AC Circuits | Sinusoids and Phasors

A sinusoid is a periodic wave signal that has the form of the trigonometric sine or cosine function. Sinusoidal current is sometimes referred to as alternating current (ac) because the current reverses with alternating positive and negative values at regular time intervals. AC circuits are driven by sinusoidal current or voltage sources.

A sinusoidal forcing function produces two types of response: a transient response and a steady-state response. The transient response, as the name suggests, decays over time while the steady-state response remains. The circuit is said to be operating at sinusoidal steady state when it reaches the point where the transient response is negligibly small compared to the steady-state response. It is the sinusoidal steady-state response that is of main interest to us in this study guide.

We start our discussion with sinusoids and phasors. We will also briefly discuss impedance and admittance. Lastly, we will show that the basic circuit laws introduced for dc circuits, Kirchhoff’s and Ohm’s, also apply to AC circuits.

Figure 1: v(t) = V_m sin t (a) as a function of t, (b) as a function of t

In the sinusoidal voltage

is the amplitude of the sinusoid

is the angular frequency measured in rad/s

is the argument of the sinusoid

From fig.1(b), the sinusoid completes one cycle every T seconds; thus, T is the period of the sinusoid. From the two plots in fig.1, we observe that or

It can also be observed that repeats values every seconds. This is shown by replacing t by t + T:

That is, is said to be periodic as v has the same value at t as it does at (t + T).

• For a periodic function for all values of t and for all integers n.
• The period of the periodic function is the time it takes for one complete cycle or the number of seconds per cycle.
• The reciprocal of is known as the frequency and is the number of cycles per second. Thus,
  
• From (1) and (2), the angular frequency is
  
  
  
  where is in radians per second (rad/s) and f is in hertz (Hz).
  
  

• A sinusoid can be expressed as a function of time

  
  
  where is the argument and is the phase. The argument and phase are both measured in radians or degrees.

  The two sinusoids in fig.2 have different phases.

  
  Figure 2: Two sinusoids with the same frequency but different phases
  • We see that reaches its minimum and maximum values first. Hence, we say that leads by or that lags by .
  • If , and are out of phase.
  • If , and are in phase; which means they reach their peak and lowest values at exactly the same time.
  • and can be compared in this way because they have the same frequency.
• When comparing two sinusoids, it is convenient to express both in either sine or cosine form. We use the following trigonometric identities:
  
  
  
  Using these identities, we get
  
  
  
  With these equations, we can express a sinusoid in sine form or cosine form.

Graphical Approach

• A graphical approach is an alternative to using trigonometric identities in comparing sinusoids.
• Consider the set of axes shown in fig.3(a). The horizontal axis indicates the magnitude of cosine, while the vertical axis indicates the magnitude of sine.
  • By convention, positive angles are measured counterclockwise from the horizontal.
  • This method can be used to compare two sinusoids. For example, we see in fig.3(a) that subtracting from the argument of gives or
    
    
    
    Similarly, adding to the argument of gives , or
    
    
    
    as shown in fig.3(b).
    
    
    Figure 3: Graphical comparison of cosine and sine: (a) and (b)
• The graphical approach also works in obtaining the sum of two sinusoids with the same frequency when one is in sine form while the other is in cosine form.
  • To add and , we note that is the magnitude of while is the magnitude of , as shown in fig.4(a). The magnitude and argument of the sum is readily obtained from the triangle. Thus,
    
    
    
    where
    
    
    
    For example, we may add and as shown in fig.4(b) and obtain .
    Figure 4: (a) Adding and , (b) adding and

• A phasor is a complex number that carries information about the amplitude and phase of a sinusoid.
• Phasors are more convenient to work with than sine and cosine functions and allow a simple means of analyzing linear circuits fed by sinusoidal sources.
• Before defining phasors and applying them to circuit analysis, we need to be familiar with complex numbers.
• As it is a complex number, a phasor can also be expressed in rectangular form, polar form, or exponential form.
  A complex number can be represented in three ways
  • 
  • 
  • 
  
  Given and ,
  
  
  
  Given and ,
  
  
  
  Thus may be written as
  
  
  
  Figure 5: Complex number on the complex plane
  
  
• Phasor representation is based on Euler’ s identity for complex numbers. In general,
  
  
  
  which shows that we can consider and as the real and imaginary parts of :
  
  
  
  
  where Re and Im stand for the real part and the imaginary part of z, respectively...
  
