Circuit Analysis: Voltage/Current Elements, Source Transformation, Equivalents, Study notes of Electrical Circuit Analysis

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Circuit Analysis Lecture

Outline:

• Generalized voltage/current elements
• Source transformation
• Thevenin & Norton equivalent circuits
• Superposition

Generalized Voltage/Current Elements

• Generalized description of circuit “element”:

i = αv + β

v = (1/α)(i − β)

(subject to sign convention & assuming α = 0)

• Example:

is z 1 z 2

z 1 : i 1 = α 1 v 1 + β 1

z 2 : i 2 = α 2 v 2 + β 2

KCL: is = i 1 + i 2

KVL: (i 1 − β 1 )/α 1 = (i 2 − β 2 )/α 2

Solve for i 1 , i 2 ...

• Main point: Can do circuit analysis with more general descriptions of “elements”

Illustration

• Compute voltage across 20Ω resistor via node analysis:

5 − va

3 − va

va

Rewrite

va

va −^3

va

• Apply source transformation:

is = 1 & Rp = 5

• Compute voltage across 20Ω resistor via node analysis:

va

va

va − 3

Same!

(α, β) computation

• Given: Mystery circuit with two nodes that satisfies

i = αv + β

but α & β unknown.

• Q: What experiments can be done to determine α & β?
• First attempt:
  • Apply v = v 1 and measure resulting i = i 1
  • Apply v = v 2 and measure resulting i = i 2
  • Solve simultaneous equations for α & β:

( i 1

i 2

v 1 1

v 2 1

α

β

α

β

v 1 1

v 2 1

i 1

i 2

• Alternatively, could have “applied” i and measured resulting v

Thevenin and Norton Equivalence Theorems

• Recap: General “element” described by
  • General “element”:

i = (isc/voc)v − isc

• Voltage source in series with resistor

i = (1/Rt)v − vt/Rt

• Current source in parallel with resistor

i = (1/Rn)v − in

Note:

• t := Thevenin
• n := Norton
• Thevenin theorem: General element is equivalent to voltage source (vt) in series with

resistor (Rt) with

voc = Rtisc

vt = voc

• Norton theorem: General element is equivalent to current source (in) parallel with

resistor (Rn) with

voc = Rnisc

is = isc

• What do we mean by “general element”? Any terminal from any (linear) circuit consisting

of resistors, independent sources, and dependent sources.

• Still to come...
  • Shortcut to compute Rt = Rn
  • What if voc = 0?

Examples

...vs...

• Compute isc:

10 − vc

vc

vc

vc = − 40 / 15

isc = − 2 / 3

• Compute voc:

10 − vc

vc

vc = −8 = voc

• Conclusion:

vt = − 8 V

Rt = voc/isc = 12Ω

or

in = − 2 / 3 A

Rn = 12Ω

What if voc = 0?

• Recall equivalence equations:
  • General “element”:

i = (isc/voc)v − isc

• Voltage source in series with resistor

i = (1/Rt)v − vt/Rt

• Current source in parallel with resistor

i = (1/Rn)v^ −^ in

• Recall derivation:

0 = αvoc + β

If voc = 0 then β = 0 = −isc Then

i = αv

• To compute α, apply i = 1A, and let v 1 A be resulting voltage
• Accordingly,

α = 1/v 1 A

• Conclusion:
  • Rt = Rn = v 1 A/ 1
  • vt = 0
  • in = 0

In other words, equivalent circuit is single resistor

Superposition

• Solution of circuit equations results in set of linear equations:

u 1 .. .

un

s 1 .. .

sm

where

• uk are either unknown voltages or currents
• sk are independent voltage or current sources
• This can be rearranged as

Au = Bs

for appropriate matrices A and B. Alternatively,

u = A

− 1 Bs

• Let C = A

− 1 B. Then

u 1 = c 11 s 1 + c 12 s 2 + ... + c 1 msm

.. .

uk = ck 1 s 1 + ck 2 s 2 + ... + ckmsm

• This reveals a linear relationship between unknown values and independent sources
• Principle of superposition: The overall effect in a circuit due to multiple independent

sources is the sum of the effects of individual sources acting one at a time.