Transformer – Two Coupled Inductors
+
+
v1
v2
-
-
N1 turns
N2 turns
|v2|/|v1| = N2/N1
EECS 42, Spring 2005
Week 4a, Slide 1
Prof. White
AC Power System
Flowing water
or
Steam produced
from
Fuel oil,
Natural gas,
Coal,
or
Nuclear energy
35,000 volts
Turbine
130,000 volts
Generator
Generating Plant
Step-up Transformer
(for efficient transmission at higher voltage, lower current)
21,000 volts
Step-down Transformer (for local
distribution)
Transmission-line support towers
120-240
volts
Step-down transformer mounted
on power pole
Customer
Simplified representation of the transmission and distribution systems that bring electric
power from a generating station to customers. In the generating station, a turbine driven by any of the
means indicated at the upper left is coupled to a generator, turning it to produce three-phase alternating
voltages.
EECS 42, Spring 2005
Week 4a, Slide 2
Prof. White
Summary of Electrical Quantities
Quantity
Variable
Unit
Unit
Symbol
Typical
Values
Defining
Relations
Charge
Q
coulomb
C
1aC to 1C
magnitude of
6.242 × 1018
electron
charges qe = 1.602x10-19 C
Current
I
ampere
A
1mA to 1kA
1A = 1C/s
Important
Equations
i = dq/dt
N node
I
n 1
Voltage
V
volt
V
1mV to 500kV
1V = 1N-m/C
0
n
N loop
V
n 1
EECS 42, Spring 2005
Week 4a, Slide 3
Symbol
n
0
Prof. White
Summary of Electrical Quantities (concluded)
Power
P
watt
W
1mW to
100MW
1W = 1J/s
P = dU/dt;
P=IV
Energy
U
joule
J
1fJ to 1TJ
1J = 1N-m
U = QV
Force
F
newton
N
Time
t
second
s
Resistance
R
ohm
W
1N = 1kgm/s2
1W to 10MW
V = IR; P =
V2/R = I2R
R

Capacitance
Inductance
C
L
EECS 42, Spring 2005
farad
henry
F
H
1fF to 5F
1mH to 1H
Week 4a, Slide 4
Q = CV; i =
C(dv/dt);U =
(1/2)CV2
v = L(di/dt);
U = (1/2)LI2

C


L


Prof. White
Types of Circuit Excitations and Their Uses
1. Steady excitation (“DC”) – steady DC voltage or
current sources. Uses: the DC part of all kinds
of communications, transistor, sensor, … circuits
2. Transient excitation -- DC voltage or current sources
are suddenly turned on or off. Uses: flashbulb
circuit where charge a capacitor and suddenly discharge it through flashbulb; put an on and an off
switching together and get a digital pulse:
+
=
3. Sinusoidal excitation (“AC”) – Uses: AC power
systems; communication systems to find frequency
response.
EECS 42, Spring 2005
Week 4a, Slide 5
Prof. White
First-Order Circuits
• A circuit which contains only sources, resistors
and an inductor is called an RL circuit.
• A circuit which contains only sources, resistors
and a capacitor is called an RC circuit.
• RL and RC circuits are called first-order circuits
because their voltages and currents are
described by first-order differential equations.
R
i
i
vs
L
–
+
EECS 42, Spring 2005
Week 4a, Slide 6
–
+
vs
R
C
Prof. White
• The natural response of an RL or RC circuit is its
behavior (i.e. current and voltage) when stored
energy in the inductor or capacitor is released to
the resistive part of the network (containing no
independent sources).
• The step response of an RL or RC circuit is its
behavior when a voltage or current source step is
applied to the circuit, or immediately after a switch
state is changed.
EECS 42, Spring 2005
Week 4a, Slide 7
Prof. White
Natural Response of an RL Circuit
• Consider the following circuit, for which the switch is
closed for t < 0, and then opened at t = 0:
t=0
Io
Ro
i
L
+
R
v
–
Notation:
0– is used to denote the time just prior to switching
0+ is used to denote the time immediately after switching
• The current flowing in the inductor at t = 0– is Io
EECS 42, Spring 2005
Week 4a, Slide 8
Prof. White
Solving for the Current (t  0)
• For t > 0, the circuit reduces to
i
Io
L
Ro
+
R
v
–
• Applying KVL to the LR circuit:
• Solution:
EECS 42, Spring 2005
i(t )  i(0)e
( R / L )t
Week 4a, Slide 9
Prof. White
Solving for the Voltage (t > 0)
i (t )  I o e
 ( R / L )t
+
Io
Ro
L
R
v
–
• Note that the voltage changes abruptly:

v (0 )  0
for t  0, v(t )  iR  I o Re
( R / L )t

 v (0 )  I o R
EECS 42, Spring 2005
Week 4a, Slide 10
Prof. White
Time Constant t
• In the example, we found that
i(t )  I o e
( R / L )t
and v(t )  I o Re
• Define the time constant
( R / L )t
L
t
R
– At t = t, the current has reduced to 1/e (~0.37)
of its initial value.
– At t = 5t, the current has reduced to less than
1% of its initial value.
EECS 42, Spring 2005
Week 4a, Slide 11
Prof. White
Transient vs. Steady-State Response
• The momentary behavior of a circuit (in response
to a change in stimulation) is referred to as its
transient response.
• The behavior of a circuit a long time (many time
constants) after the change in voltage or current is
called the steady-state response.
EECS 42, Spring 2005
Week 4a, Slide 12
Prof. White
Review (Conceptual)
• Any* first-order circuit can be reduced to a
Thévenin (or Norton) equivalent connected to
either a single equivalent inductor or capacitor.
RTh
RTh
L
VTh
–
+
ITh
C
– In steady state, an inductor behaves like a short circuit
– In steady state, a capacitor behaves like an open circuit
EECS 42, Spring 2005
Week 4a, Slide 13
Prof. White
Natural Response of an RC Circuit
• Consider the following circuit, for which the switch is
closed for t < 0, and then opened at t = 0:
+
Vo 
Ro
C
t=0
+
v
–
R
Notation:
0– is used to denote the time just prior to switching
0+ is used to denote the time immediately after switching
• The voltage on the capacitor at t = 0– is Vo
EECS 42, Spring 2005
Week 4a, Slide 14
Prof. White
Solving for the Voltage (t  0)
• For t > 0, the circuit reduces to
i
+
Vo 
+
Ro
C
v
R
–
• Applying KCL to the RC circuit:
• Solution:
EECS 42, Spring 2005
v(t )  v(0)e
 t / RC
Week 4a, Slide 15
Prof. White
Solving for the Current (t > 0)
i
+
Vo 
+
Ro
C
v
R
v(t )  Vo e  t / RC
–
• Note that the current changes abruptly:
i (0 )  0
v Vo t / RC
for t  0, i (t )   e
R R
Vo

 i (0 ) 
R
EECS 42, Spring 2005
Week 4a, Slide 16
Prof. White
Time Constant t
• In the example, we found that
v(t )  Vo e
t / RC
Vo t / RC
and i (t )  e
R
• Define the time constant
t  RC
– At t = t, the voltage has reduced to 1/e (~0.37)
of its initial value.
– At t = 5t, the voltage has reduced to less than
1% of its initial value.
EECS 42, Spring 2005
Week 4a, Slide 17
Prof. White
Natural Response Summary
RL Circuit
RC Circuit
i
L
+
v R
C
R
–
• Inductor current cannot
change instantaneously

•

v (0 )  v (0 )
t / t
v(t )  v(0)e t /t
i (0 )  i (0 )
i(t )  i(0)e
L
• time constant t 
R
EECS 42, Spring 2005
Capacitor voltage cannot
change instantaneously

•
Week 4a, Slide 18
time constant

t  RC
Prof. White