  
• Given a sinusoid , we can use eq.(11.a) to express as
  
  
  
  Then we take out the time factor , and what remains is the phasor corresponding to the sinusoid:
  
  
  
  where
  
  
  
  
  • is thus called the .
  • To summarize, we can get the phasor corresponding to a sinusoid by expressing the sinusoid in the cosine form so that it can be written as the real part of a complex number. Suppressing the time factor transforms the sinusoid from the time-domain to the phasor-domain:
    
    
    
    
  • Using the sine form will also work:
  • To find the sinusoid corresponding to a phasor , we multiply the phasor by the time factor ejt and take the real part. The time-domain representation is the cosine function with the same magnitude as the phasor and with t plus the phase of the phasor as the argument.
  • Note that in eq. (15) the frequency is not explicitly shown in the phasor domain representation because is constant. The phasor domain is also called the frequency domain because the response depends on .
    
    

    Table 1. Sinusoid-phasor transformation

    Time domain representation	Phasor domain representation

• A phasor is a vector as it has a magnitude and “direction” (phase).

Phasor Diagram

• Phasors and are graphically represented in fig.(6)
  
  Figure 6: A phasor diagram with and

Transformation

The derivative of v(t) is transformed to the phasor or frequency domain as

The integral of v(t) is transformed to the phasor or frequency domain as

Equations (16) and (17) apply in obtaining the steady-state solution, where the initial values of the variable involved are not necessarily known.

Differences between v(t) and

1. is the time domain representation, while is the phasor domain representation
2. is time-dependent, while is time-independent
3. is always real and has no complex term, while is complex

We should note that phasor analysis applies only for constant frequency; we use phasor analysis to manipulate two or more sinusoidal signals only if they are of the same frequency.

• To apply a voltage or current in the phasor or frequency domain to circuits involving the passive elements R, L, and C, we need to transform the voltage-current relationship from the time-domain to the frequency domain for each element.
• Assume the passive sign convention.

Resistor

• Suppose the current through a resistor is , from Ohm’s law the voltage across it is given by
  
  
  
  The phasor form of the voltage is
  
  
  
  The current in phasor form is . Hence,
  
  
  
  showing that the resistor in the phasor domain remains to be Ohmic, as in the time domain.
  
  

  Figure 7: Voltage-current correlations for a resistor in the (a) time-domain and (b) phasor domain

• The voltage and current are in phase, as shown in the phasor diagram in fig. 8.
  
  

  Figure 8: Phasor diagram for the resistor

Inductor

• Let the current through the inductor be , the voltage across the inductor is

  
  

  

  
  

  Recall from eq.(6) that , hence the voltage can be written as

  
  

  

  
  

  which transforms to the phasor

  

  
  

  But and from eq.(10), . Thus
  
  

  
  

  This shows that the voltage has a magnitude of and a phase of . The voltage and current out of phase, and the current lags the voltage by .
  
  

  Figure 9: Voltage-current correlations for an inductor in the (a) time domain and (b) phasor domain

  Figure 10: Phasor diagram for the inductor; lags by

Capacitor

• Assuming that the voltage through the capacitor is , the current through the capacitor is
  

  
  

  

  
  

  As in the steps that we took for the inductor or by applying eq.(16) on eq.(25), we obtain

  

  
  

  This shows that the current and voltage are out of phase. Specifically, the current leads the voltage by .

  
  

  

  Figure 11: Voltage-current correlations for a capacitor in the (a) time domain and (b) phasor domain

  Figure 12: Phasor diagram for the capacitor leads by

Table 2 summarizes the time domain and phasor domain representations of the circuit elements.

