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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Steady-State Sinusoidal Analysis

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Steady-State Sinusoidal Analysis 1. Identify the frequency, angular frequency, peak value, rms value, and phase of a sinusoidal signal. 2. Solve steady-state ac circuits using phasors and complex impedances. 3. Compute power for steady-state ac circuits. 4. Find Thévenin and Norton equivalent circuits. 5. Determine load impedances for maximum power transfer.

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING The sinusoidal function v(t) = V M sin t is plotted (a) versus t and (b) versus t.

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING The sine wave V M sin ( t + ) leads V M sin t by radian

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING FrequencyAngular frequency

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING A graphical representation of the two sinusoids v 1 and v 2. The magnitude of each sine function is represented by the length of the corresponding arrow, and the phase angle by the orientation with respect to the positive x axis. In this diagram, v1 leads v2 by 100 o + 30 o = 130 o, although it could also be argued that v 2 leads v 1 by 230 o. It is customary, however, to express the phase difference by an angle less than or equal to 180 o in magnitude.

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING 4-6 Euler’s identity In Euler expression, A cos t = Real (A e j t ) A sin t = Im( A e j t ) Any sinusoidal function can be expressed as in Euler form.

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING The complex forcing function V m e j ( t + ) produces the complex response I m e j ( t + ). It is a matter of concept to make use of the mathematics of complex number for circuit analysis. (Euler Identity)

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING The sinusoidal forcing function V m cos ( t + θ) produces the steady-state response I m cos ( t + φ). The imaginary sinusoidal forcing function j V m sin ( t + θ ) produces the imaginary sinusoidal response j I m sin ( t + φ ).

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Re(V m e j ( t + ) ) Re(I m e j ( t + ) ) Im(V m e j ( t + ) ) Im(I m e j ( t + ) )

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Phasor Definition

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING A phasor diagram showing the sum of V 1 = 6 + j8 V and V 2 = 3 – j4 V, V 1 + V 2 = 9 + j4 V = Vs Vs = Ae j θ A = [9 2 + 4 2 ] 1/2 θ = tan -1 (4/9) Vs = 9.85 24.0 o V.

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Adding Sinusoids Using Phasors Step 1: Determine the phasor for each term. Step 2: Add the phasors using complex arithmetic. Step 3: Convert the sum to polar form. Step 4: Write the result as a time function.

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Using Phasors to Add Sinusoids

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING 7.3997.29 14.1906.23 5660.814.. 30104520 21s j jj VVV

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Phase Relationships To determine phase relationships from a phasor diagram, consider the phasors to rotate counterclockwise. Then when standing at a fixed point, if V 1 arrives first followed by V 2 after a rotation of θ, we say that V 1 leads V 2 by θ. Alternatively, we could say that V 2 lags V 1 by θ. (Usually, we take θ as the smaller angle between the two phasors.)

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING To determine phase relationships between sinusoids from their plots versus time, find the shortest time interval t p between positive peaks of the two waveforms. Then, the phase angle isθ = (t p /T ) × 360°. If the peak of v 1 (t) occurs first, we say that v 1 (t) leads v 2 (t) or that v 2 (t) lags v 1 (t).

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING COMPLEX IMPEDANCES

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING (b) (a) (c) In the phasor domain, (a) a resistor R is represented by an impedance of the same value; (b) a capacitor C is represented by an impedance 1/j C; (c) an inductor L is represented by an impedance j L.

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Zc is defined as the impedance of a capacitor The impedance of a capacitor is 1/j C. It is simply a mathematical expression. The physical meaning of the j term is that it will introduce a phase shift between the voltage across the capacitor and the current flowing through the capacitor.

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Z L is defined as the impedance of an inductor The impedance of a inductor is j L. It is simply a mathematical expression. The physical meaning of the j term is that it will introduce a phase shift between the voltage across the inductor and the current flowing through the inductor.

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Complex Impedance in Phasor Notation

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Kirchhoff’s Laws in Phasor Form We can apply KVL directly to phasors. The sum of the phasor voltages equals zero for any closed path. The sum of the phasor currents entering a node must equal the sum of the phasor currents leaving.

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Circuit Analysis Using Phasors and Impedances 1.Replace the time descriptions of the voltage and current sources with the corresponding phasors. (All of the sources must have the same frequency.) 2. Replace inductances by their complex impedances Z L = jωL. Replace capacitances by their complex impedances Z C = 1/(jωC). Resistances remain the same as their resistances.

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING 3. Analyze the circuit using any of the techniques studied earlier performing the calculations with complex arithmetic.

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Solve by nodal analysis

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Vs= - j10, Z L =j L=j(0.5×500)=j250 Use mesh analysis,

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING AC Power Calculations

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING THÉVENIN EQUIVALENT CIRCUITS

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING The Thévenin voltage is equal to the open-circuit phasor voltage of the original circuit. We can find the Thévenin impedance by zeroing the independent sources and determining the impedance looking into the circuit terminals.

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING The Thévenin impedance equals the open-circuit voltage divided by the short-circuit current.

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING Maximum Power Transfer If the load can take on any complex value, maximum power transfer is attained for a load impedance equal to the complex conjugate of the Thévenin impedance. If the load is required to be a pure resistance, maximum power transfer is attained for a load resistance equal to the magnitude of the Thévenin impedance.

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Department of Electronic Engineering BASIC ELECTRONIC ENGINEERING

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