lect3eee 2021 voltage and current division; superposition dr. holbert january 23, 2008
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Lect3 EEE 202 2
Single Loop Circuit
• The same current flows through each element of the circuit—the elements are in series
• We will consider circuits consisting of voltage sources and resistors
+–
VS
R
R
R
I
Lect3 EEE 202 3
Solve for I
• In terms of I, what is the voltage across each resistor? Make sure you get the polarity right!
• To solve for I, apply KVL around the loop
+–
VS
R
R
R
I + –
I R
+
–
I R
I R
+
–
N Total Resistors
IR + IR + … + IR – VS = 0
I = VS / (N R)
Lect3 EEE 202 4
In General: Single Loop
• The current i(t) is:
• This approach works for any single loop circuit with voltage sources and resistors
• Resistors in series
sresistanceofsumsourcesvoltageofsum
RV
tij
Si )(
jNseries RRRRR 21
Lect3 EEE 202 5
Voltage Division
Consider two resistors in series with a voltage v(t) across them:
R1
R2
–
v1(t)
+
+
–
v2(t)
+
–
v(t)21
11 )()(
RR
Rtvtv
21
22 )()(
RR
Rtvtv
Lect3 EEE 202 6
In General: Voltage Division
• Consider N resistors in series:
• Source voltage(s) are divided between the resistors in direct proportion to their resistances
j
iSR R
RtVtV
ki)()(
Lect3 EEE 202 8
Apply KCL with Ohm’s Law
212121
11
RRV
R
V
R
VIII
I R1 R2 V
+
–
I1 I2
21
21
21
111
RR
RRI
RR
IV
Lect3 EEE 202 9
Equivalent Resistance
If we wish to replace the two parallel resistors with a single resistor whose voltage-current relationship is the same, the equivalent resistor has a value of:
Definition: Parallel - the elements share the same two end nodes
21
21
RR
RRReq
Lect3 EEE 202 10
21
2
1
21
21
11 RR
RI
R
RRRR
I
R
VI
Now to find I1
• This is the current divider formula
• It tells us how to divide the current through parallel resistors
Lect3 EEE 202 11
Three Resistors in Parallel
I= I1 + I2 + I3
11 R
VI
22 R
VI
33 R
VI
I R2 V
+
–
R1
I1 I2
R3
I3
Lect3 EEE 202 13
Equivalent Resistance (Req)
321
1111
RRR
Req
iMpar RRRRR
11111
21
Which is the familiar equation for parallel resistors:
Lect3 EEE 202 14
Current Divider
• This leads to a current divider equation for multiple parallel resistors
• For 2 parallel resistors, it reduces to a simple form
• Note this equation’s similarity to the voltage divider equation
j
parSR R
RII
j
Lect3 EEE 202 16
Apply KCL at the Top Node
21212121
11
RRV
R
V
R
VIIII ss
21
2121 RR
RRIIV ss
Is1 Is2 VR1 R2
+
–
I1 I2
Lect3 EEE 202 17
Multiple Current Sources
• We find an equivalent current source by algebraically summing current sources
• As before, we find an equivalent resistance
• We find V as equivalent I times equivalent R
• We then find any necessary currents using Ohm’s law
Lect3 EEE 202 18
In General: Current Division
• Consider N resistors in parallel:
• Special Case (2 resistors in parallel)
iNpar
j
parSR
RRRRR
R
Rtiti
kj
11111
)()(
21
21
2)()(1 RR
Rtiti SR
Lect3 EEE 202 19
Superposition
“In any linear circuit containing multiple independent sources, the current or voltage at any point in the circuit may be calculated as the algebraic sum of the individual contributions of each source acting alone.”
Lect3 EEE 202 20
How to Apply Superposition
• To find the contribution due to an individual independent source, zero out the other independent sources in the circuit
– Voltage source short circuit
– Current source open circuit
• Solve the resulting circuit using your favorite technique(s)
Lect3 EEE 202 21
Superposition of Summing Circuit
+
–
Vout
1k
1k
1k
V1 V2
+–
+–
+
–
V’out
1k
1k
1k
V1
+
–
V’’out
1k
1k
1k
V2++–
+–
Lect3 EEE 202 22
Use of Superposition
V’out = V1/3
V’’out = V2/3
Vout = V’out + V’’out = V1/3 + V2/3
+
–
V’out
1k
1k
1k
V1
+
–
V’’out
1k
1k
1k
V2++–
+–
Lect3 EEE 202 23
Superposition Procedure
1. For each independent voltage and current source (repeat the following): a) Replace the other independent voltage sources with
a short circuit (i.e., V = 0). b) Replace the other independent current sources with
an open circuit (i.e., I = 0). Note: Dependent sources are not changed!
c) Calculate the contribution of this particular voltage or current source to the desired output parameter.
2. Algebraically sum the individual contributions (current and/or voltage) from each independent source.