Download Circuit Theory Chapter 4

Chap. 4 Circuit Theorems
Review of Chapter 3
• Nodal analysis
– Voltage variables (node)
– Current law (KCL)
– Voltage sources  Supernode
• Mesh analysis
– Current variables (mesh)
– Voltage law (KVL)
– Current sources  Supermesh
𝑣1
𝑅
𝑣2
𝑣1 − 𝑣2
𝑅
+ 𝑅𝑖 −
𝑅
𝑖
Linearity Property
• Linearity property is a combination of both the
homogeneity (scaling) property and the additivity
property.
– Homogeneity Property: If the input (also called the excitation) is
multiplied by a constant, then the output (also called the
response) is multiplied by the same constant.
v = iR
kv = kiR
– Additivity Property: The response to a sum of inputs is the sum
of the responses to each input applied separately.
If v1 = i1R and v2 = i2R
then v = (i1+i2)R = i1R + i2R = v1 + v2
Linear circuit
• A Linear circuit is one whose output is linearly related (or
directly proportional) to its input.
– A linear circuit consists of only linear elements, linear dependent
sources and independent sources.
Ex. A resistor is a linear element because v-i relationship
satisfies the linearity property (v: output, i: input).
※ The relationship between power and voltage (or current)
is nonlinear.
Example 4.1 (For linearity property)
Example 4.2 (For linearity property)
Superposition Principle
•
The superposition principle states that the voltage
across (or current through) an element in a linear circuit
is the algebraic sum of the voltage across (or currents
through) that element due to each independent source
acting alone.
•
Assuming a linear circuit consisting of M independent
voltage sources 𝑣𝑚 and N independent current sources
𝑖𝑛 , the voltage across (or current through) an element in
𝑁
the linear circuit is given as 𝑀
𝐴
𝑣
+
𝑚=1 𝑚 𝑚
𝑛=1 𝐵𝑛 𝑖𝑛
where the coefficients 𝐴𝑚 and 𝐵𝑛 are given by the
circuit.
Superposition Principle
•
Keep these in mind:
– We consider one independent source at a time while all other
independent sources are turned off. This implies that we replace
every voltage source by 0V (or a short circuit), and every
current source by 0A (or an open circuit). This way we obtain a
simpler and more manageable circuit.
– Dependent sources are left intact because they are controlled
by circuit variables.
Steps to Apply Superposition Principle
1. Turn off all independent sources except one source.
Find the output (voltage or current) due to that active
source using the techniques covered in Chap. 2 and 3.
2. Repeat step 1 for each of the other independent
sources.
3. Find the total contribution by adding algebraically all the
contributions due to the independent sources.
※ It is not applicable to the effect on power due to each
source.
Example 4.3 (For superposition principle)
Find v.
Example 4.3 (For superposition principle)
Example 4.4 (For superposition principle)
Find i0.
Example 4.4 (For superposition principle)
Example 4.5 (For superposition principle)
Example 4.5 (For superposition principle)
Source Transformation
• A source transformation is the process of replacing a
voltage source in series with a resistor R by a current
source is in parallel with a resistor R, or vice versa.
• Keep these in mind:
– The arrow of the current source is directed toward the positive
terminal of the voltage source.
– Source transformation is not possible when R = 0, which is the
case with an ideal voltage source. Similarly, an ideal current
source with R = ∞ cannot be replaced by a finite voltage source.
Source Transformation
• V-I characteristics
𝑖
𝑣
𝑣 = 𝑅𝑖 + 𝑣𝑠
𝑣𝑠 = 𝑅𝑖𝑠
𝑖
𝑣
𝑣 = 𝑅𝑖 + 𝑅𝑖𝑠
Fig. 4.15 Transformation of independent sources
vs = isR
is = vs/R
Fig. 4.16 Transformation of dependent sources
vs = isR
is = vs/R
Example 4.6
Find v0.
Example 4.6
Example 4.7
Find vx.
Example 4.7
Thevenin’s theorem
• Thevenin’s theorem states that a linear two-terminal
circuit can be replaced by an equivalent circuit consisting
of a voltage source VTh in series with a resistor RTh,
where VTh is the open-circuit voltage at the terminals and
RTh is the input or equivalent resistance at the terminals
when the independent sources are turned off.
𝑉 = 𝐴𝐼 + 𝐵
𝑉 = −𝑅𝑇ℎ 𝐼 + 𝑉𝑇ℎ
Thevenin’s theorem
• What’s the relationship between v and i?
