Power computationsece.tamu.edu/~jose-silva-martinez/courses/ECEN489... · •Effective value of a voltage or current is also known as the root-mean-square (rms) value •Effective

1ECEN 489: Power Management Circuits and Systems

Power computations


Power and Energy

• Instantaneous power p(t) is defined as the product of instantaneous voltage v(t) and instantaneous current i(t).

𝑝 𝑡 = 𝑣(𝑡)𝑖(𝑡)

• Average power P is the time average of p(t) over one or more periods

𝑃 =1

𝑇න𝑡𝑜

𝑡𝑜+𝑇

𝑝(𝑡) 𝑑𝑡

T is the period of the power waveform


Power and Energy

• Energy, or work, is the integral of instantaneous power

𝑊 = න𝑡1

𝑡2

𝑝(𝑡) 𝑑𝑡

• Power has units of watts and energy has units of joules.

• Power can also computed from energy per period

𝑃 =𝑊

𝑇


Power and Energy

Example


Power and Energy

Example contd

• Energy absorbed by the device in one period

• Average power is

= 60𝑊

𝑃 =𝑊

𝑇=

1.2𝐽

0.02𝑠= 60𝑊


Power and Energy

• For applications such as battery charging circuits and DC power supplies, average power absorbed by a dc voltage source 𝑣 𝑡 = 𝑉𝑑𝑐 that has a periodic current i(t) is 𝑃𝑑𝑐

• Average power absorbed by a dc voltage source is the product of the voltage and the average current

𝑃𝑑𝑐 = 𝑉𝑑𝑐𝐼𝑎𝑣𝑔• Average power absorbed by a dc source 𝑖 𝑡 = 𝐼𝑑𝑐

𝑃𝑑𝑐 = 𝑉𝑎𝑣𝑔𝐼𝑑𝑐


Inductors

• For periodic currents and voltages,

𝑖 𝑡 + 𝑇 = 𝑖 𝑡𝑣 𝑡 + 𝑇 = 𝑣 𝑡

• Voltage-current relationship for the inductor

𝑖 𝑡𝑜 + 𝑇 =1

𝐿න𝑡𝑜

𝑡𝑜+𝑇

𝑉𝐿 𝑡 𝑑𝑡 + 𝑖(𝑡𝑜)

• Energy stored in an inductor

𝑤 𝑡 =1

2𝐿𝑖2(𝑡)

• For periodic inductor current, stored energy at the end of one period is the same as at the beginning

• No net energy transfer indicates that the average power absorbed by an inductor is zero for steady-state periodic operation

𝑃𝐿 = 0

𝑖 𝑡𝑜 + 𝑇 − 𝑖 𝑡𝑜 =1

𝐿න𝑡𝑜

𝑡𝑜+𝑇

𝑉𝐿 𝑡 𝑑𝑡

𝑎𝑣𝑔 𝑉𝐿 𝑡 = 𝑉𝐿 =1

𝑇න𝑡𝑜

𝑡𝑜+𝑇

𝑉𝐿 𝑡 𝑑𝑡 = 0

For periodic currents, the average voltage across an inductor is zero


Capacitors• Voltage-current relationship for the inductor

𝑣 𝑡𝑜 + 𝑇 =1

𝐶න𝑡𝑜

𝑡𝑜+𝑇

𝑖𝑐 𝑡 𝑑𝑡 + 𝑣(𝑡𝑜)

• Energy stored in an capacitor

𝑤 𝑡 =1

2𝐶𝑣2(𝑡)

• For a periodic capacitor voltage, the stored energy is the same at the end of a period as at the beginning.

• No net energy transfer indicates that the average power absorbed by the capacitor is zero for steady-state periodic operation

𝑃𝑐 = 0

𝑣 𝑡𝑜 + 𝑇 − 𝑣 𝑡𝑜 =1

𝐶න𝑡𝑜

𝑡𝑜+𝑇

𝑖𝑐 𝑡 𝑑𝑡 = 0

𝑎𝑣𝑔 𝑖𝑐 𝑡 = 𝐼𝐶 =1

𝑇න𝑡𝑜

𝑡𝑜+𝑇

𝑖𝐶 𝑡 𝑑𝑡 = 0

For periodic voltages, the average current in a capacitor is zero


Inductors and Capacitors

ExampleDetermine the voltage, instantaneous power, and

average power for the inductor

• Remember average inductor voltage is zero

• 𝑝 𝑡 = 𝑣(𝑡)𝑖(𝑡)

