Transcript Phys 345 Electronics for Scientists

Announcements
• First Assignment posted:
– Due in class in one week (Thursday Sept 15th)
http://www.physics.udel.edu/~jholder/Phys645/index.htm
Lecture 3 Overview
• Loop Analysis with KVL & KCL
• Mesh Analysis
• Thevenin/Norton equivalent circuits
Circuit analysis method 2a: KVL and KCL
Kirchoff’s Voltage Law: Loop analysis
The sum of the voltages around a closed loop must be zero
• Draw the current direction in every
branch (arbitrary) and label the voltage
directions (determined by the defined
current direction). Voltage on a voltage
source is always from positive to negative
end.
• Define either clockwise or counterclockwise as positive direction for
summing voltages. Once the direction is
defined, use the same convention in every
loop. Voltage across a resistor is +’ve if
voltage direction the same as current
direction, -’ve otherwise
• Apply KVL
Kirchoff’s Voltage Law: Multiloop
The sum of the voltages around a closed loop must be zero
• Draw the current direction (arbitrary)
and label the voltage directions
(determined by the defined current
direction).
• Define either clockwise or counterclockwise as positive voltage direction.
the direction is defined, use the
R3 Once
same convention in every loop.
• Apply KVL
   Vr  V1  V2  0
 V2  V3  0
  Ir  IR1  I 2 R2
 I 2 R2  I 3 R3  0
Say r=1Ω, R1=3Ω, R2=5Ω, R3=10Ω, ε=3V
What are the currents?
Kirchoff’s Current Law
The sum of the current at a node must be zero: Iin=Iout
I=I2+I3
(1)
ε=Ir+IR1+I2R2 (2)
I3R3-I2R2 =0 (3)
R3
Say r=1Ω, R1=3Ω, R2=5Ω, R3=10Ω, ε=3V
1I- 1I2- 1I3 = 0
(4)
4I+5I2+ 0I3= 3
(5)
0I- 5I2+10I3= 0 (6)
Last note on KCL KVL analysis
• If solutions to currents or voltages are
negative, this just means the real direction
is opposite to what you originally defined
• To deal with current sources: current is
known, but assign a voltage across it
which has to be solved
Another Sample Problem: Multiple Sources
Method 2b: Mesh Analysis
Example: 2 meshes
(Mesh is a loop that does not contain other loops)
Step 1: Assign mesh currents clockwise
Step 2: Apply KVL to each mesh
•
The self-resistance is the effective resistance of the resistors in series within
a mesh. The mutual resistance is the resistance that the mesh has in common
with the neighbouring mesh
• To write the mesh equation, evaluate the self-resistance, then multiply by the
mesh current
• Next, subtract the mutual resistance multiplied by the current in the
neighbouring mesh for each neighbour.
• Equate the above result to the driving voltage: taken to be positive if it tends
to push current in the same direction as the assigned mesh current
Mesh1: (R1+R2)I1
Mesh2:
- R2I2
=ε1-ε2
-R2I1+ (R2+R3)I2
=ε2-ε3
Step 3: solve currents
Method 2b: Mesh Analysis
Another mesh analysis example
Find the currents in each branch
Step 1: Replace any combination of resistors in
series or parallel with their equivalent resistance
Step 2: Assign mesh currents clockwise
Step 3: Write the mesh equations for each mesh
Left mesh: 11I1-6I2=9
Right mesh: -6I1+18I2=9
Note:suppressed “k” for each resistor, so answer is in mA
Step 4: Solve the equations
Solution: I1=4/3mA
=1.33mA
I2=17/18mA =0.94mA
Mesh analysis with a current source
Magnitude of current in branch containing current source is IS ,
(although if the current flow is opposite to the assigned current
direction the value will be negative).
This works only if the current source is not shared by any other mesh
For a shared current source, label it with an unknown voltage.
Mesh analysis with mixed sources
• Find Ix
• Identify mesh currents and label accordingly
• Write the mesh equations
Mesh1:
I1
=-2
Mesh2: -4I1+8I2
-4I4=12
Mesh3:
8I3
=-12
Mesh4: -2I1-4I2
+6I4=10
I1= -2.0A
I2= 1.5A
I3= -1.5A
I4= 2.0A
Ix=I2-I3
Ix=3.0A
Method 3: Thevenin and Norton
Equivalent Circuits
Any network of sources
and resistors will appear
to the circuit connected
to it as a voltage source
and a series resistance
vTH= open circuit voltage at
terminal (a.k.a. port)
RTH= Resistance of the
network as seen from port
(Vm’s, In’s set to zero)
Norton Equivalent Circuit
Any network of sources
and resistors will appear
to the circuit connected
to it as a current source
and a parallel resistance
Calculation of RT and RN
• RT=RN ; same calculation (voltage and current sources set to zero)
• Remove the load.
• Set all sources to zero (‘kill’ the sources)
– Short voltage sources (replace with a wire)
– Open current sources (replace with a break)
Calculation of RT and RN continued
• Calculate equivalent resistance seen by the load
Calculation of VT
• Remove the load and calculate the open circuit voltage
VOC  VR 2 
R2
VS
R1  R2
Calculation of IN
• Short the load and calculate the short circuit current
(R1+R2)i1 - R2iSC = vs
-R2i1 + (R2+R3)iSC = 0
KCL
Source Transformation
Summary: Thevenin’s Theorem
• Any two-terminal linear circuit can be replaced with a voltage source
and a series resistor which will produce the same effects at the
terminals
• VTH is the open-circuit voltage VOC between the two terminals of the
circuit that the Thevenin generator is replacing
• RTH is the ratio of VOC to the short-circuit current ISC; In linear circuits
this is equivalent to “killing” the sources and evaluating the
resistance between the terminals. Voltage sources are killed by
shorting them, current sources are killed by opening them.
Summary: Norton’s Theorem
• Any two-terminal linear circuit can be replaced with a current source
and a parallel resistor which will produce the same effects at the
terminals
• IN is the short-circuit current ISC of the circuit that the Norton
generator is replacing
• Again, RN is the ratio of VOC to the short-circuit current ISC; In linear
circuits this is equivalent to “killing” the sources and evaluating the
resistance between the terminals. Voltage sources are killed by
shorting them, current sources are killed by opening them.
• For a given circuit, RN=RTH
Maximum Power Transfer
• Why use Thevenin and Norton equivalents?
– Very easy to calculate load related quantities
– E.g. Maximum power transfer to the load
• It is often important to design circuits that
transfer power from a source to a load.
There are two basic types of power transfer
– Efficient power transfer (e.g. power utility)
– Maximum power transfer (e.g.
communications circuits)
• Want to transfer an electrical signal
(data, information etc.) from the source
to a destination with the most power
reaching the destination. There is limited
power at the source and power is small
so efficiency is not a concern.
Maximum Power Transfer: Impedance matching
2
 vT 
 RL
p  i RL  
 RT  RL 
2
Differentiate using quotient rule:
du
dv
u
d u
 dx 2 dx
dx v
v
v
Set to zero to find maximum:
2
dp
2  ( RT  RL )  2 RL ( RT  RL ) 
  0
 vT 
4
dRL
( RT  RL )


vT2 ( RT  RL ) 2 2 RL vT2 ( RT  RL )

( RT  RL ) 4
( RT  RL ) 4
so maximum power transfer occurs when
RT  RL
http://circuitscan.homestead.com/files/ancircp/maxpower1.htm
and
vT2
pmax 
4 RL