Discussion Question 7A
P212, Week 7
Two Methods for Circuit Analysis
Method 1: Progressive “collapsing” of circuit elements
In a previous discussion, we learned how to analyse circuits involving batteries and capacitors. Our method was
to progressively collapse groups of capacitors (connected in series or in parallel) into effective capacitors. Once
the circuit became simple enough, we could calculate everything about it: charge and voltage. Then we worked
backwards, breaking up each combination and calculating the charge and voltage on the individual capacitors.
This week, we will analyze circuits involving resistors. As you know, the formulas for combining resitors in series
and in parallel are “opposite” to those for capacitors. We also need an expanded set of rules for “breaking up”
combinations of devices:
•
For devices connected in parallel: the voltage across them is always the same.
•
For devices connected in series: the charge and current is always the same.
But otherwise, the procedure is the same!
(a) Can you explain the rules stated above in terms of physical principles? They have simple physical origins, so
they’re easy to remember.
(b) Consider the circuit shown at right. All the resistors Ri have the same value R. Find the current I4 through
resistor R4 (including its direction) and the electric potential VA at the indicated point A. Be sure to express your
answers in terms of the given parameters E (battery “EMF” = voltage) and R (resistance of each resistor).
A
R1
E
R3
R2
1
R4
Discussion Question 7A
P212, Week 7
Two Methods for Circuit Analysis
Method 2: Kirchhoff’s Laws
This week, we will also practice a new and more powerful method for circuit analysis. The new method is based
on Kirchoff’s two Laws:
•
Kirchhoff’s Current Law (KCL): The sum of the currents flowing into any node (junction) in the circuit
equals the sum of the currents flowing out of that node.
•
Kirchhoff’s Voltage Law (KVL): The sum of the potential changes V around any loop is zero. And here are
the rules for how you sum up those V’s as you go around a loop in the circuit:
➢ Crossing a battery from the negative to the positive terminal = a potential rise → add +E
➢ Crossing a battery from the positive to the negative terminal = a potential drop → add -E
➢ Crossing a resistor in the same direction as the current flow = a potential drop → add –IR
➢ Crossing a resistor against the direction of the current flow = a potential rise → add +IR
(c) Can you explain these rules on physical principles?
Now, let’s apply these laws to analyze the same circuit as on the
previous page. Once more, your job is to find the current I4 and the
potential VA. Let’s go through the procedure in detail:
A
R1
(d) Identify how many different currents there are in the circuit.
These are often called branch currents. Label them on the figure,
from I1 to In. Be sure to label them with arrows, indicating the
direction of each current! If the directions you chose are wrong, no
problem, your currents will simply come out with minus signs.
E
R3
R4
R2
(e) Next, apply KCL to every node in the circuit. How many nodes are there in this circuit? Write down a KCL
equation for each one.
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Discussion Question 7A
P212, Week 7
Two Methods for Circuit Analysis
(f) Examine the equations you just wrote down … how many of them are independent? Here’s the rule: if you
have Nnodes nodes in a circuit, KCL will only give you Nnodes-1 independent equations. In other words, one of the
equations gives you no new information. Use all but one of your equations to ‘get rid’ of as many of the
unknown currents as possible, by writing expressions for them in terms of the remaining currents.
(g) How many unknown currents are left? Let’s call this number n. To determine these n currents, we will use
KVL. Apply KVL to n loops in the circuit … this will give you the n equations you need to determine all the branch
currents in the circuit.
A
R1
E
R3
R4
R2
(h) Solve your complete set of KCL and KVL equations! Remember, your job is to find I4 and VA.
For reference, here is a summary of the steps to take in analyzing a circuit using Kirchhoff’s Laws:
1. Identify all the different branch currents in the circuit, and label them with arrows to indicate direction.
Let NI be this number of currents.
2. Apply KCL to all but one of the nodes in the circuit. This will give you (Nnodes-1) equations.
3. Apply KVL to NI-(Nnodes-1) loops in the circuit. Together with your KCL equations, this gives you a total of
NI equations → enough to determine all the different currents.
4. Solve your complete set of NI equations to determine all the branch currents. Knowing that, and good
old “V = IR”, you can calculate everything about your resistor network!
