Kirchoff’s Voltage Law (KVL)
Describe a circuit or elements of a circuit by applying Kirchoff’s voltage law (a.k.a. “Kirchoff’s loop rule” in official AP material).
Energy changes in simple electrical circuits may be represented in terms of charges moving through electric potential differences within circuit elements. \[\begin{equation} \Delta U_E = q \Delta V \end{equation}\]
Kirchoff’s voltage law (Kirchhoff’s loop rule) is a consequence of the conservation of energy.
Kirchhoff’s voltage law states that the sum of potential differences across all circuit elements ina single closed loop must equal zero. \[\begin{equation} \Sigma \Delta V = 0 \end{equation}\]
The values of electric potential at points in a circuit can be represented by a graph of electric potential as a function of position within a loop.
Series circuits
Two elements are in series when there is only one path (branch) for the current to flow through them. For example, in figure 1, \(R_1\), \(R_2\), and \(R_3\) are in series.
(0,2) to [V=V, i=i] (0,0); (0,2) to [R=R1, i=i] (4,2); (4,2) to [R=R2, i=i] (4,0); (4,0) to [R=R3, i=i] (0,0);
The current \(i\) is the same in all parts of the circuit in figure 1. This means that the current flowing through \(R_1\) is the same as the current flowing through \(R_2\), is the same as the current flowing through \(R_3\), and is the same as the current supplied by voltage source \(V\). The reason this is the case is conservation of charge; the charges are flowing around the circuit and are not being created, destroyed, or leaked out similar to water flowing in a single pipe.
For this series circuit, conservation of energy also applies in a form typically stated as Kirchoff’s Voltage Law (KVL): the sum of the voltage drops along any closed path is equal to zero. Thus, \(V = v_1+v_2+v_3\).
(0,2) to [V=V, i=i] (0,0); (0,2) to [R=R1, v=v1, i=i] (4,2); (4,2) to [R=R2, v=v2, i=i] (4,0); (4,0) to [R=R3, v=v3, i=i] (0,0);
There is a voltage drop across each resistor in series. In effect, each drop reduces the voltage available for the next component in the circuit, analagous to pressure drops along a length of piping.
For resistors, Ohm’s Law gives \(v=iR\); combining the above yields: \[\begin{align} V &= i R_1 + i R_2 + i R_3 \quad \text{then, dividing by $i$}\\ \frac{V}{i} &= R_1 + R_2 + R_3 \\ R_{eq} &= R_1 + R_2 + R_3 \end{align}\] This last bit shows that when resistances are connected in series, the total equivalent resistance in the circuit is equal to the sum of the resistances. This also suggests we can simplify circuits during analyses; if we are only interested in the “world” from the point of view of voltage source \(V\), we could replace everything with an equivalent resistance \(R_{eq} = R_1+R_2+R_3\).
Parallel circuits
A parallel circuit is a circuit in which components are connected in different branches between the same two nodes (e.g. across the same voltage source). In figure 2, resistors \(R_1\), \(R_2\), and \(R_3\) are in parallel with each other.
(0,2) to [V=V, i=i] (0,0); (0,2) to (2,2); (2,2) to (4,2); (4,2) to (6,2); (0,0) to (2,0); (2,0) to (4,0); (4,0) to (6,0); (2,2) to [R=R1, v=v, i=i1] (2,0); (4,2) to [R=R2, v=v, i=i2] (4,0); (6,2) to [R=R3, v=v, i=i3] (6,0);
The voltages at each node are the same, therefore the voltages across all the parallel components are the same, e.g. \(v_1=v_2=v_3=V\).
In each parallel branch, the current can be different from the other branches. However, at each node, as a consequence of charge conservation, the sum of the currents (in and out) must be zero; this is known as Kirchoff’s Current Law (KCL). For figure 2, \(i=i_1+i_2+i_3\) (regardless of whether the resistances are equal). This is analogous to the flow of water in a single pipe that encounters a header and flows into several pipes.
By Ohm’s Law, \(i_1 = V/R_1\), \(i_2=V/R_2\), and \(i_3=V/R_3\). This means that branches with higher resistances see less current. In other words, current “seeks the path of least resistance”; and is shunted to branches with lower resistance. With the same applied voltage, a branch that has less resistance allows more current through it than a branch with higher resistance.
Combining the above results by dividing the currents by the voltage and applying Ohm’s Law: \[\begin{align} i &= i_1 + i_2 + i_3 \quad \text{divide both sides by V} \\ \frac{i}{V} &= \frac{i_1}{V} + \frac{i_2}{V} + \frac{i_3}{V} \\ \frac{1}{R_{eq}} &= \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \quad \text{invert both sides} \label{eq:conductances}\\ R_{eq} &= \frac{1}{\frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3}} \label{eq:resistances} \end{align}\] In other words, the conductances add when in parallel (eq \[eq:conductances\]), while the total equivalent resistance is the inverse of the sum of the inverses of the individual resistances in parallel (eq \[eq:resistances\]).
Simplified formulas
It is left as an exercise for the student to prove the following.
For two resistors in series: \[\begin{equation} R_{eq} = R_1+R_2 \quad \text{for $R_1$ and $R_2$ in series} \end{equation}\]
For two resistors in parallel: \[\begin{align} R_{eq} &= R_1 || R_2 \quad \text {this means ``$R_1$ in parallel with $R_2$''}\\ &= \frac{R_1 R_2}{R_1+R_2} \quad \text{for $R_1$ and $R_2$ in parallel} \end{align}\]
For \(n\) equal resistors in parallel, the total resistance is equal to the resistance of one of the resistors divided by the number of resistors: \[\begin{equation} R_{eq} = \frac{R}{n} \quad \text{for $n$ equal resistors $R$} \end{equation}\]
In some cases with two parallel resistors, it is useful to find what size \(R_x\) to connect in parallel with a known \(R\) in order to obtain a required value of \(R_T\): \[\begin{align} R_T &= \frac{R R_x}{R+R_x} \\ R_x &= \frac{R R_T}{R-R_T} \end{align}\]
See also
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