Wheatstone Bridge Equivalent Circuits

1
Wheatstone Bridge / Equivalent Circuits
Homework handout Barnaal 12 (pg48) Lab2
Wheatstone Bridge/ Equiv. Circuits Read
Oscilloscope Handout Barnaal 16-20
2
Wheatstone Bridge
Used to measure an unknown resistor Rx Resistor
Rk is known, and the two resistors R1 and R2 have
a known ratio R2 /R1, although their individual
values may not be known. A galvanometer (a
sensitive voltmeter) G measures the voltage
difference VAB between points A and B. Either the
known resistor Rk or the ratio R2 /R1 is adjusted
until the voltage difference VAB is zero and no
current flows through G. When VAB 0, the bridge
is said to be "balanced".
unknown
known
Since VAB 0, the voltage drop from C to A must
equal the voltage drop from C to B, VCA VCB.
Likewise, we must have VAD VBD. So we can write
3
Circuit Analysis Loaded Voltage Divider (1)
R1
i1
RS
i
i1
VS
R2
i2
RLoad
Use KCL and KVL to find VLoad
?
Brute Force Analysis
4
Circuit Analysis Loaded Voltage Divider (2)
Result
Reff
i
Veff
RLoad
Note A systematic application of KVL and KCL
with Ohms Law will always yield a complete
solution for the voltages and currents. In most
problems, however, one might only be interested
in the value of a single voltage or current.
There is a better way
5
Thevenin method easier!
Veff open circuit potential at the divider
output
R1
i1
RS
VS
R2
i2
6
Thevenin method (2)
Reff open circuit resistance with Vs replaced
by a short circuit
R1
RS
R2
7
Thevenins Theorem
a
a
b
b

• Any linear network may be simplified to a
  two-terminal circuit consisting of a single
  voltage source, ETh , in series with a single
  resistor, RTh.
• ETh is the equivalent open-circuit voltage across
  terminals a and b, and RTh is the equivalent
  resistance seen between the same terminals.
• (Note EThVeff and RThReff from previous
  example)
• Any mess of batteries and resistors can be
  mimicked with one battery and one resistor in
  series with it

8
Procedure for converting any circuit to its
Thevenin equivalent

• Remove the load from the circuit.
• Set all sources in the circuit to zero by
  replacing them with wires.
• RTh is obtained by calculating the resistance
  across the open-terminals ab.
• Replace the sources removed in step 2 and obtain
  Eth by calculating the open-circuit voltage
  across terminals ab.

9
Thevenins Theorem
a
a
R
Network A
Network B
Th
E
Th
R
L
b
b
a
a
Network A (with independent sources removed)
Network A

E
Th
R
Th
-
b
b
10
Example problem (with your neighbor)
Find the Thevenin equivalent circuit! Note the
load has been already removed!
RTh?
R1
A
A
V0
ETh?
R2
B
B
R3
11
Nortons Theorem

• Any linear network may be simplified to a
  two-terminal circuit consisting of a single
  current source, IN , and a single shunt resistor,
  RN.
• IN is the equivalent short-circuit current
  between terminals a and b, and RN is the
  equivalent resistance seen between the same
  terminals.

12
Procedure for converting any circuit to its
Norton equivalent

• Remove the load from the circuit.
• Set all sources in the circuit to zero.
• RN is obtained by calculating the resistance
  across the open-terminals ab. (note RNRTh)
• Replace the sources removed in step 2 and obtain
  IN by calculating the short-circuit current
  between terminals ab.

13
Nortons Theorem
a
a
Network A
Network B
I
N
RN
R
L
b
b
a
a
Network A (with independent sources removed)
Network A
IN
R
N
b
b
14
Example problem (with your neighbor)
Find the Norton equivalent circuit! Note the
load has been already removed!
R1
A
A
V0
IN?
R2
RN?
B
B
R3
(double-check your answer with earlier example)
15
Voltage source
V0

• a perfect voltage source is a two-terminal black
  box that maintains a fixed voltage drop across
  its terminals, regardless of the load resistance.
  This means that it must supply a current I V /
  R when a resistance R is attached to its
  terminals
• a real voltage source can only supply a finite
  maximum current, and behaves like a perfect
  voltage source with a small resistance in series

16

• Real voltage sources (like batteries) have
  internal resistance

Rs


Vs
RL
Depends on current leaving battery!
-
-
-
Battery
Current in circuit
A voltage source likes an open-circuit load and
hates a short-circuit load
17
Current source

• a perfect current source is a two-terminal black
  box that maintains a constant current through the
  external circuit, regardless of load resistance
  or applied voltage. In order to do this it must
  be capable of supplying any necessary voltage
  across its terminals.
• real current sources have a limit to the voltage
  they can provide, they do not provide an absolute
  constant output current
• A current source likes short-circuit load and
  hates open-circuit load

18
Source Conversion

• A voltage source can be converted into a current
  source
• The voltage source is represented by its Thevenin
  equivalent with VEth, Rs RTh
• The current source is represented by its Norton
  equivalent with IIN, RsRNRTh
• Note INETh/RN , RNRTh

Ideal constant-current source
I
RS
I
E
RS
E IRS
I E/RS
Source conversion
19
Voltage Transfer
Current Transfer
i
RS


i
iS
VS
RL
RS
V
V
RL
-
-
(current divider)
(voltage divider)
Reference Senturia
20
Power Transfer
RS

i
VS
V
RL
-
Power dissipated in RL
Maximum power transfer for
(show in homework)
Then
21
Maximum Power Transfer Theorem
a
b

• A load resistance will receive maximum power from
  a circuit when the resistance of the load is
  exactly the same as the Thevenin (or Norton)
  resistance looking back at the circuit.
• The maximum power delivered to the load is

22
Superposition Theorem
E
E
I

I
R1
R1
R2
R1
R2
R2

• The total current through or voltage across a
  resistor or branch may be determined by summing
  the effects due to each independent source.
• Replace all voltage sources with a short circuit
  and all current sources with an open circuit,
  except for the one source to be examined.

23
Superposition Theorem (2)
I
KCL
KVL
(2 equations with 2 unknowns i1 i2)
brute force
Solve and find
24
Superposition Theorem (3)
(same result)