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ECE 221
Electric Circuit Analysis I
Chapter 4
Circuit Elements
Herbert G. Mayer, PSU & CCUT
Status 10/3/2015
Taken with permission from PSU Prof. Phillip Wong
Syllabus

Resistor

Inductor

Capacitor

Voltage and Current Sources

Voltage and Current Dividers
Resistor

A resistor is a passive electronic component that
obeys Ohm’s Law. It has resistance R

SI Unit: ohm ()

Symbol:

Impedance:

I-V relationship: v(t )  R i (t )
i(t)
v(t)
R
ZR
1
i (t )  v(t )
R
2
3
Resistors Connected in Series (end-to-end)

If N resistors are connected in series, with the
i-th resistor having a resistance Ri , then the
equivalent resistance Req is:
N
Req   Ri
i 1
R1
R2
R1
R2

R3

Req = R1 + R2
Req = R1 + R2 + R3
4
Properties of Serial Resistances (DC):
Iin →
R1
R2
R3
→
→
→
I1
I2
I3
→ Iout

The amount of current Iin entering one end of a
series circuit is equal to the amount of current Iout
leaving the other end.

The current is the same through each resistor in the
series and is equal to Iin.
Iin = Iout = I1 = I2 = I3
5

The amount of voltage drop across each resistor in a
series circuit is given by Ohm’s Law.
I
→
R1
R2
–
+
V1
R3
–
+
V2
–
+
→I
V3
V1 = IR1 , V2 = IR2 , V3= IR3

By convention, the resistor terminal that the current
enters is labeled “+”, and the terminal the current
exits is labeled “–”
6
Example:
I
A
2
6
3
a) Calculate the current I that
flows through the resistors.
b) Find the voltage drop across
the 6  resistor.
B
VAB = 3 V
Assume 3 significant figures.
Solution:
a) Approach – Use Ohm’s law: I 
VAB
Req
Calculate the equivalent resistance: Req  2  6  3   11 
3V
 0.273 A
Calculate the current: I 
11 
b) Use Ohm’s law
again:
V6  IR6  0.273 A 6    1.64 V
7
Resistors Connected in Parallel (side-by-side)
If N resistors
are in parallel, with the i-th resistor having a
resistance Ri , then the equivalent resistance is:

1
Req    
 i 1 Ri 
N
1
1
R1
R1
R2
R2
R3

1
1 
RR
Req      1 2
R1  R2
 R1 R2 

1
1
1 

Req   
 
 R1 R2 R3 
1
8
Properties of Parallel Resistances (DC):
R1
+ V1 –
Iin →
R2
+ V2 –
→ Iout
R3
+ V3 –

The amount of current Iin entering one end of a
parallel circuit is equal to the amount of current Iout
leaving the other end

For parallel resistors, the voltage drop across each
resistor is the same
Iin = Iout and
V1 = V2 = V3
9

The amount of current through each resistor in a
parallel circuit is given by Ohm’s Law.
R1
I1
+ V –
R2
I→
I2
+ V –
R3
I3

→I
I1 = V / R1
I2 = V / R2
I3 = V / R3
+ V –
The sum of the currents through each resistor is
equal to the original amount of current entering or
leaving the parallel circuit
I1 + I 2 + I3 = I
10
Example:
Ix
2
6
I =4A
A
B
3
VAB
a) Find the current Ix through
the 2  resistor.
b) What is the power dissipated
by the 3  resistor?
Assume 3 significant figures.
Solution:
a) Approach – Use Ohm’s law: VAB  IReq
1
1 1 1
Calculate the equivalent resistance: Req        1 
 2 6 3
Calculate the voltage drop: VAB  4 A1   4 V
Find the current: I x 
VAB
 2.00 A
2
2
VAB
 5.33 W
b) Use power equation: P 
3
11
Inductor & Capacitor

An inductor is a passive component that stores
energy within a magnetic field. It has inductance L
SI unit: henry (H)
Impedance: Z  jL
Symbol:

A capacitor is a passive component that stores
energy within an electric field. It has capacitance C
SI unit: farad (F)
Symbol:
non-polarized
1
Impedance: Z   j C 
+
polarized
12
DC Voltage & Current Sources


An ideal DC voltage source
outputs a constant voltage
regardless of the amount of
current through it
Vs +–
An ideal DC current source
outputs a constant current
regardless of the amount
of voltage across it
Is
I
I
+
Vs
–
V
13
Voltage & Current Dividers
I
+
V0
–
+
I
V1 –
R1
R2
+
V2
–
Voltage Divider
+
I1
I2
VS
G1
G2
–
Current Divider
R1
V1 
V0
R1  R2
G1
I1 
I
G1  G2
R2
 I1 
I
R1  R2
R2
V2 
V0
R1  R2
G2
I2 
I
G1  G2
R1
 I2 
I
R1  R2
14
Example:
Vs = 8 V
Ra = 1 Ω
Va
Rb = 3 Ω
Vb
+
–
What is the voltage drop across Ra?
Ra
Va 
VS
Ra  Rb
1
8 V   2 V
Va 
1 3 
What is the voltage drop across Rb?
Rb
Vb 
VS
Ra  Rb
3
8 V   6 V
Va 
1 3 
15
Example:
The voltage Vg from a signal generator must be
reduced by a factor of three before entering an
audio amplifier. Determine the resistor values of a
voltage divider that performs this task
R1
Max input = 1 V
Peak output = 3 V
Signal
Generator
Vout
Vg
R2

Vg

R1  R2
3
Vg
R2
Vout
3R2  R1  R2
Audio
Amplifier
 R1  2R2
16