The second systematic technique to determine all currents and voltages in a circuit
Circuits With Multiple Sources Suppose you have the following circuit diagram: How do you determine...
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Transcript of Circuits With Multiple Sources Suppose you have the following circuit diagram: How do you determine...
Circuits With Multiple Sources
• Suppose you have the following circuit diagram:• How do you determine the
current throughout the circuit?• How do you determine the
voltage throughout the circuit?• Cannot use equivalent
circuits to simplify this…what to do???
- -
Kirchoff’s Circuit Laws
• Gustav Kirchoff– generalized observations of circuits• Kirchoff’s Junction Rule (aka: Kirchoff’s
current law)• Current into node = Current leaving
node• Iin = Iout
• Kirchoff’s Loop Rule (aka: Kirchoff’s voltage law)• Sum of voltage around any closed
loop must be zero• Vloop = 0 V
Image obtained from: http://www-history.mcs.st-and.ac.uk/BigPictures/Kirchhoff_4.jpeg
Applying Kirchoff’s Laws
• Step 1: Decide on directions for current• Electron flow notation:
electrons move away from -, move toward +• Every node must have at
least 1 current going in and 1 going out• # currents = # wire lines
between nodes
- -I1
I1
I1
I2
I2
I3
I3
I3
Applying Kirchoff’s Laws
• Step 2: Sign conventions for components• Electron flow notation:
electrons move away from -, move toward +• Every component needs a
(-) end and a (+) end - -I1
I1
I1
I2
I2
I3
I3
I3
-
--+
+
+
Applying Kirchoff’s Laws
• Step 3: Perform Junction Rule for every node• Iin = Iout
• Node A: I1 + I2 = I3
• Node B: I3 = I1 + I2
• Good! We’re consistent! - -I1
I1
I1
I2
I2
I3
I3
I3
-
--+
+
+
A
B
Applying Kirchoff’s Laws
• Step 4: Perform Loop Rule for every loop• Vloop = 0 V• Going from - to +: negative
voltage (V being lost)• Going from + to -: positive
voltage (V being gained)• Remember: V = IR
- -I1
I1
I1
I2
I2
I3
I3
I3
-
--+
+
+
A
B
Loop 1 Loop 2
Loop 3
Applying Kirchoff’s Laws
• Loop 1:
-I1R2 - ε2 - I1R1 + ε1 = 0 V• Loop 2:
-I3R3 + ε2 = 0 V• Loop 3:
-I1R2 - I3R3 - I1R1 + ε1 = 0 V- -I1
I1
I1
I2
I2
I3
I3
I3
-
--+
+
+
A
B
Loop 3
Loop 1 Loop 2
Applying Kirchoff’s Laws
• Let’s say we have the following:• ε1 = 6.0 V• ε2 = 1.5 V• R1 = 2.0 Ω• R2 = 3.0 Ω• R3 = 5.0 Ω
What is the current passing through the battery ε2?
- -I1
I1
I1
I2
I2
I3
I3
I3
-
--+
+
+
A
B
Loop 3
Loop 1 Loop 2
Applying Kirchoff’s Laws
• Looking at diagram, I2 is current passing through ε2
• How to find I2?• I3 = I1 + I2 so I2 = I3 - I1
• How to find I3?• -I3R3 + ε2 = 0 V so I3 = ε2/R3
• How to find I1?• -I1R2 - ε2 - I1R1 + ε1 = 0 V
soI1 = (ε1 - ε2) / (R1 + R2)
What we know:I3 = I1 + I2
-I1R2 - ε2 - I1R1 + ε1 = 0 V
-I3R3 + ε2 = 0 V
-I1R2 - I3R3 - I1R1 + ε1 = 0 V
Applying Kirchoff’s Laws
• I3 = ε2/R3
• I1 = (ε1 - ε2) / (R1 + R2)
I1 = (6.0 V - 1.5 V) / (2.0 Ω + 3.0 Ω)
I1 = 4.5 V / 5.0 Ω
I1 = 0.90 A
• I2 = I3 - I1
What we know:I3 = I1 + I2
-I1R2 - ε2 - I1R1 + ε1 = 0 V
-I3R3 + ε2 = 0 V
-I1R2 - I3R3 - I1R1 + ε1 = 0 V
Oh no! A negative current! What does that mean?
= 1.5 V / 5.0 Ω = 0.30 A
= 0.30 A - 0.90 A = -0.60 A
Negative current: goes in opposite direction of what you said it did
Try Another!
Information:• ε1 = 120 V• ε2 = 120 V• R1 = 1.0 Ω• R2 = 2.0 Ω• R4 = 4.0 Ω• R5 = 5.0 Ω• R6 = 6.0 Ω
Find currents across: R1, R4
Let’s Step It Up A Bit!
Information:• ε1 = 120 V• ε2 = 120 V• R1 = 1.0 Ω R5 = 5.0 Ω• R2 = 2.0 Ω R6 = 6.0 Ω• R3 = 3.0 Ω R8 = 8.0 Ω• R4 = 4.0 Ω
Find currents across: R3, R6
19.0 A
15.9 A
