Chapter 10 Sinusoidal SteadyState Analysis Charles P Steinmetz
- Slides: 54
Chapter 10 Sinusoidal Steady-State Analysis
Charles P. Steinmetz (1865 -1923), the developer of the mathematical analytical tools for studying ac circuits. Courtesy of General Electric Co.
Heinrich R. Hertz (1857 -1894). Courtesy of the Institution of Electrical Engineers. cycles/second Hertz, Hz
Sinusoidal Sources Amplitude Period = 1/f Phase angle Angular or radian frequency = 2 pf = 2 p/T Sinusoidal voltage source vs Vm sin( t ). Sinusoidal current source is Im sin( t ).
Example v i + i circuit v element _ Voltage and current of a circuit element. The current leads the voltage by radians OR The voltage lags the current by radians
Example 10. 3 -1 Find their phase relationship and Therefore the current leads the voltage by
Recall Triangle for A and B of Eq. 10. 3 -4, where C .
Example 10. 3 -2 B A A B
Steady-State Response of an RL circuit An RL circuit. From #8	 Substitute the assumed solution into 10. 4 -1 Coeff. of cos Coeff. of sin Solve for A & B
Steady-State Response of an RL circuit (cont. ) Thus the forced (steady-state) response is of the form
Complex Exponential Forcing Function Input Response magnitude phase frequency Exponential Signal Note
Complex Exponential Forcing Function (cont. ) try We get where
Complex Exponential Forcing Function (cont. ) Substituting for A We expect
Example We replace Substituting ie
Example(cont. ) The desired answer for the steady-state current interchangeable Or
Using Complex Exponential Excitation to Determine a Circuit’s SS Response to a Sinusoidal Source Write the excitation as a cosine waveform with a phase angle Introduce complex excitation Use the assumed response Determine the constant A
Obtain the solution The desired response is Example 10. 5 -1
Example 10. 5 -1(cont. )
Example 10. 5 -1(cont. ) The solution is The actual response is
The Phasor Concept A sinusoidal current or voltage at a given frequency is characterized by its amplitude and phase angle. Magnitude Thus we may write Phase angle unchanged
The Phasor Concept(cont. ) A phasor is a complex number that represents the magnitude and phase of a sinusoid. phasor The Phasor Concept may be used when the circuit is linear , in steady state, and all independent sources are sinusoidal and have the same frequency. A real sinusoidal current phasor notation
The Transformation Time domain Transformation Frequency domain
The Transformation (cont. ) Time domain Transformation Frequency domain
Example Substitute into 10. 6 -2 Suppress
Example (cont. )
Phasor Relationship for R, L, and C Elements Time domain Frequency domain Resistor Voltage and current are in phase
Inductor Time domain Frequency domain Voltage leads current by
Capacitor Time domain Frequency domain Voltage lags current by
Impedance and Admittance Impedance is defined as the ratio of the phasor voltage to the phasor current. Ohm’s law in phasor notation phase magnitude or polar exponential rectangular
Graphical representation of impedance Resistor R Inductor L Capacitor 1/ C
Admittance is defined as the reciprocal of impedance. conductance In rectangular form Resistor Inductor Capacitor susceptance G 1/ L C
Kirchhoff’s Law using Phasors KVL KCL Both Kirchhoff’s Laws hold in the frequency domain. and so all the techniques developed for resistive circuits hold Superposition Thevenin &Norton Equivalent Circuits Source Transformation Node & Mesh Analysis etc.
Impedances in series Admittances in parallel
Example 10. 9 -1 KVL R = 9 W, L = 10 m. H, C = 1 m. F i = ?
Example 10. 9 -2 KCL v=?
Node Voltage & Mesh Current using Phasors va = ? vb = ?
KCL at node a KCL at node b Rearranging Admittance matrix
If Im = 10 A and Using Cramer’s rule to solve for Va Therefore the steady state voltage va is
Example 10. 10 -1 v=? use supernode concept as in #4
Example 10. 10 -1 (cont. ) KCL at supernode Rearranging
Example 10. 10 -1 (cont. ) Therefore the steady state voltage v is
Example 10. 10 -2 i 1 = ?
Example 10. 10 -2 (cont. ) KVL at mesh 1 & 2 Using Cramer’s rule to solve for I 1
Superposition, Thevenin & Norton Equivalents and Source Transformations Example 10. 11 -1 i=? Consider the response to the voltage source acting alone = i 1
Example 10. 11 -2 (cont. ) Substitute
Example 10. 11 -2 (cont. ) Consider the response to the current source acting alone = i 2 Using the principle of superposition
Source Transformations
Example 10. 11 -2 IS = ?
Example 10. 11 -3 ? Thevenin’s equivalent circuit
Example 10. 11 -4 Thevenin’s equivalent circuit
Example 10. 11 -4 Norton’s equivalent circuit ?
Phasor Diagrams A Phasor Diagram is a graphical representation of phasors and their relationship on the complex plane. Take I as a reference phasor The voltage phasors are
Phasor Diagrams (cont. ) KVL For a given L and C there will be a frequency w that Resonant frequency Resonance
Summary Sinusoidal Sources Steady-State Response of an RL Circuit for Sinusoidal Forcing Function Complex Exponential Forcing Function The Phasor Concept Impedance and Admittance Electrical Circuit Laws using Phasors
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