Kirchoff's Current Law (KCL):
At a junction or node, since there can be no net accumulation of charge, the sum of the currents must be zero. To set the sum equal to zero, we have to remember to assign the signs of the currents into the node as positive and the currents out of the node as negative. Alternatively, we can state the Junction Rule (or KCL) as:
S i into node = S i away from nodeHere is an example:
|i1 - i2 - i3
i1 = i2 + i3
In circuit diagrams, nodes are sometimes indicated by a black dot, but more often not. What's more, since electrical circuits tend to involve a lot of wires, often wires cross each other without connecting. Usually it's clear what is meant, as in the first four figures below. In those figures, connections are indicated by a solid dot or simply by two solid lines connecting; crosses (meaning wires which go over one another, but are not physically connected) are indicated by a break in the line, or a deviation of the line. However, occasionally (especially on that ratty circuit diagram on the back of your dryer or furnace) you will find a diagram in which solid lines crossing without a dot or a break really mean crosses (no connection), while all connections have a solid dot. You can recognize the latter "sloppy" diagrams because they show no wires with breaks or deviations in them.
Kirchoff's Voltage Law (KVL):
Voltage relationships are a bit trickier. You can use Kirchoff's loop rule (as we did for hydraulic systems) which says that the sum of the voltages added up around any closed loop is zero:
|You can apply Kirchoff's loop rule to any loop in a circuit.
It doesn't have to be isolated from other elements, or from other loops.
Shown is just a small chunk of a larger circuit, and there's only one complete
loop visible, but you can imagine that each of the elements could be considered
to be part of other loops as well.
With the signs of the voltages carefully considered, Kirchoff's loop rule says
V1 + V2 - V3 = 0You can think about going around the loop adding up voltages, with a minus sign whenever you hit the pointy end of an arrow.
Typically you won't draw current arrows at random -- you'll draw them predominantly from one side toward the other, here north to south.
Although it is hard to read above, in this figure many of the wires are labeled with a specific potential. The potential of those certain wires usually has some meaning - it represents a signal from one part of the circuit to another, or it is a common voltage source for a lot of components. So, in addition to labeling the voltage across components, you often see the so-called voltage "of" certain wires labeled. Voltage is still an across variable! The voltage "of" (really, potential of) a wire means the voltage of the wire relative to ground. When we encounter such diagrams, we can retreat from writing loop rules and do the following:
Series and Parallel Rules
When the voltages of certain wires are named, we can return to the series and parallel rules stated earlier to obtain equations relating the voltages in a circuit:
For N elements connected in series:
V = V1 + V2 + V3 + . . . + VN = Va - Vbwhere, Va and Vb are given potentials of the left and right wires (their voltages relative to ground).
While, for N elements connected in parallel:
V = V1 = V2 = V3 = . . . = VN = Va - Vb
We can analyze any segment of a circuit using these rules, similar to how we related velocities in segments of mechanical systems. Here is an example:
|Note that there are two "named" voltages of wires here. These are voltages
relative to ground. Ground is the symbol at the bottom, and the voltage
of that wire is of course zero.
Picking any two of the "named" wires, you can use their voltage difference with the series and parallel rules to write an equation relating the voltages across elements between those chosen "named" wires. For instance, picking +12 volts and ground, we can write
+12 - 0 = Vc + V1 + V2or picking Vbus and ground,
Vbus - 0 = V1 + V2 = V3Warning -- you do still have to be careful about signs.
Finally, notice that I named too many voltages. You can tell right away from the diagram that Vbus = V3. So it would have been more efficient just to call it Vbus, and not bother to name V3.
|This circuit looks pretty complicated. Five resistors, two capacitors,
and two busses with named voltages. Hopefully the values of the components
is given. Here I've just given them names (R1, etc)
|A first step is to pick directions for currents.
Take a look at all the nodes and make sure a direction has been chosen
for every current impinging on any node.
Note that I did not specify a positive direction of current in the red wire ... this is because the red wire contains no elements. In fact, this wire simply connects R3, R4, R5 and C1 together and can be drawn as an elongated node, which we will do in the next picture.
Rather than give each current its own name, we'll use the convention that iR1 means the current of the arrow associated with R1.
(There are so many currents and voltages, that it's really worthwhile to be frugal about names.)
|We could write the name of the voltage across each component next to
that component, but it's getting crowded there. So let's just use the notation
that VC1 means the voltage across capacitor C1.
For nodes that involve three or more currents, assign a "named" voltage (relative to ground), taking advantage of the voltages that are already named by the existing components.
Two of the nodes were already named: +12 and ground. For the remaining two nodes, I recycled the existing names VR2 and VC1, so I didn't have to invent any new names.
iR1 = iR2 + iR3
12 = VR1 + VR2
Next collect the constitutive laws of every component.
VR1 = R1 iR1
VC1 = (1/C1) qC1
Writing State Equations:
If there are capacitors, however, the voltages and currents will be functions of time. Now there is motion. The equations of motion are differential equations. You will have one state equation for each capacitor (just like in mechanical systems we have one diffeq for each spring), and these differential equations will be coupled (also like in mechanical systems). To get started building these diffeqs, use the definition of charge, q'=i, to change the capacitor constitutive laws into the beginnings of diffeqs (called differential equation kernals):
q'C1 = iC1
There are now 16 equations: 3 for nodes, 4 for loops, 5 for resistors, 2 for capacitors, and 2 definitional equations relating charge and current in the capacitors. And there are now 16 variables: a current and a voltage for each of 7 components, and a charge for each capcitor.
If you don't stop at this point and count variables and equations in this way, you are likely to proceed into a lot of hopeless algebra. Hopeless because you missed a node or loop equation! So catch your mistake early, at the counting stage.
Now the task is to eliminate all the dynamic variables from the right side of (2) except the state variables You should wind up with two equations relating the time derivatives of the state variables, q'C1 and q'C2, to the state variables themselves, qC1 and qC2:
q'C1 = blah-blah-blah about qC1 and qC2
q'C1 = yak-yak-yak about qC1 and qC2