𝑖
Linear
two-terminal
circuit
+
𝑣
𝑣𝑠
𝑖𝑠
−
Thevenin’s theorem
• Apply superposition principle
Linear
two-terminal
circuit
+
𝑣
𝑣𝑠
𝑖𝑠
𝑖
−
𝑣 = 𝐴𝑖 + 𝐵𝑣𝑠 + 𝐶𝑖𝑠
Thevenin’s theorem
• Replace a linear circuit with the equivalent circuit
𝑉𝑇ℎ
𝐵𝑣𝑠 + 𝐶𝑖
𝑠
𝑅𝐴𝑇ℎ
Linear
two-terminal
circuit
𝑖
+
𝑣
𝑣𝑠
𝑖𝑠
−
𝑅𝑇ℎ
+ 𝑠𝑉+
𝑣𝑣==𝐴𝑖
+ 𝑖𝐵𝑣
𝑇ℎ𝐶𝑖𝑠
𝑉𝑇ℎ = 𝐵𝑣𝑠 + 𝐶𝑖𝑠
𝑅𝑇ℎ = 𝐴
How to find VTh
• VTh : open circuit voltage across the terminals a & b
𝑖=0
𝑖=0
𝑣𝑜𝑐 = 𝑉𝑇ℎ
𝑣𝑜𝑐 = 𝑅𝑇ℎ 𝑖 + 𝑉𝑇ℎ
⇒ 𝑣𝑜𝑐 = 𝑉𝑇ℎ
𝑖=0
𝑣𝑜𝑐 = 𝑉𝑇ℎ
How to find RTh
• RTh : input resistance at the terminals when the independent sources are turned off
𝑖
(𝑣𝑠 = 0, 𝑖𝑠 = 0)
𝑅𝑖𝑛 = 𝑅𝑇ℎ
𝑖
+
+
𝑣
𝑣
−
−
𝑉𝑇ℎ = 𝐵𝑣𝑠 + 𝐶𝑖𝑠 = 0
𝑅𝑖𝑛 = 𝑅𝑇ℎ
𝑣 = 𝑅𝑇ℎ 𝑖 + 𝑉𝑇ℎ
⇒ 𝑣 = 𝑅𝑇ℎ 𝑖
𝑉𝑇ℎ = 𝐵𝑣𝑠 + 𝐶𝑖𝑠 = 0
𝑅𝑖𝑛
How to find RTh if the network has dependent sources?
1. Turn off all independent sources.
2. Dependent sources are not to be turned off.
3. Apply a voltage source v0 at terminals a and b
4. Determine the resulting current i0.
5. Then RTh = v0 / i0.
(𝑣𝑠 = 0, 𝑖𝑠 = 0)
How to find RTh if the network has dependent sources?
Alternatively,
1. Turn off all independent sources.
2. Dependent sources are not to be turned off.
3’. Insert a current source i0 at terminals a-b
4’. Find the terminal voltage v0.
5’. Again RTh = v0 / i0.
(𝑣𝑠 = 0, 𝑖𝑠 = 0)
Thevenin’s Theorem:
Replacement of Large Circuit by a Single Independent Voltage source and a Single Resistor
Original circuit with a load
Thevenin equivalent circuit
VTh
IL 
RTh  RL
RL
VL  RL I L 
VT h
RT h  RL
Example 4.8
Find the current through RL.
Example 4.8
Replace a linear circuit by the equivalent circuit
𝑅𝑇ℎ
𝑉𝑇ℎ
Example 4.8
Find VTh
VTh : open circuit voltage across the terminals a & b
𝑖=0
+
𝑣𝑜𝑐 = 𝑉𝑇ℎ
−
𝑖=0
𝑅𝑇ℎ
+
𝑉𝑇ℎ
𝑣𝑜𝑐 = 𝑉𝑇ℎ
−
Example 4.8
Find VTh
Apply mesh analysis
𝑖=0
+
𝑣𝑜𝑐 = 𝑉𝑇ℎ
−
𝑖=0
𝑅𝑇ℎ
+
𝑉𝑇ℎ
𝑣𝑜𝑐 = 𝑉𝑇ℎ
−
Example 4.8
Find RTh
RTh : input resistance at the terminals when the independent sources are turned off
𝑖
𝑅𝑇ℎ
+
𝑉𝑇ℎ = 0
𝑣
−
𝑅𝑇ℎ
𝑣 = 𝑅𝑇ℎ 𝑖
∴ 𝑅𝑇ℎ = 𝑣/𝑖
Example 4.8
• Apply Thevenin Theorem
Thevenin equivalent circuit
Example 4.9
Example 4.9
Find VTh
VTh : open circuit voltage across the terminals a & b
𝑖=0
+
𝑣𝑜𝑐 = 𝑉𝑇ℎ
−
𝑖=0
𝑅𝑇ℎ
+
𝑉𝑇ℎ
𝑣𝑜𝑐 = 𝑉𝑇ℎ
−
Example 4.9
Find RTh
𝑅𝑇ℎ
𝑉𝑇ℎ = 0
𝑣0 = 𝑅𝑇ℎ 𝑖0
𝑣0 1
∴ 𝑅𝑇ℎ =
=
𝑖0 𝑖0
Example 4.10
※ No independent source⇒ No VTh
Example 4.10
• Find RTh
𝑣0 = 𝑅𝑇ℎ 𝑖0
𝑣0
∴ 𝑅𝑇ℎ =
= 𝑣0
𝑖0
𝑖0 = 1
Example 4.10
• Apply Thevenin’s Theorem
Norton’s theorem
• Norton’s theorem states that a linear two-terminal circuit
can be replaced by an equivalent circuit consisting of a
current source IN in parallel with a resistor RN, where IN is
the short-circuit current through the terminals and RN is
the input or equivalent resistance at the terminals when
the independent sources are turned off.