Average inductor power is 0


Energy recovery

• For 0 < 𝑡 < 𝑡1

𝑣𝐿 = 𝑉𝐶𝐶

𝑖𝑠(𝑡) = 𝑖𝐿(𝑡)


Energy recovery

For 𝑡1 < 𝑡 < 𝑇

• Initial condition for inductor current

• Average power supplied by dc source


Effective values: RMS

• Effective value of a voltage or current is also known as the root-mean-square (rms) value

• Effective value of a periodic voltage waveform is based on the average power delivered to a resistor

• For a periodic voltage across a resistor, effective voltage is the voltage that is as effective as the dc

voltage in supplying average power

𝑃 =𝑉𝑑𝑐

2

𝑅; 𝑃 =

𝑉𝑒𝑓𝑓2

𝑅• Average resistor power

⇒



• Example: Find the rms of a periodic pulse waveform



• If a periodic voltage is the sum of two periodic voltage waveforms

𝑣 𝑡 = 𝑣1 𝑡 + 𝑣2(𝑡), the rms value of v(t)

• For orthogonal functions



Examples

=


Apparent power and power factor

• Apparent power is the product of rms voltage and rms current magnitudes

𝑆 = 𝑉𝑟𝑚𝑠𝐼𝑟𝑚𝑠

• Power factor is the ratio of average power to apparent power

𝑝𝑓 =𝑃

𝑆=

𝑃

𝑉𝑟𝑚𝑠𝐼𝑟𝑚𝑠

𝑝𝑓 = cosΦ

• Lagging power factor– Current lags behind voltage

– Circuit is inductive

• Leading power factor– Current leads voltage

– Circuit is capacitive

• Unity power factor– Current and voltage in phase

– ideal

lagging

leading

In phase


Power computations for sinusoidal

ac circuits

• For linear circuits that have sinusoidal sources, all steady-state voltages and currents are sinusoids

• For

𝑣 𝑡 = 𝑉𝑚cos(ω𝑡 + θ)𝑖 𝑡 = 𝐼𝑚cos(ω𝑡 + Φ)

• Instantaneous power

𝑝 𝑡 = [𝑉𝑚cos ω𝑡 + θ ][𝐼𝑚 cos ω𝑡 + Φ ]

𝑝 𝑡 =𝐼𝑚𝑉𝑚2

[cos 2ω𝑡 + θ + Φ + cos θ − Φ ]

• Average power

𝑃 =1

𝑇න0

𝑇

𝑝 𝑡 𝑑𝑡 =𝐼𝑚𝑉𝑚2

1

𝑇න0

𝑇

[cos 2ω𝑡 + θ + Φ + cos θ − Φ ]𝑑𝑡

𝑃 =𝐼𝑚𝑉𝑚2

cos θ − Φ

𝑃 = 𝑉𝑟𝑚𝑠𝐼𝑟𝑚𝑠 cos θ − Φ

Power factor

θ − Φ is the phase angle between voltage and current


Power computations for sinusoidal

ac circuits

• Remember: In the steady state, no net power is absorbed by an inductor or a capacitor

• Reactive power is commonly used in conjunction with voltages and currents for inductors and capacitors

• Reactive power is characterized by energy storage during one-half of the cycle and energy retrieval during the other half

𝑄 = 𝑉𝑟𝑚𝑠𝐼𝑟𝑚𝑠 sin θ − Φ• Complex power combines real and reactive powers for ac circuits

𝑆 = 𝑃 + 𝑗𝑄 = 𝑉𝑟𝑚𝑠𝐼𝑟𝑚𝑠∗

• Apparent power in ac circuits is the magnitude of complex power

𝑆 = 𝑆 = 𝑃2 + 𝑄2


Power computations for nonsinusoidal

periodic waveforms

• Power electronics circuits typically have voltages and/or currents that are periodic but not sinusoidal

• Applying some special relationships for sinusoids to waveforms that are not sinusoids can cause errors

• Option to use Fourier series to describe nonsinusoidal periodic waveforms in terms of a series of sinusoids