Table 2. Voltage-current relationships

Element	Time domain	Frequency domain

Impedance

• The impedance is a measure of the opposition to electrical flow. For ac systems, it is the phasor voltage divided by the phasor current , in ohms ().
• The equations
  
  

  
  

  can be rewritten as the ratio of the phasor voltage to the phasor current as shown in eq.(28):

  
  

  

  
  

  From these three expressions, Ohm’s law in phasor form for any type of element is

  

  
  

  where is the impedance, a frequency-dependent quantity measured in ohms.

  
  

• Although the impedance is the ratio of two phasors and is a complex number, it is not a phasor because it does not correspond to a real-valued sinusoidal signal.
• In rectangular form, the impedance may be expressed as
  

  
  

  

  
  

  is the resistance while is the reactance.

  
  

  The reactance, , is a (positive) magnitude value but when used as a vector, a is associated with inductance and a is associated with capacitance. Thus,

  
  

  • impedance is inductive or lagging since current lags voltage, while
  • impedance is capacitive or leading since current leads voltage.
• Impedance and reactance, like resistance, are measured in ohms ().
• In polar form, the impedance is written as
  

  
  

  

  
  

  From equations (30) and (31), it can be inferred that

  

  
  

  Where

  

  
  

  and

  
• The impedances of resistors, inductors, and capacitors are given by eq.(28). Consider these two extreme cases of angular frequency :
  • when confirming that the inductor acts like a short circuit at dc (zero frequency), while the capacitor acts like an open circuit.
  • when showing that the inductor is an open circuit at high frequencies, while the capacitor is a short circuit.

    Figure 13: Circuit elements at dc and high frequencies: (a) inductor, (b) capacitor

• It is sometimes convenient to work with admittance, which is the reciprocal of impedance.

Admittance

• The admittance is the inverse of impedance, and is measured in siemens (S).

• The ratio of the phasor current to the phasor voltage across an element

  
• is a complex quantity and can be written as

  
  

  

  
  

  where is the conductance and is the susceptance.

• Admittance, conductance, and susceptance are all measured in Siemens (or mhos).

• From eq.(30) and eq.(35),

  

  Rationalizing and equating the real and imaginary parts gives

  

  
  

  Table 3. Impedances and admittances of passive elements

  Element	Impedance	Admittance

• Both KVL and KCL apply in the phasor domain.

• For KVL, let be the voltages around a closed loop.

  Then

  
  
  

  Each voltage may be written in cosine form and expressed as

  
  
  If we let , then
  
  
  
  Because ,
  
  
  
  

  Kirchhoff’s voltage law applies to phasors.

• In the same manner, we can show that KCL holds for phasors. If we let be the current passing through a closed surface in a network at time , then
  
  
  
  If are the phasor forms of the sinusoids then
  
  
  

  This is Kirchhoff’s current law in the phasor domain.

Impedances in series

N series-connected impedances are shown in fig.14. The current I flows through all the impedances. Applying KVL around the loop gives

Figure 14: impedances connected in series

At the input terminals, the equivalent impedance is

The total or equivalent impedance of impedances connected in series is the sum of the individual impedances. This is similar to resistances connected in series.

If , the current through the two impedances is

Because

then

which is the voltage-division principle.

Impedances in parallel

For parallel-connected impedances, the voltage across each impedance is the same. Using KCL,

Figure 15: impedances connected in parallel

The equivalent impedance is

and the equivalent admittance is

If , the equivalent impedance becomes

Also, since

the currents in the impedances are

which is the current-division principle.

and

The wye-to-delta and delta-to-wye transformations applied to resistive circuits are also valid for impedances. With reference to Fig. 16, the conversion formulas are as follows.

Figure 16: Superimposed Y and networks

conversion:

conversion:

A delta or wye circuit is balanced if it has equal impedances in all branches i.e. it provides equal, or balanced, load to all three phases. When a circuit is balanced, the equations become

where

Voltage division, current division, circuit reduction, impedance equivalent, and Y-∆ transformation concepts all apply to ac circuits.