EECS 40
Fall 2002 Lecture 8
Copyright, Regents University of California
S. Ross and W. G. Oldham
1
EECS 40
Fall 2002 Lecture 8
Copyright, Regents University of California
S. Ross and W. G. Oldham
SERIES ELEMENTS
KCL tells us that all of the elements in a single branch carry
the same current.
We say these elements are in series.
Current entering node = Current leaving node
i1 = i 2
2
EECS 40
Fall 2002 Lecture 8
Copyright, Regents University of California
S. Ross and W. G. Oldham
RESISTORS IN SERIES
Circuit with several resistors in series: Find “equivalent resistance”
I
R1
VSS
+

R2
R3
R4
• KCL tells us same current
flows through every resistor
• KVL tells us
I  R1  I  R2  I  R3  I  R4  VSS  0
• Clearly,
VSS  I (R1  R2  R3  R4 )
Thus, equivalent resistance of resistors in series is the sum
3
EECS 40
Fall 2002 Lecture 8
S. Ross and W. G. Oldham
Copyright, Regents University of California
VOLTAGE DIVIDER
Circuit with several resistors in series
• We know
I
VSS +

R1
R2
R3
R4
+V
 1
I  VSS /(R1  R2  R3  R4 )
• Thus,
+V
 3
V1 
R1
 VSS
R1  R2  R3  R 4
and
V3 
R3
 VSS
R1  R2  R3  R 4
etc…
4
EECS 40
Fall 2002 Lecture 8
S. Ross and W. G. Oldham
Copyright, Regents University of California
WHEN IS VOLTAGE DIVIDER FORMULA CORRECT?
I
V
SS
+

R1
R2
I
+

V2
R3
R4
R
2
V 
V
2
SS
R R R R
1 2 3 4
Correct if nothing else
connected to nodes
V +
SS
R1
R2
R3
R4
+V
2
I
3
R5
R
2
V ≠
V
2
SS
R R R R
1 2 3 4
because R5 removes condition of
resistors in series I3  I
5
EECS 40
Fall 2002 Lecture 8
S. Ross and W. G. Oldham
Copyright, Regents University of California
MEASURING CURRENT
To measure current in a circuit, insert DMM (in current mode) into
circuit, in series with measured element.
But ammeters change the circuit. Ammeters are characterized by
their “ammeter input resistance,” Rin. Ideally this should be very
low. Typical value 1W.
Real
Ammeter
?
Ideal
Ammeter
Rin
6
EECS 40
Fall 2002 Lecture 8
S. Ross and W. G. Oldham
Copyright, Regents University of California
MEASURING CURRENT
Potential measurement error due to non-zero input resistance:
Imeas
I
R1
R1
+
V _
V
_+
undisturbed circuit
R in
R2
R2
V1
I
R1  R2
ammeter
with ammeter
V1
Imeas 
R1  R2  Rin
Example: V = 1 V: R1= R2 = 500 W, Rin = 1W
1V
1V
I
 1mA, Imeas 
 0.999 mA
500W  500W
500W  500W  1W
7
EECS 40
Fall 2002 Lecture 8
S. Ross and W. G. Oldham
Copyright, Regents University of California
PARALLEL ELEMENTS
KVL tells us that any set of elements which are connected at
both ends carry the same voltage.
We say these elements are in parallel.
KVL clockwise,
start at top:
Vb – Va = 0
Va = Vb
8
EECS 40
Fall 2002 Lecture 8
S. Ross and W. G. Oldham
Copyright, Regents University of California
RESISTORS IN PARALLEL
Resistors in parallel can be made into one equivalent resistor
x
• KCL tells us
ISS
I2
I1
R1
ground
R2
Iss = I1 + I2
• The two resistors are
in parallel; they have
the same voltage
VX = I1 R1 = I2 R2
Iss = VX / R1 + VX / R2
VX = Iss R1 R2 / (R1+R2)
Generally, Req = (R1-1 + R2-1 + R3-1 + …)-1
9
EECS 40
Fall 2002 Lecture 8
Copyright, Regents University of California
S. Ross and W. G. Oldham
IMPORTANT FACTS
• For resistors in series:
– Current through Req is equal to the current through
each of the original resistors (all have same current)
– Voltage over Req is the sum of the voltages over the
original resistors
• For resistors in parallel:
– Current through Req is equal to the sum of the
currents through each of the original resistors
– Voltage over Req is equal to the voltage over the
original resistors (all have same voltage)
10
EECS 40
Fall 2002 Lecture 8
S. Ross and W. G. Oldham
Copyright, Regents University of California
CURRENT DIVIDER
There is a simple equation for the way current splits between
two parallel resistors: x
• Remember
I1
ISS
I2
R1
VX = I1 R1 = ISS Req
R2
ground
V
I1  X
R1
 R1R 2 


ISS Req
R1 R 2 
R2


 ISS
 ISS
R1
R1
R1  R2
11
EECS 40
Fall 2002 Lecture 8
S. Ross and W. G. Oldham
Copyright, Regents University of California
REAL VOLTMETERS
How is voltage measured? Digital multimeter (DMM) in parallel
with measured element.
Connecting a real voltmeter across two nodes changes the
circuit. The voltmeter may be modeled by an ideal voltmeter
(open circuit) in parallel with a resistance: “voltmeter input
resistance,” Rin. Typical value: 10 MW
Real
Voltmeter
Ideal
Voltmeter
Rin
12
EECS 40
Fall 2002 Lecture 8
S. Ross and W. G. Oldham
Copyright, Regents University of California
REAL VOLTMETERS
Computation of voltage
(uses ideal voltmeter)
Measurement of voltage
(with loading by real voltmeter)
R1
VSS
+-
R1
R2
 R2 
V2  VSS 

R

R
 1
2
+
V2
-
VSS
+-
R2
Rin
+
V2
-
 R2 || Rin 
V2  VSS 

R
||
R

R
 2
in
1
Example: VSS  10V, R2  100K, R1  900K  V2  1V
But if Rin  10M, V2  0.991V, a 1% error
13
EECS 40
Fall 2002 Lecture 8
S. Ross and W. G. Oldham
Copyright, Regents University of California
CAPACITORS IN PARALLEL
+
i(t)
V

Ceq
|(
C2
|(
C1
|(
i(t)
+
V(t)

Equivalent capacitance defined by
dV
i(t)  Ceq
dt
i( t )  C1
dV
dV
 C2
dt
dt
Ceq  C1  C2
14
EECS 40
Fall 2002 Lecture 8
+ V1  + V2 
i(t)
S. Ross and W. G. Oldham
Copyright, Regents University of California
|(
|(
C1
C2
CAPACITORS IN SERIES
+ Veq 
|(
Equivalent to
i(t)
C1C2
Ceq 

1
1
C1  C2

C1 C2
Ceq
1
15
EECS 40
Fall 2002 Lecture 8
Copyright, Regents University of California
S. Ross and W. G. Oldham
DIGITAL CIRCUIT: DRAM
t 0
+
VC CC
R
initially
uncharged
CB
-
16