Kirchoff’s Current Law (KCL)
Describe a circuit or elements of a circuit by applying Kirchoff’s current law (a.k.a. “Kirchoff’s junction rule” in official AP material).
Kirchhoff’s Current Law (Kirchoff’s junction rule) is a consequence of the conservation of electric charge.
Kirchoff’s Current Law states that the total amount of charge entering a node per unit time must equal the total amount of charge exiting that junction per unit time. \[\begin{equation} \sum I_{in} = \sum I_{out} \end{equation}\]
Voltage divider circuit
Consider the circuit shown below, called a voltage divider:
(vin) at (2,2) ; (gndl) at (2,0) ; (vout) at (6,2) ; (gndr) at (6,0) ;
(0,2) to [V=] (0,0); (0,2) to [short, i=i] (vin); (vin) to [R=R1, i=i] (4,2); (4,2) to [R=R2, i=i] (4,0); (4,0) to (gndl); (gndl) to (0,0); (vin) to [open, v=vin] (gndl); (4,2) to (vout); (4,0) to (gndr); (vout) to [open, v=vout] (gndr);
Let us solve for the transfer function from \(v_{in}\) to \(v_{out}\). \(R_1\) and \(R_2\) are in series, thus the total resistance looking into \(v_{in}\) is: \[\begin{equation} R_{eq} = R_1+R_2 \end{equation}\] Therefore, the current in the circuit is \[\begin{equation} i=\frac{v_{in}}{R_1+R_2} \end{equation}\] \(v_{out}\) is the voltage drop across \(R_2\). Applying Ohm’s Law: \[\begin{equation} v_{out} = i R_2 = v_{in} \frac{R_2}{R_1+R_2} \end{equation}\] Therefore: \[\begin{equation} \frac{v_{out}}{v_{in}} = \frac{R_2}{R_1+R_2} \end{equation}\]
By changing the values of \(R_1\) and \(R_2\) we can get different values of voltage out, anything from 0 V for \(R_2 = \qty{0}{\ohm}\) to \(v_{in}\) for \(R_1 = \qty{0}{\ohm}\).
Rheostats, potentiometers, variable resistors
The device shown in figure \[fig:rheostat\] is called a rheostat.
It implements the voltage divider discussed above, in a way that the resistances can be varied by moving the slider or wiper along the device from one terminal to the other. Rheostats (and similar devices called potentiometers) are variable resistors and can be used to vary the amount of resistance in a circuit.
The circuit diagram symbol (fig \[fig:potsymbol\]) for a potentiometer or rheostat reflects its construction. In both devices, a sliding arm makes contact with the stationary resistance (fig \[fig:potinternals\]). As the sliding arm or wiper is moved, the resistances between the center wiper terminal and the end terminals change.
(0,0) to \[american potentiometer\] (2,0);
The rating of a variable resistor is the resistance of the entire stationary resistance from one end terminal to the other end terminal (typically amounts like 10 kΩ or 100 kΩ. Device specifications also include the number of turns or the length of travel of the wiper. Devices are available in which the resistance profile (“taper”) changes linearly as well as logarithmically with wiper travel.
Rheostats are often used to control very high currents such as though found in motor and lamp loads (at the expense of high quiescent power usage, i.e. the device must always burn some of the available power to accomplish control, even when doing nothing.) Their construction makes it easy to visualize how the device operates.
As discussed below, potentiometers can be used to vary the value of voltage applied to a circuit. They are often used as control devices such as the knobs found in amplifiers, radio, television sets and electrical instruments.
More on potentiometers
The physical realization for ES202 is a 3-pin device with three leads, as shown in figure \[fig:pot\]. An equivalent circuit for the device is shown in figure \[fig:poteq\].
image image
Turning the adjustment screw moves the wiper arm and divides the total resistance (\(R_{1-3}\)) proportionally. Let \(K\) be the fraction of distance from the bottom to the wiper arm, so that \(0\leq K \leq 1\), and \(R_{1-3}\) be the total resistance of the device: \[\begin{align} (1-K) R_{1-3} + K R_{1-3} &= R_{1-3} \\ R_{1-2} &= (1-K) R_{1-3} \\ R_{2-3} &= K R_{1_3} \end{align}\]
(0,0) to\[american potentiometer,n=mypot, o-o\] (2,0) (0,0) node\[anchor=south\] 1 (2,0) node\[anchor=south\] 3 (mypot.wiper) node\[ocirc, label=above:2\] ;
Two simple uses become apparent. The device can be used as a variable resistor (fig 1) by shorting across pins 1 and 2, which reduces the equivalent resistance between pins 1 and 3 by the amount that has been shorted out. The equivalent resistances that can be achieved in this way vary between 0 Ω (a short) and \(R_{1-3}\).
(0,0) coordinate (t1) (2,0) coordinate (t3) (t1) to[american potentiometer,n=mypot, o-o] (t3) (mypot.wiper) to [short, o-] ++(-1,0) to [short] (t1) (mypot.wiper) node[anchor=south] 2 (t1) node[anchor=north] 1 (t3) node[anchor=south] 3
(4,0) to[variable american resistor, o-o] ++(2,0) (4,0) node[anchor=north] 1 (6,0) node[anchor=south] 3 ;
Alternatively, the device can be used to get a variable voltage out by configuring it as a voltage divider (fig 2).
(0,0) node[ground] (gnd) (0,3) coordinate (V) (2,3) coordinate (t1) (2,0) coordinate (t3) (4,1.5) coordinate (vout) (4,0) coordinate (gndout)
- to [V=vin]
(gnd) (V) to [short, i=i] (t1)
(t1) to [american potentiometer, n=mypot, o-o] (t3) (mypot.wiper) to
[short, o-o] (vout) (t3) to [short, o-o] (gndout) (t3) to (gnd)
(t1) node[anchor=south] 1 (t3) node[anchor=225] 3 (mypot.wiper) node[anchor=south] 2 (vout) to [open, v^=vout] (gndout)
;
For the circuit shown, the output voltage \(v_{out}\) varies between 0 V and \(v_{in}\), according to the position of the wiper (terminal 2). Such an arrangement might be used to achieve linear or rotary position sensing (such as in a joystick).
See also
- list here