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Kirchhoff's Laws#

Kirchhoff's laws are two rules which tell us how to find the currents and voltages throughout a given electronic circuit.

Kirchhoff's Current Law#

Theorem: Kirchhoff's Current Law (KCL)

Let \(N\) be a node in a lumped circuit.

If \({}_{\text{in}}I_1, \dotsc, {}_{\text{in}}I_p\) are all currents whose chosen reference arrows point into \(N\) and \({}_{\text{out}}I_1, \dotsc, {}_{\text{out}}I_q\) are all currents whose chosen reference arrows point away from \(N\), then:

\[\sum_{k=1}^p {}_{\text{in}}I_k = \sum_{k=1}^q {}_{\text{out}}I_k\]

Example

KCL Standard Form

\[{}_{\text{in}} I_1 = {}_{\text{out}}I_1 + {}_{\text{out}}I_2 + {}_{\text{out}}I_3\]
Proof

TODO

This can be stated alternatively if we label all currents as \(I_1, \cdots, I_n\) and then write the equation as

\[\sum_{k = 1}^n s_k \cdot I_k = 0,\]

where we define \(s_k\) according to either one of the following conventions:

\[\begin{aligned}s_k &\overset{\text{def}}{=} \begin{cases}-1 \qquad \text{if the chosen reference arrow of } I_k \text{ points towards the node} \\ 1 \qquad \text{if the chosen reference arrow of } I_k \text{ points away from the node} \end{cases} \\ &\text{or} \\ s_k &\overset{\text{def}}{=} \begin{cases}-1 \qquad \text{if the chosen reference arrow of } I_k \text{ points away from the node} \\ 1 \qquad \text{if the chosen reference arrow of } I_k \text{ points towards the node} \end{cases} \end{aligned}\]

The first convention stems from the fact that

\[\sum_{k=1}^p {}_{\text{in}}I_k = \sum_{k=1}^q {}_{\text{out}}I_k \iff \sum_{k=1}^q {}_{\text{out}}I_k - \sum_{k=1}^p {}_{\text{in}}I_k = 0\]

and the second convention is due to the fact that

\[\sum_{k=1}^p {}_{\text{in}}I_k = \sum_{k=1}^q {}_{\text{out}}I_k \iff \sum_{k=1}^p {}_{\text{in}}I_k - \sum_{k=1}^q {}_{\text{out}}I_k = 0\]

Example

KCL Sign Convention Form

\[\begin{aligned}- I_4 + I_3 + I_2 + I_1 &= 0 \\ I_4 - I_3 - I_2 - I_1 &= 0 \end{aligned}\]

Kirchhoff's Voltage Law#

Theorem: Kirchhoff's Voltage Law (KVL)

Consider any number of points (any points, not necessarily nodes) in a lumped circuit. Draw a closed loop passing through them andcChoose an orientation (either clockwise or counterclockwise) for it. Now consider all voltages between adjacent points.

If \({}_{\text{along}} V_1, \dotsc, {}_{\text{along}} V_p\) are the voltages whose chosen reference arrows point along the orientation of the loop and if \({}_{\text{against}} V_1, \dotsc, {}_{\text{against}} V_q\) are the voltages whose chosen reference arrows point opposite the orientation of the loop, then:

\[\sum_{k = 1}^p {}_{\text{along}} V_k = \sum_{k = 1}^q {}_{\text{against}} V_k\]

Example

KVL Standard Form

\[{}_{\text{along}} V_1 + {}_{\text{along}} V_2 + {}_{\text{along}} V_3 = {}_{\text{against}} V_1 + {}_{\text{against}} V_2\]
Proof

TODO

This can be stated alternatively if we label the voltages as \(V_1, \dotsc, V_n\) and then write the equation as

\[\sum_{k = 1}^n s_k \cdot V_k = 0,\]

where we define \(s_k\) according to one of the following conventions:

\[\begin{aligned} s_k &= \begin{cases}-1 & \text{if the reference arrow of } U_k \text{ points against the chosen loop direction} \\ 1 & \text{if the reference arrow of } U_k \text{ points along the chosen loop direction}\end{cases} \\ &\text{or} \\ s_k &= \begin{cases}1 & \text{if the reference arrow of } U_k \text{ points against the chosen loop direction} \\ -1 & \text{if the reference arrow of } U_k \text{ points along the chosen loop direction}\end{cases} \end{aligned}\]

The first convention stems from the fact that

\[\sum_{k = 1}^p {}_{\text{along}} V_k = \sum_{k = 1}^q {}_{\text{against}} V_k \iff \sum_{k = 1}^p {}_{\text{along}} V_k - \sum_{k = 1}^q {}_{\text{against}} V_k = 0\]

and the second convention is due to the fact that

\[\sum_{k = 1}^p {}_{\text{along}} V_k = \sum_{k = 1}^q {}_{\text{against}} V_k \iff \sum_{k = 1}^q {}_{\text{against}} V_k - \sum_{k = 1}^p {}_{\text{along}} V_k = 0\]

Example

KVL Sign Convention Form

\[\begin{aligned}- V_1 - V_2 + V_3 + V_4 + V_5 &= 0 \\ V_1 + V_2 - V_3 - V_4 - V_5 &= 0\end{aligned}\]

Tip: Connections and KVL

Kirchhoff's voltage law also applies when only some or none of the points are physically connected by wires because you can always draw a closed loop between any finite number of points (the orange line):

General KVL