This

**preview**shows pages 1-3. to view the full**9 pages of the document.**I. CIRCUIT BASICS

• Electrical quantities

◦ Current: dt

dq

I= [Units: C/s = Amps (A)] ◦ Voltage: dq

dw

V= [Units: J/C = Volts (V)]

◦ Power: VIP dt

dq

dq

dw

dt

dw ===

[Units: J/s = Watts (W)]

avg power: ∫

=T

TdttVtIP 0

1)()(

P = IV > 0: power delivered

P = IV < 0: power extracted

• Primitive circuit elements

◦ Voltage Source ◦ Current Source

◦ Resistor – follows Ohm’s Law: V IR

=

(note polarity)

R = resistance [Units: V/A = Ohms (Ω)]

G = 1/R = conductance [Units: Siemens (S)]

Resistor power dissipation: R

RIIVP === V2

2

∑

=

=n

k

keq RR

1

• Circuit definitions

◦ Node – point where 2 or more circuit elements are connected

◦ Series elements – same current flows through all elements

◦ Parallel elements – same voltage across all elements

II. CIRCUIT ANALYSIS BASICS

• KCL (Kirchhoff’s Current Law)

◦ Sum of all currents entering a node = 0

◦ Sum of all currents leaving a node = 0

◦ Σ(currents in) = Σ(currents out)

• KVL (Kirchhoff’s Voltage Law)

◦ Sum of voltage drops around a loop = 0

◦ Sum of voltage rises around a loop = 0

◦ Σ(voltage drops) = Σ(voltage rises)

• Series resistors: • Parallel resistors: ∑

=

=n

kkeq RR 1

11

2

R+

1

RReq =

21

21

21 || RR

RR

RRReq +

==

321

1111

RRRReq

++=

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

• Voltage divider • Current divider

S

V

RR

R

V

21

2

2+

= S

V

RRR

R

V

321

3

3++

= S

I

RR

R

I

21

1

2+

= S

I

RRR

R

I

321

3

3111

1

++

=

• Source combinations (series voltage sources and parallel current sources)

III. CIRCUIT ANALYSIS METHODS

• Nodal Analysis – finds unknown node voltages in a circuit; once all node voltages are known,

currents can be found through IV relationships of circuit elements (e.g., Ohm’s Law)

1. Choose a reference node (“ground”)

2. Define unknown voltages (those not fixed by voltage sources)

3. Write KCL at each unknown node, expressing current in terms of node voltages

- use IV relationships of the circuit elements (e.g., I=V/R for resistors)

4. Solve the set of independent equations (N eqn’s for N unknown node voltages)

• Supernode – for a floating voltage source (where both terminals are unknown voltages), define

a supernode around the source, write KCL at supernode, and use the voltage source equation

xyF

y

x

VVV

R

V

R

V

II

−=

+=+ 21

21

• Superposition – In any linear circuit containing multiple independent sources, any I or V in the

circuit can be calculated as the sum of the individual contributions of each source acting alone

o Linear circuit – circuit with only independent sources and linear elements (linear RLC,

linear dependent sources). Linear elements have linear IV characteristics.

1. Leave one source on and turn off all other sources

Æ replace voltage source with short circuit (V=0)

Æ replace current source with open circuit (I=0)

2. Find the contribution from the “on” source

3. Repeat for each independent source.

4. Sum the individual contributions from each source to obtain the final result

Note: Superposition doesn’t work for power, since power is nonlinear (P=I2R=V2/R)

Only pages 1-3 are available for preview. Some parts have been intentionally blurred.

• Thevenin/Norton Equivalent Circuit Models – Any linear 2-terminal network of independent

sources and linear resistors can be replaced by an equivalent circuit consisting of 1 independent

voltage source in series with 1 resistor (Thevenin) or 1 independent current source in parallel

with 1 resistor (Norton). The circuit models have the same IV characteristics.

◦ Three variables: Vth=Voc, Rth=RN, IN=Isc.

◦ Thevenin/Norton relationship: Vth=INRth Æ only 2 of the 3 variables are required

◦ Vth = Voc: open-circuit voltage – Leave the port open (IL=0) and solve for Voc.

◦ IN = Isc: short-circuit current – Short the port (VL=0) and solve for IN.

◦ Rthc: Thevenin/Norton resistance – Turn off all independent sources (leave the dependent

sources alone). If there are no dependent sources, simplify the resistive network using

series and parallel reductions to find the equivalent resistance. If dependent sources are

present, attach Itest or Vtest and use KCL/KVL to find Rth=Vtest/Itest.

test

test

th I

V

R=

note the direction of Itest

and the polarity of Vtest

• Source Transformations – conversion between Thevenin and Norton equivalent circuits

• Maximum Power Transfer Theorem

Æ power transferred to load resistor RL

is maximized when RL=Rth

• Load-line Analysis – graphical method solving circuits with 1 nonlinear circuit element

Æ graph the IV curves for the nonlinear circuit element and the Thevenin/Norton equivalent of

the rest of the circuit on the same axes; the operating point is where the two curves intersect

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