Kirchhoff's Laws

Here are some terminologies for electric circuits:

In a circuit diagram, both the direction of a current through, and the polarity of a voltage across an element can be arbitrarily labeled. The actual direction and polarity will be determined by the sign of the specific values obtained after the circuit is solved. For example, a current labeled in the left-to-right direction with a negative value is actually flowing in the right-to-left direction.

Kirchhoff's Laws

Example 1:

Kirchhoff.png

Assume currents flowing into the node are positive and those leaving the node negative, the KCL states: $4+5-3-4-2=0$.

Assume the current flows around the loop in clockwise direction, the KVL states: $-12+3+4+5=0$.

Example 2: Given the circuit below, find $V_2$, $V_0$, $I_2$, $R_1$ and $R_2$.

Kirchhoff1.gif

According to Ohm's law, we have $I_2=3/2=1.5A$.

Apply KVL to the loop on the right to get:

$\displaystyle V_2-5+3=0,\;\;\;\;\;\; V_2=2V$ (77)

According to Ohm's law, we have $R_2=V_2/I_2=2V/1.5A=1.33\Omega$.

Apply KCL to the middle node on top to get:

$\displaystyle 2-I_1-I_2=2-I_1-1.5=0,\;\;\;\;\;I_1=0.5A$ (78)

Again by Ohm's law, we get $R_1=5V/0.5A=10\Omega$.

Apply KVL to the loop on the left to get:

$\displaystyle 3\times 2 +5-V_0=0,\;\;\;\;\;\;V_0=11V$ (79)

The series and parallel combinations of circuit components

  Resistor $R$ Inductor $L$ Capacitor $C$
Governing Equation $v=Ri=i/G$, $i=v/R=Gv$ $v=L\;di/dt$, $i=\int v\; dt/L$ $v=\int i\;dt/C$, $i=C\;dv/dt$
Series connection $R_s=R_1+R_2$, $1/G_s=1/G_1+1/G_2$ $L_s=L_1+L_2$ $1/C_s=1/C_1+1/C_2$
Parallel connection $1/R_p=1/R_1+1/R_2$, $G_p=G_1+G_2$ $1/L_p=1/L_1+1/L_2$ $C_p=C_1+C_2$

Example 3 Consider the following six circuits as either current or voltage dividers.

RCLPS.png