NDSU
4. Circuits I
pg 1
Circuits I
Circuits I Concepts
Circuit Symbols
Conservation of current
Conservation of Voltage
Resistors in series and parallel
Matlab Functions
matrix addition, multiplication, subtraction
Circuit Symbols
Standard symbols for circuit elements are:
I
+
+
R
Vr
V
+
+
-
V
V
-
0V
Resistor
DC Voltage Source
AC Voltage Source
Ground
Note that current flows into the + side the voltage across a resistor (termed passive sign convention).
If the resistor dissipates heat (gets hot), current flows into the + side
If the resistor produces energy (i.e. is a battery), the current flows out of the + side.
Circuits I Concepts
Current is the flow of electrons. As such
Matter can neither be created or destroyed (conservation of energy)
Current can neither be created or destroyed (conservation of current)
What this means is that the current flowing into a node must equal the current flowing out of that node.
Conservation of Current: The current into a node must equal the current out of the node.
Example: Find the currents in the following circuit:
NDSU
4. Circuits I
pg 2
60A
80A
100A
Ia
Ic
Ib
From conservation of current
Current In = Current Out
100A = Ia + 80A
Ia = 20A
80A = Ic + 60A
Ic = 20A
60A = Id
Ib = Ic + Id = 80A
Id
NDSU
4. Circuits I
pg 3
Conservation of Voltage: The net voltage around any loop must add to zero
Example: Find the unknown voltages.
V1
+
-
+
V4
Ib
V3
V2
-
+
+
Id
-
+
55V
-
+
80V
+
Ia
-
+
60V
45V
Ic
-
-
Solution: Find a loop where you know all but one voltage, such as Ia. Summing the voltages around the
loop with a minus sign if you hit the - input first
Ia:
-80 + V2 + 60 = 0
V2 = 20V
Ic:
-60 + V3 + 45 = 0
V3 = 15V
Id:
-45 - V4 + 55 = 0
V4 = 10V
Ib:
V1 + V4 - V3 - V2 = 0
V1 = 25V
NDSU
4. Circuits I
pg 4
Simplifying Resistor Circuits: The fundamental equations for Circuits is
V=IR
12V
+
Iin
200
Iout
Resistors in Series: Resistors in series add.
Example 1:
R1
+
12V
-
R2
I
+
V3
R3
-
The current for resistors in series is the same for all three resistors (current in = current out)
The voltage across all three are:
V = IR 1 + IR 2 + IR 3
V = I ⋅ (R 1 + R 2 + R 3 )
V = I ⋅ R net
Example 2: Find the total resistance if R1 = 100, R2 = 200, R3 = 300 Ohms,.
Solition: The total resistance is the sum of the resistors
R net = 100 + 200 + 300
R net = 600Ω
NDSU
4. Circuits I
pg 5
Voltage Division: The voltage across R3 is
V 3 = IR 3
R
V 3 = ⎛⎝ R 1 +R 32 +R 3 ⎞⎠ 12V
The voltage across a resistor for resistors in series is
The resistance you’re measuring across ⎞
V R = ⎛⎝
The total resistance
⎠ ⋅The input voltage
Resistors in Parallel add as the inverse of the sum of the inverses:
Example
Ia
Vin
I1
I2
+
-
R1
R2
Current In = Current Out
Ia = I1 + I2 + I3
From V = IR
V
V
V
I a = ⎛⎝ Rin1 ⎞⎠ + ⎛⎝ Rin2 ⎞⎠ + ⎛⎝ Rin3 ⎞⎠
I a = ⎛⎝ R11 + R12 + R13 ⎞⎠ V in
V in = I a ⎛⎝ R11 + R12 + R13 ⎞⎠
−1
V in = I a R net
Example: Find the net resistance if R1 = 100, R2 = 200, R3 = 300
1
1
1 ⎞
R net = ⎛⎝ 100
+ 200
+ 300
⎠
R net = 54.54Ω
−1
I3
R3
NDSU
4. Circuits I
pg 6
Example: Determine the resistance of the following network from A to B
200
100
300
A
100
300
200
100
200
300
B
Solution: The three resistors to the right are in series. They add
200
100
A
100
900
200
100
200
B
900 in parallel with 200 is
1
1 ⎞
900 200 = ⎛⎝ 900
+ 200
⎠
−1
= 163.63Ω
200
100
A
163.63
100
100
B
200, 163, and 200 in series is 563.63 Ohms
563.63 Ohms in parallel with 100 Ohms is 84.93 Ohms
200
NDSU
4. Circuits I
pg 7
100
A
84.83
100
B
(100, 84.93, 100) in series is 284.83 Ohms
ans:
The total resistanec is 284.83 Ohms
NDSU
4. Circuits I
pg 8
Homework:
1) Use conservation of current to find the unknown currents
Ia
15
75
Ie
Ib
22
Id
Ic
100
28
27
16
If
12
84
67
Ig
2) Use conservation of voltages to find the unknown votlages
V3
+
-
+
15V
20V
V2
+
+
-
+
-
V4
-
+
+
100V
+
50V
V1
-
-
-
NDSU
4. Circuits I
pg 9
3) Find the resistance between A and B. Assume all resistors are 100 Ohms.
R
R
R
A
R
R
R
R
R
R
R
R
R
B
4) Find R so that the resistance between A and B is 200 Ohms
200
100
300
A
50
B
500
400
R
600
700