𝑣 = 𝐴𝑖 + 𝐵
𝑣 = 𝑅𝑁 𝑖 + 𝑅𝑁 𝐼𝑁
𝑖
𝑖
+
𝑣
−
+
𝑣
−
Norton’s theorem
• What’s the relationship between v and i?
𝑖
Linear
two-terminal
circuit
+
𝑣
𝑣𝑠
𝑖𝑠
−
Norton’s theorem
• Apply superposition principle
𝑖
Linear
two-terminal
circuit
𝑣𝑠
𝑣
𝑖𝑠
𝑖 = 𝐷𝑣 + 𝐸𝑣𝑠 + 𝐹𝑖𝑠
Norton’s theorem
• Replace a linear circuit with the equivalent circuit
𝑖
Linear
two-terminal
circuit 1
𝐼
−(𝐸𝑣𝑠 + 𝐹𝑖𝑠𝑁)
𝑅𝑁
𝐷
𝑣𝑠
𝑖𝑠
+
𝑣
−
1 𝑅𝑁 𝑖1+ 𝑅𝑁 𝐼𝑁
𝑣=
𝑣 = 𝑖 − (𝐸𝑣𝑠 + 𝐹𝑖𝑠 )
𝐼𝑁 =
𝐷 −(𝐸𝑣
𝐷 𝑠 + 𝐹𝑖𝑠 )
𝑅𝑁 = 1/𝐷
How to find IN
• IN : short circuit current from terminals a to b
𝑖𝑠𝑐 = −𝐼𝑁
𝑖𝑠𝑐 = −𝐼𝑁
+
+
𝑣𝑣 =
= 00
−
−
+
+
𝑣𝑣==00
−−
𝑣 = 𝑅𝑁 𝑖𝑠𝑐 + 𝑅𝑁 𝐼𝑁
𝑣=0
⟹ 𝑖𝑠𝑐 = −𝐼𝑁
How to find RN
• RN : input resistance at the terminals when the independent sources are turned off
𝑖
𝑖
+
𝑣
(𝑣𝑠 = 0, 𝑖𝑠 = 0)
+
𝑣
𝐼𝑁 = 0
−
𝑅𝑖𝑛 = 𝑅𝑁
𝑣 = 𝑅𝑁 𝑖 + 𝑅𝑁 𝐼𝑁
⇒ 𝑣 = 𝑅𝑁 𝑖
𝐼𝑁 = −(𝐸𝑣𝑠 + 𝐹𝑖𝑠 ) = 0
−
𝑅𝑖𝑛 = 𝑅𝑁
𝑅𝑖𝑛
Thevenin’s vs. Norton’s
• Source transform
𝑖
+
𝑣
−
𝑣 = 𝑅𝑇ℎ 𝑖 + 𝑉𝑇ℎ
𝑅𝑇ℎ = 𝑅𝑁
𝑉𝑇ℎ = 𝑅𝑁 𝐼𝑁
𝑖
+
𝑣
−
𝑣 = 𝑅𝑁 𝑖 + 𝑅𝑁 𝐼𝑁
To determine the Thevenin or Norton equivalent
circuit, we need to find:
𝑖=0
𝑣𝑜𝑐 = 𝑉𝑇ℎ
𝑖𝑠𝑐 = −𝐼𝑁
+
𝑣𝑣==00
−−
𝑖
+
𝑣
(𝑣𝑠 = 0, 𝑖𝑠 = 0)
−
𝑅𝑇ℎ = 𝑅𝑁
voc
RT h 
 RN
isc
Example 4.11
Find the Norton equivalent circuit.
Example 4.11
• Find RN
Example 4.11
• Find IN
Example 4.11
• Find VTh
Example 4.12
Find RN and IN.
Example 4.12
• Find RN
𝑣0 = 𝑅𝑁 𝑖0
𝑣0 1
∴ 𝑅𝑁 =
=
𝑖0 𝑖0
Example 4.12
• Find IN
Power delivered to the load as a function of RL
• Maximum power is transferred to the load when the load
resistance equals the Thevenin resistance as seen from
the load (RL = RTh).
2
 VT h 
 RL
p  i 2 RL  
 RT h  RL 
dp
0
dRL
RL  RT h
pmax
V 2T h

4RT h
Example 4.13
For maximum power transfer, RL=?
Example 4.13
• Find RTh
Example 4.13
• Find VTh
Example 4.13
• Apply Thevenin’s Theorem
Homework
• Problem 4.3, 4.12, 4.26, 4.37, 4.39, 4.42, 4.63, 4.64,
4.67
• Due date: April 12, 2013 (before class starts)