• A periodic function f(t) can be expressed in trigonometric form

• Sines and cosines can be combined into one sinusoid

or

• rms value of f(t)



periodic waveforms

• Average power

• Average power for nonsinusoidal periodic voltage and current waveforms

𝑃 =1

𝑇න0

𝑇

𝑝(𝑡) 𝑑𝑡

or

Total average power is the sum of the powers at the frequencies in the Fourier series



periodic waveforms

• For a nonsinusoidal source applied to a linear load, use superposition

to determine power absorbed by load

•Nonsinusoidal periodic voltage is equivalent to the series combination

of the Fourier series voltages

= load



periodic waveforms

Example

Determine the power absorbed by the load

𝑣 𝑡 = 10 + 20 cos 2π60𝑡 − 25𝑜 + 30 cos(4π60𝑡 + 20𝑜) 𝑉

• Current at source frequency should be computed separately

𝐼𝑜 =𝑉𝑜𝑅=10

5= 2𝐴

• Amplitudes of ac current terms computed from phasor analysis



periodic waveforms

Example• Load current

• Power at each frequency in Fourier series

• Total power

• Power absorbed by load can computed from 𝐼𝑟𝑚𝑠2𝑅



periodic waveforms

• If a sinusoidal voltage source is applied to a nonlinear load, current

waveform can be represented as a Fourier series

Only nonzero power term is at the frequency of the applied voltage

Power factor, 𝑝𝑓 =𝑃

𝑆=

𝑃

𝑉𝑟𝑚𝑠𝐼𝑟𝑚𝑠

𝑝𝑓 =𝑉1,𝑟𝑚𝑠𝐼1,𝑟𝑚𝑠 cos(θ1−Φ1)

𝑉1,𝑟𝑚𝑠𝐼𝑟𝑚𝑠=𝐼1,𝑟𝑚𝑠 cos(θ1−Φ1)

𝐼𝑟𝑚𝑠



periodic waveforms

• Distortion factor- ratio of the rms value of the fundamental frequency to the total rms value

• Distortion factor represents the reduction in power factor due to the nonsinusoidal property of the current

• Total harmonic distortion (THD) is another term used to quantify the nonsinusoidal property of a waveform

• THD is the ratio of the rms value of all the nonfundamental frequency terms to the rms value of the

fundamental frequency term

• Distortion factor can be expressed as

• Power factor can also be expressed as

• THD can be expressed as Distortion factor can also be expressed as



periodic waveforms

Example

• A sinusoidal voltage source of v(t) =100 cos(377t) V is applied to a nonlinear load,

resulting in a nonsinusoidal current which is expressed in Fourier series form

Determine

• power absorbed by the load

• power factor of the load

• distortion factor of the load current

• total harmonic distortion of the load current



periodic waveforms

Solution

• v(t) =100 cos(377t) and

• Power absorbed by the load- power absorbed at each frequency in Fourier series

• rms voltage • Power factor=𝑃

𝑆=

𝑃

𝑉𝑟𝑚𝑠𝐼𝑟𝑚𝑠=

650

70.7∗14.0= 0.66

or



periodic waveforms

Solution

• v(t) =100 cos(377t) and

• Distortion factor can be expressed as

• Total harmonic distortion of load current


Summary

• Instantaneous power p(t) is 𝑝 𝑡 = 𝑣 𝑡 𝑖 𝑡

• Average power P 𝑃 =1

𝑇𝑡𝑜𝑡𝑜+𝑇 𝑝(𝑡) 𝑑𝑡

• For inductors and capacitors that have periodic voltages and currents, the average power is zero.

Instantaneous power is generally not zero because the device store energy and then returns energy

to the circuit.

• For periodic currents, the average voltage across an inductor is zero.

• For periodic voltages, the average current in a capacitor is zero.

• For nonsinusoidal periodic waveforms, average power may be computed from the basic definition,

or the Fourier series method may be used. The Fourier series method treats each frequency in the

series separately and uses superposition to compute total power.

• Power factor is the ratio of average power to apparent power

• Apparent power is the product of rms voltage and current

• The rms value is the root-mean-square or effective value of a voltage or current waveform

Documents

Power computationsece.tamu.edu/~jose-silva-martinez/courses/ECEN489... · •Effective value of a voltage or current is also known as the root-mean-square (rms) value •Effective