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Discussion Question 7B
P212, Week 7
Mixing Methods in Resistor Network Analysis
R2
B
R3
R1
I4
E1 = 10 V
E2 = 20 V
all Ri = 10 
R4
E2
R5
E1
A
In the previous question, you analyzed a “2-loop” circuit. In such cases, the linear algebra is relatively easy. But
circuits of 3 loops or more are another story, the linear algebra can become really time consuming! Consider
the circuit below. Your tasks are:
(a) Find the current I4 through resistor R4.
(b) Find the potential difference VB – VA.
This circuit looks like a truly horrible 3-loop affair … but it isn’t! With two batteries arranged as shown, we will
need to use method 2, Kirchoff’s Laws. But we can greatly simplify our lives by first applying method 1, the
“collapse” of series and parallel resistor combinations as much as possible.
Go for it! Solve for the quantities requested in (a) and (b) using a combination of methods 1 and 2.
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Discussion Question 7B
P212, Week 7
Mixing Methods in Resistor Network Analysis
Finally, for practice, here are two examples of real “3-loop” circuits. In these rather complex cases, your task is
merely to (c) identify all the different branch currents and (d) write down enough equations to solve for all of
them. You may assume that all resistors Ri have the same value R. Solving the equations is just algebra … we
won’t bother with that part here.
(c)
R4
R2
R1
R6
E2
E1
R3
2
R5
Discussion Question 7B
P212, Week 7
Mixing Methods in Resistor Network Analysis
(d)
R1
R2
R4
R3
R5
R6
E1
E2
R7
E3
R8
3
E4
R9
Discussion Question 7C
P212, Week 7
RC Circuits
The circuit shown initially has the capacitor uncharged, and the switch connected to neither terminal.
At time t = 0, the switch is thrown to position a.
C
a
E
b
R2
R1
E = 12 V
C = 5 F
R1 = 3 
R2 = 6 
(a) At t = 0+, immediately after the switch is thrown to position a, what are the currents I1 and I2
across the two resistors?
What does the uncharged capacitor look like to the rest of the circuit at time 0? Does it offer any
resistance to the flow of charge? (Why or why not?)
(b) After a very long time, what is the instantaneous power P dissipated in the circuit?
After a very long time, what will have happened to the capacitor? Now what will it look like to the
rest of the circuit?
(c) After a very long time, what is the Q charge on the capacitor?
To determine Q, you need the voltage across the capacitor ...
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Discussion Question 7C
P212, Week 7
RC Circuits
Next, after a very long time T, the switch is thrown to position b.
C
a
E
b
R1
R2
(d) What is the time constant  that describes the discharging of the capacitor?
We have a nice formula available for time constants:  = RC. But the R in the formula refers to the
total resistance through which the capacitor discharges. Redrawing your circuit might help you to
determine this R .
(e) Write down an equation for the time dependence of the charge on the capacitor, for times t > T.
Your answer for Q(t) should depend only on the known quantities E, R1, R2, C, and T.
You know the general form for the time dependence of a discharging capacitor. All you have to do is
fix the constants in this expression to match the charge at t = T and at t = ∞.
(f) What is the charge Q20 on the capacitor 20 sec after time T?
(g) What is the current through R 2 20 sec after time T?
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Discussion Question 7D
P212, Week 7
RC Circuits
The circuit shown initially has the capacitor uncharged,
and the switch open. At time t = 0, the switch is thrown.
Write all answers in terms of E , R, and C as needed.
C
R
E
R
(a) At 𝑡 = 0+ , immediately after the switch is thrown
what is the current 𝐼(0+ )?
I
(b) After a very long time, what is the current 𝐼∞ ?
(c) After a very long time, what is the charge on the capacitor 𝑄∞ ?
(d) Assume 𝑄(𝑡) = 𝑄∞ (1 − 𝑒𝑥𝑝( − 𝛽𝑡)) gives the charge on the capacitor as a function of time. Use
𝑑𝑄
[ 𝑑𝑡 ]
𝑑𝑄
0+
= 𝐼(0+ ) and your answers to (a) and (c) to compute 𝛽. Why does [ 𝑑𝑡 ]
0+
= 𝐼(0+ )? What does
your 𝛽 imply for the effective resistance that you use in 𝜏 = 𝑅effective C?
(e) Use KVL for a loop that includes the capacitor and the battery to compute 𝐼(𝑡). Check that your
answers are consistent with your answers to parts (a) and (b).
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