Transcript Voltage Current Dividers Impedance

Objective of Lecture
 Explain mathematically how a voltage that is applied
to components in series and how a current that enters
the a node shared by components in parallel is
distributed among the component.
 Chapter 9.7 Fundamentals of Electric Circuits
 Chapters 15.4, 15.6, 16.3, 16.5, 17.2, 17.5, 17.7 Principles of
Electric Circuits
 Chapters 10.3, 10.5-6, 12.3, 12.5-6, 13.2, and 13.5 Electric
Circuit Fundamentals
Voltage Dividers
Impedances in series share the same current
Voltage Dividers
From Kirchhoff’s Voltage Law and Ohm’s Law
0   VS  V1  V2
V1  IZ1
and
Therefore, V2 
Z1
V1 
Vs
Z1  Z 2
V2 
Z2
Vs
Z1  Z 2
V2  IZ2
V1
Z2
Z1
Voltage Division
The voltage associated with one impedance Zn in a
chain of multiple impedances in series is:


 Z 
Vn   S n  Vt ot al or
 Z 
s 
 
s 1

 Zn 
Vn  
 Vt ot al
 Z eq 
where Vtotal is the total of the voltages applied across
the impedances.
Voltage Division
 Because of changes in phase angle of the voltage that
occur with inductors and capacitors, the calculation of
the percentage of the total voltage associated with a
particular impedance, Zn, is not directly related to the
percentage of the magnitude of that particular
impedance, Zn, relative to the total equivalent
resistance, Zeq.
 Zn = Zn jn
 Zeq = Zeq jeq
Current Division
All components in parallel share the same voltage
Current Division
All components in parallel have the same voltage across
them.
From Kirchhoff’s Current Law and Ohm’s Law
:
0  IS  I1  I2  I3
VS  I1Z1
VS  I2 Z 2
Vs  I3 Z 3
Current Division
+
Vin
I1 
I2 
_
I3 
Z2 Z3
Z1  Z 2 Z 3
Z1 Z 3
Z 2  Z1 Z 3
Z1 Z 2
Z 3  Z1 Z 2
IS
IS
IS
Current Division
+
Vin
_
Z eq
Z2Z3
where Z eq  Z 2 Z 3 
and I1 
IS
Z2  Z3
Z1  Z eq
Current Division
The current associated
with one component Z1
in parallel with one other
component is:
 Z2 
I1  
 It ot al
 Z1  Z 2 
 The current associated
with one component Zm
in parallel with two or
more components is:
 Z eq 
Im  
 It ot al
 Zm 
where Itotal is the total of the currents entering the
node shared by the components in parallel.
Impedance and Admittance
 Given the equations for voltage and current division, it
is sometimes easier to calculate how voltage is divided
among components using admittances and to
calculate how current is divided among components
using impedances when using phasor notation.
Summary
 The equations used to
calculate the voltage
across a specific
component Zn in a set of
components in series are:
 Zn 
Vn  
 Vt ot al
 Z eq 
 Yeq 
Vn  
 Vt ot al
 Yn 
 The equations used to
calculate the current
flowing through a specific
component Zm in a set of
components in parallel are:
Im 
Z eq
Zm
It ot al
Ym
Im 
It ot al
Yeq