how to solve current circuits

how to solve current circuits


1 Answers

Aman Bansal
592 Points
11 years ago

Dear Balakrishna,

Solving Circuits with Kirchoff Laws

Example 1: Find the three unknown currents ($I_1,I_2,I_3$) and three unknown voltages ( $V_{ab}, V_{bd}, V_{cb}$) in the circuit below:


Note: The direction of a current and the polarity of a voltage source can be assumed arbitrarily. To determine the actual direction and polarity, the sign of the values also should be considered. For example, a current labeled in left-to-right direction with a negative value is actually flowing right-to-left.


  • The Branch-Current Method:
    • Step 1: Label each unknown branch current arbitrarily with a reference direction. If the calculated value is negative, the actual direction of the current is opposite to the reference direction.
    • Step 2: Define unknown voltages in terms of the assumed currents to obtain three element equations


      \begin{displaymath}\begin{array}{l} V_{ab}=2I_1,\;\;\mbox{or}\;\;\;V_{ba}=-2I_1... ...3 \\ V_{cb}=4I_2,\;\;\mbox{or}\;\;\;V_{bc}=-4I_2 \end{array} \end{displaymath}


    • Step 3: Apply KCL to node b: 


      \begin{displaymath}\sum I_b=0=+I_1+I_2-I_3\;\;\;\;\;\;\;\;\;\;(1) \end{displaymath}


      We could also apply KCL to node d: 


      \begin{displaymath}\sum I_d=0=-I_1-I_2+I_3 \end{displaymath}


      but this equation is not independent as it adds no information.
    • Step 4: Apply KVL to loops abda and bcdb:


      \begin{displaymath}\begin{array}{r} \sum V_{abda}=0=V_{ab}+V_{bd}+V_{da}\;\;\;... ...cdb}=0=V_{cb}+V_{bd}+V_{dc}\;\;\;\;\;\;\;\;\;\;(3) \end{array} \end{displaymath}


      We could also apply KVL to loop abcda: 


      \begin{displaymath}\sum V_{abcda}=0=V_{ab}+V_{bc}+V_{cd}+V_{da} \end{displaymath}


      But this equation is not independent as it can be obtained as the difference between the previous two equations. The above three independent equations (one for current, two for voltage) are called connection equations.
    • Step 5: We now have three simultaneous equations with three unknown branch currents: 


      \begin{displaymath}\begin{array}{r} \sum I_b=0=+I_1+I_2-I_3\;\;\;\;\;\;\;\;\;\;... ..._{bd}+V_{dc}=4I_2+8I_3-20\;\;\;\;\;\;\;\;\;\;(3'') \end{array} \end{displaymath}


      which can be rewritten as 


      \begin{displaymath}\left\{ \begin{array}{rrrrr} I_1 & +I_2 & -I_3& = & 0\;\;\;\... ..._2& +8I_3 & = & 20\;\;\;\;\;\;\;\;\;\;(3'') \end{array} \right. \end{displaymath}


    • Step 6: Solving the equations, we get the three unknown currents: 


      \begin{displaymath}I_1=4,\;\;\;I_2=-1,\;\;\;I_3=3 \end{displaymath}


      and then we get the three unknown voltages: 


      \begin{displaymath}V_{ab}=8,\;\;\;V_{bd}=24,\;\;\;V_{cb}=-4 \end{displaymath}


    Note that equations from KCL at node d and KVL to other loops are not independent. In general, if a circuit has n nodes and b branches, then there are $(n-1)$ independent node equations and $l=b-(n-1)$ independent loop equations. In other words, the sum of the number of independent loops and the number of independent nodes is always the same as the number of branches, i.e., the number of equations is always equal to the number of unknowns in the branch current method.


  • The Loop/Mesh-Current Method (based on KVL):
    1. Define a loop current around each loop in clockwise direction (although it could be arbitrary). Assume there are $l$ independent loops in the circuit, then we have $l$ loop currents as the unknown variables.
    2. Apply KVL around each of the loops in the same clockwise direction to obtain $l$ equations. While calculating the voltage drop across each resistor shared by two loops, both loop currents (in opposite positions) should be considered.
    3. Solve the equation system with $l$ equations for the $l$ unknown loop currents.

    Now we can resolve the problem above using loop-current method:


    For the first circuit, we apply KVL to the two loops to get 


    \begin{displaymath}\sum V_{abda} =0=-32+2I_a+8(I_a-I_b) \end{displaymath}



    \begin{displaymath}\sum V_{bcdb} =0=8(I_b-I_a)+4I_b+20 \end{displaymath}


    Rewrite these as: 


    \begin{displaymath}\left\{ \begin{array}{rrrr} 10I_a&-8I_b&=&32  -8I_a &+12I_b&=&-20 \end{array} \right. \end{displaymath}


    which can be solved to get $I_a=I_1=4$$I_b=-I_2=1$, and $I_3=I_a-I_b=3$.


  • The Node-Voltage Method (based on KCL):
    1. Assume there are n nodes in the circuit. Select one node as the ground, i.e., the reference point for all voltages of the circuit. The voltage at each of the remaining n-1 nodes is an unknown to be obtained.
    2. Apply KCL to each of the n-1 nodes to obtain n-1 equations.
    3. Solve the equation system with n-1 equations for the n-1 unknown node voltages.

    Assume node $d$ is the ground, and consider just $V_b=V_{bd}$ as the only unknown in the problem. Apply KCL to node $b$, we have 


    \begin{displaymath}\sum I_b=0=I_1+I_2-I_3 \end{displaymath}




    \begin{displaymath}I_1=\frac{V_a-V_b}{R_{ab}}=\frac{32-V_b}{2},\;\;\;\; I_2=\fr...{20-V_b}{4},\;\;\;\; I_3=\frac{V_b}{R_{bd}}=\frac{V_b-0}{8} \end{displaymath}


    This equation of one unknown $V_b$ can be solved to get $V_b=24$, and all other unknown currents and voltages can be found easily.


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Aman Bansal

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