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Circuit Analysis (Equation) with Multiple Voltage Sources

  1. Sep 16, 2015 #1
    1. The problem statement, all variables and given/known data
    Derive the differential equation that relates the voltage (V) to the current (I) entering the circuit.
    Screen Shot 2015-09-16 at 11.29.49 AM.png



    2. Relevant equations
    Let:

    ic = C dV/dt = current traveling through the capacitor (C)
    ik = (V+Ek) / Rk = current traveling through the potassium (K) channel
    iNa = (V-ENa) / RNa = current traveling through the sodium (Na) channel
    iCl = (V+ECl) / RCl = current traveling thought the chlorine (Cl) channel

    I = ic + ik + iNa + iCl = total current


    3. The attempt at a solution

    I = ic + ik + iNa + iCl
    I = C dV/dt + (V+Ek) / Rk + (V-ENa) / RNa + (V+ECl) / RCl

    My confusion is from the signs of the batteries. What is the convention (or physical meaning) when you are subtracting the voltages. For example, in the potassium (K) channel, it is V - (-Ek); while in the sodium (Na) channel, the signs on the battery are reversed. How do I know when a voltage source is positive or negative in relation to the current flow?

    Thank you!
     
  2. jcsd
  3. Sep 16, 2015 #2
    Your equations look correct.

    The voltage polarities are usually given. They are usually the consequence of some physical (generator) or chemical (battery) situation. When the current which is conventionally considered a positive charge flow is taken from the negative side of a voltage source to the positive side the voltage drop is taken as negative while if the current is "flowing" from positive to negative the voltage drop is taken as positive

    .
     
  4. Sep 16, 2015 #3

    CWatters

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    You didn't say how you arrived at your equations for Ik, INA and ICL?

    One way is to use KVL. KVL states that the voltages around a loop sum to zero. However before you can apply KVL you have to define which direction you mean by +ve current. Lets do KVL for the potassium channel...

    First I arbitrarily define +ve IK as current flowing down through Rk.
    Then I arbitrarily decide to sum the voltages around the loop clockwise and get...

    V + (-IkRk) + EK = 0

    Rearrange and you get..

    IK = (EK + V)/RK

    which is the same as you got.

    The essential thing is to mark the diagram with arrows showing your definition of +ve current at the outset and stick to it when you write the equations. It doesn't matter which direction you choose as +ve just as long as you are consistent when you write the equations. eg If you assume +ve is down through RK then there will be a voltage drop in RK when you travel clockwise summing the voltages. That's why I wrote -IkRk. If I had defined +ve current as flowing up through RK then I would have written +IkRk in the equation.

    In short it doesn't matter which direction you choose as +ve current flow OR which way around a loop you sum the voltages as long as you are consistent. It all comes out in the wash.

    Try it for the other branches.
     
  5. Sep 16, 2015 #4

    CWatters

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    PS Your equations are correct if you defined +ve current flow as down through the relevant resistor. Suppose you solved all the equations and one came out as -ve. What would that mean if you haven't defined which direction is +ve?
     
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