^2+B1*x+B0=0")

This yields three real roots, a,b, and c, and so we now have three of the
required constants. The roots will always be real with this circuit because
there are no oscillatory components. They are also negative.

Next, since we now know a,b, and c we can calculate the other three
constants A,B, and C from these equations:

V=1
A=(A2*a*a+A1*a+A0)/(B3*a*(a-b)*(a-c))
B=(A2*b*b+A1*b+A0)/(B3*b*(b-a)*(b-c))
C=(A2*c*c+A1*c+A0)/(B3*c*(c-a)*(c-b))

and now all we do is insert all of the constants into the time solution and
we have the solution for v2(t) which is the voltage across the resistor R2 at
any given time t:

v2(t)=A*(e^(a*t)-1)+B*(e^(b*t)-1)+C*(e^(c*t)-1)

We can now insert any time t and get the solution. If we change any of the
circuit values however we need to go back and recalculate all of the constants
again first.

Also,

v2(t,V)=A*(e^(a*t)-1)+B*(e^(b*t)-1)+C*(e^(c*t)-1) with V=1,

and with another V we get:

v2(t,V)=V*(A*(e^(a*t)-1)+B*(e^(b*t)-1)+C*(e^(c*t)-1))

and that's it.

Of course we could also use the state vector differential equation in the
previous post too putting it into the form x'=Ax+Bu and then solving
after calculating the state transition matrix.

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#### flat5

##### Member
My Jr. Collage teacher was right to advise me to drop engineering in 1970

#### steveB

##### Well-Known Member
Hi i have been asked to analyse the attached circuit. I need to find the voltage across the resistor Rc as a function of V and time. I dont know what method to use, any help would be much appreciated.
The problem you have given is written as a steady state problem. There is no information about how things will change in time. In other words, we need to know if the voltage is off and then turned on at time=0. Also, one would need to know the initial values of the capacitor voltages.

Basically, all voltages and currents are constant (in time) based on the way the problem is presented right now. Was more information given?

#### Bob Scott

##### New Member
My Jr. Collage teacher was right to advise me to drop engineering in 1970
You'd be surprised at how many working electrical engineers are lost at the sight of op-amps.

#### MrAl

##### Well-Known Member
The problem you have given is written as a steady state problem. There is no information about how things will change in time. In other words, we need to know if the voltage is off and then turned on at time=0. Also, one would need to know the initial values of the capacitor voltages.

Basically, all voltages and currents are constant (in time) based on the way the problem is presented right now. Was more information given?
Hi,

You are right of course, but usually when circuits are presented like this
it means to find the response with a step change in input voltage from

0 volts
at t=0-
to V volts
at t=0+

and with all initial states equal to zero.

When initial conditions need to be considered they are usually
given with the problem question, such as:

Find the voltage v2 at time t with initial conditions
v1=0.3v, v2=0.1v, and v3=0.2v
or whatever.

The problem statement:
"I need to find the voltage across the resistor Rc as a function of V and time."

led me to believe he wanted the time response, not some steady state solution
at t=+inf. Since initial conditions were not stated, they were simply left out.

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