ELEC1103_Fundamentals Of Elec And Electronic Engineering_2014 Semester 1_Ch4-additional

I
=
1 L
2 IL
V
1 L
+
VL2
Circuit with current source set to zero(OPEN) Linearity or the principle of source superposition says:
1 2 IL = IL + IL
VL = VL1 + VL2
211
Linearity can never be verified absolutely empirically on a system, but it could be disproved by a single counter example. However, mathematically it can be verified for the models that we use.
source to zero and solving the circuit.
ELEC1103
210
SUPERPOSITION
Ch4. VII. Source Superposition
SOURCE SUPERPOSITION Circuit with voltage source set to zero (SHORT CIRCUITED)
ELEC1103
213
SUPERPOSITION
Ch4. VII. Source Superposition
+
Thus, by superposition we have:
Req = 3 + 3 || 6 [k ] R = 6 + (3 || 3) [k ] eq
Contribution of v1 Contribution of v2
ELEC1103 – Semester 1
Chapter 4: Additional Circuit Analysis Methods
Dr. Craig Jin 16 April 2012
ELEC1103
202
ADDITIONAL CIRCUIT CONCEPTS
Ch4. V. Circuit Analysis Techniques – Additional Concepts & Techniques › The next slides introduce concepts that are fundamental throughout engineering and science:
7 −5 −2 i1 10 −5 20 −10 i = 0 2 −2 −10 22 i3 0
You can think of linearity as meaning that the operation of addition or multiplication in the input to the system influences the output of the system in a similar manner. We sometimes refer to the linearity requirement as a superposition requirement. In other words, people will say that linearity means that superposition holds. What they mean, is exactly the mathematical definition given above.
ELEC1103
205
LINEARITY
Ch4. VI. Linearity › Linearity:
Using nodal analysis for resistive circuits one obtains models that have the form:
Av = Bs
Using mesh analysis for resistive circuits one obtains models that have the form:
We set to zero the voltage source
Now we set to zero the current source
V0 = V0' + V0" = ?
ELEC1103
215
SUPERPOSITION
Ch4. VII. Source Superposition
We set to zero the voltage source
209
SUPERPOSITION
Ch4. VII. Source Superposition
VS
+ -
Consider the circuit shown to the left. Linearity says that
IL + VL
_
VL = a1VS + a2 I S
CONTRIBUTION BY VS
ELEC1103
204
LINEARITY
Ch4. VI. Linearity › Linearity:
Sometimes, it is convenient to break the requirements for linearity into two parts:
A (circuit) model described by y = Tu is linear if and only if 1. T (u1 + u2 ) = Tu1 + Tu2 , ∀u1 , u2 additivity 2. T (α u ) = α Tu , ∀α , ∀u homogeneity
Ai = Cs
where s is a vector of values for independent sources. Thus, we have that the voltages or currents are linearly dependent on the values of the independent sources.
1 VL
circuit
CONTRIBUTION BY I S
2 VL
IS
V 1L can be computed by setting the current
source to zero and solving the circuit.
V 2 L can be computed by setting the voltage
1 2 6 = I 0 2= [ mA ] 1 1 [ mA ] 3 + 3 6
2 = V = [ mA ] 6 [ kΩ] 4 [ V ] 3
ELEC1103
This means V0 is V1 + 1.5 mA x 2 kΩ = 6 V. This is off by a factor of 2.
207
LINEARITY
Ch4. VI. Linearity: homogeneity › Another example of using homogeneity in a circuit analysis:
ELEC1103
203
LINEARITY
Ch4. VI. Linearity › Linearity:
A (circuit) model described by y = Tu is linear if and only if T u y T (α1u1 + α 2u2 ) = α1Tu1 + α 2Tu2 for all possible input pairs u1 , u2 and all possible scalars α1 , α 2 .
ELEC1103
206
LINEARITY
Ch4. VI. Linearity: homogeneity › Using homogeneity in a circuit analysis:
Scaling the input results in the same scaling for the output and vice versa scaling the output means the same scaling is required for the input. Let’s apply homogeneity in a circuit problem: Work the circuit backward: Assume Vout is 1 V. Then reverse the voltage divider circuit to obtain: 6 = V1 = Vout 3 [ V ] 2 Thus, I1 is 1mA, I2 is 0.5 mA, and I0 is 1.5 mA. Thus, scaling using homogeneity tells us that Vout is really 2 V because 12 V is twice 6 V.
ELEC1103
SUPERPOSITION
Ch4. VII. Source Superposition
A voltage source set to zero acts like a short circuit.
A current source set to zero acts like an open circuit.
- We will introduce the concept of Linearity which has the properties of additivity and homogeneity. - We will introduce the concept of superposition in linear systems