# Steady-State Sinusoidal Analysis - PowerPoint PPT Presentation

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Steady-State Sinusoidal Analysis . 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 .

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### 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.

The sinusoidal function v(t) = VM sin  t is plotted (a) versus  t and (b) versus t.

The sine wave VM sin ( t + ) leads VM sin  t by  radian

Frequency

Angular frequency

A graphical representation of the two sinusoids v1 and v2.

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 100o + 30o = 130o, although it could also be argued that v2 leads v1 by 230o.

It is customary, however, to express the phase difference by an angle less than or equal to 180oin magnitude.

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.

4-6

The complex forcing function Vm e j ( t + ) produces the complex response Ime j (t + ).

It is a matter of concept to make use of the mathematics of complex number for circuit analysis. (Euler Identity)

The sinusoidal forcing function Vm cos ( t + θ) produces the steady-state response Im cos ( t + φ).

The imaginary sinusoidal forcing function j Vm sin ( t + θ) produces the imaginary sinusoidal response j Im sin ( t + φ).

Re(Vm e j ( t + ) )  Re(Ime j (t + ))

Im(Vm ej ( t + ) )  Im(Ime j (t + ))

### Phasor Definition

A phasor diagram showing the sum of

V1 = 6 + j8 V and V2 = 3 – j4 V,

V1 + V2 = 9 + j4 V = Vs

Vs = Ae j θ

A = [9 2 + 4 2]1/2

θ= tan-1 (4/9)

Vs = 9.8524.0o V.

### 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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### Phase Relationships

To determine phase relationships from a phasor diagram, consider the phasors to rotate counterclockwise.

Then when standing at a fixed point,

if V1 arrives first followed by V2 after a rotation of θ , we say that V1 leads V2 by θ .

Alternatively, we could say that V2 lags V1 by θ . (Usually, we take θ as the smaller angle between the two phasors.)

To determine phase relationships between sinusoids from their plots versus time, find the shortest time interval tpbetween positive peaks of the two waveforms.

Then, the phase angle isθ = (tp/T )× 360°.

If the peak of v1(t)occurs first, we say that v1(t)leads v2(t)or that v2(t)lags v1(t).

### COMPLEX IMPEDANCES

(a)

(b)

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.

(c)

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.

ZL 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.

Complex Impedance in Phasor Notation

### 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.

### Circuit Analysis Using Phasors and Impedances

• 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

• ZL= jωL.

• Replace capacitances by their complex impedances

• ZC = 1/(jωC).

• Resistances remain the same as their resistances.

3. Analyze the circuit using any of the techniques studied earlier performing the calculations with complex arithmetic.

Solve by nodal analysis

Vs= - j10, ZL=jL=j(0.5×500)=j250

Use mesh analysis,

### THÉVENIN EQUIVALENT CIRCUITS

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.

The Thévenin impedance equals the open-circuit voltage divided by the short-circuit current.

### 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.