CA2627 Building Science
Lecture 04 Single phase AC circuits
Instructor: Jiayu Chen Ph.D.
Review of Complex Variables
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Real Number vs. Complex Number
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Review of Complex Variables
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Review of Complex Variables
Rectangular, exponential and polar forms
The complex number may be represented in 3 forms:
• Rectangular coordinates;
• Exponential;
• Polar form on the complex plane;
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Review of Complex Variables
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Review of Complex Variables
conjugate
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Review of Complex Variables
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Review of Complex Variables
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Phasors and the Frequency Domain
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Phasors and the Frequency Domain
Take a look at these applet demonstrations:
Phasor:
http://ptolemy.eecs.berkeley.edu/eecs20/berkeley/phasors/demo/phasors.html
R, C, L phasor relations:
http://www.walter-fendt.de/ph14e/accircuit.htm
Phasor:
http://ptolemy.eecs.berkeley.edu/eecs20/berkeley/phasors/demo/phasors.html
http://www.circuit-magic.com/phasor_diagraPm.htm
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Phasors and the Frequency Domain
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Phasors and the Frequency Domain
Normally we work with the real parts (remember, cos and sin are phase shifted by 90 )
The magnitude Im and the phase angle θ, along with knowledge of ω, completely specify
the response, and contains the equivalent information as i(t).
The phasor concept may be used when the circuit is linear, the steady-state response is
sought, and all independent sources are sinusoidal and have the same frequency.
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Phasors and the Frequency Domain
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Phasors and the Frequency Domain
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Phasors and the Frequency Domain
Impedance
From the summary table, the relationships in the frequency domain for the phasor current
and phasor voltage of a capacitor, inductor, and resistor appear similar to Ohm's law for
resistors.
The impedance of an element is defined as the ratio of the phasor voltage to the phasor
current.
The impedance concept is equivalent to stating that capacitors and inductors act as
frequency-dependent resistors, that is, as resistors whose resistance is a function of
the frequency of the sinusoidal excitation.
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Phasors and the Frequency Domain
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Phasors and the Frequency Domain
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Phasors and the Frequency Domain
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Phasors and the Frequency Domain
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Phasors and the Frequency Domain
Kirchhoff Laws using phasors in the frequency domain
Kirchhoff voltage law (KVL)
holds in the frequency domain
with phasor voltages.
The sum of the phasor
voltages in a closed path
is zero.
Kirchhoff current law (KCL)
holds in the frequency domain
with phasor currents.
At a node, the sum of the
phasor currents is zero.
Since both the KVL and the KCL hold in the frequency domain, all the techniques
of analysis we developed for resistive circuits hold for phasor currents and
voltages. For example, we can use the principle of superposition, source
transformations, Thévenin and Norton equivalent circuits, and node voltage and mesh
current analysis.
All these methods apply as long as the circuit is linear and we only have one frequency.
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Phasors and the Frequency Domain
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Phasors and the Frequency Domain
Series and Parallel Impedance
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Phasors and the Frequency Domain
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Phasors and the Frequency Domain
Two methods, the node equations and the mesh equations, are also valid.
The node equations are a set of simultaneous equations in which the unknowns are the
node voltages.
1. Expressing the element voltages and currents in terms of the node voltages.
2. Applying KCL at the nodes of the ac circuit.
After writing and solving the node equations, we can determine all of the voltages and
currents of the ac circuit using Ohm’s and Kirchhoff’s laws.
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Phasors and the Frequency Domain
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Phasors and the Frequency Domain
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Phasors and the Frequency Domain
The mesh equations are a set of simultaneous equations in which the unknowns are the
mesh currents. We write the mesh equations by
1. Expressing the element voltages and currents in terms of the mesh currents.
2. Applying KVL to the meshes of the ac circuit.
After writing and solving the mesh equations, we can determine all of the voltages and
currents of the ac circuit using Ohm’s and Kirchhoff’s laws.
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Phasors and the Frequency Domain
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Phasors and the Frequency Domain
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Phasors and the Frequency Domain
Superposition
For a linear circuit containing two or more independent sources, any circuit voltage or
current may be calculated as the algebraic sum of all the individual currents or voltages
caused by each independent source acting alone.
The superposition principle is particularly useful if a circuit has two or more sources
acting at different frequencies. The circuit will have one set of impedance values at one
frequency and a different set of impedance values at another frequency. We can
determine the phasor response at each frequency. Then we find the time response
corresponding to each phasor response and add them. Note that superposition, in the
case of sources operating at different frequencies, applies to time responses only. We
cannot superpose the phasor responses.
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Phasors and the Frequency Domain
Thevenin equivalent and Norton equivalent
Thévenin's and Norton's theorems apply to phasor current or voltages and impedances
in the same way that they do for resistive circuits.
Thévenin theorem states that any circuit can be replaced by an equivalent circuit with
an ideal voltage source equal to the open-circuit voltage and an equivalent impedance
in series equal to the Thevenin equivalent impedance.
Source transformations
Transform a voltage source and its associated series impedance to a current source and
its associated parallel impedance, or vice versa.
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Phasors and the Frequency Domain
Phasor diagram
Phasors representing the voltage or current of a circuit are time quantities transformed
or converted into the frequency domain. Phasors are complex numbers and can be
portrayed in a complex plane.
The relationship of phasors on a complex plane is called a phasor diagram.
A phasor diagram is a graphical representation of phasors and their relationship on the
complex plane.
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Thank You!
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