POWER ELECTRONIC PRINCIPLES
CIRCUIT BASICS - PASSIVES
Prior to studying converter systems using rectifiers and silicon controlled rectifiers it is
necessary to consider ,or reconsider as the case may be, the basic waveforms that will be
encountered as they are of prime importance in determining the system performance and in the
selection of the devices.
Loads are typically inductive hence a review of series L-R circuits is required as neither the
steady state sinusoidal ac nor the exponential dc analysis is appropriate.
1. Series R-L Switching current
i (t )

R
v(t )  VM sin t
  2    50
L
Z

R 2  (L) 2


 L 
tan 1 

 R 
  turn on delay from voltage zero
FIGURE 1
The general expression for the current at delayed turn-on  for a series R-L circuit shown in
Fig.1 is given by equation 1
R
V
V
i(t )  M sin( t   )  M sin(   ) e  L e
Z
Z

R t
L
 t
V
V
i (t )  M sin( t   )  M sin(   ) e tan e tan
Z
Z
(1)
Equation 1 shows the current to consist of a steady state term and an energizing transient term.
The transient term which has a rate of decay dependent on R and L may be positive, negative or
zero dependent on the load phase angle ( ) and the turn-on delay ( ) from voltage zero.
The steady state and transient values at the instant of turn-on cancel resulting in the initial
current being zero, a basic property of inductive circuits.
EET 307
POWER ELECTRONICS -1
CIRCUIT BASICS - PASSIVES
1
Prof R T Kennedy
2. AC R-L CIRCUIT ENERGISING TRANSIENTS
2.1 Series R-L Circuit Switch -On at Voltage Zero   0
v (t )
i (t )
0
t

R
V
V
i(t )  M sin( t   )  M sin(   ) e  L e
Z
Z
v(t )  VM sin  t
R t
L
FIGURE 2
Fig.2 represents the complete current response i(t) after turn-on at voltage zero (  0) and
shows an initial transition period equal to the sum of the steady state and transient terms before
the steady state value is reached. In the steady state the current is phase shifted from the voltage
and lags the voltage by phase angle  .
v (t )
i (t )
0
VM
Z
0
 VM
Z
 0
t
itr (t )
iss (t )
R
V
iss (t )  M sin(  t   )
Z
V
itr (t )   M sin(    ) e  L e
Z
R t
L
FIGURE 3
At switch on the inductor current must start at zero yet the steady state curve iss(t) of Fig.3 shows
the current magnitude is  VM at t = 0.
Z
A positive decaying transient current transient itr(t) of magnitude  VM at switch on results in the
Z
current starting (as required) at zero at voltage zero and a non sinusoidal current waveform
occurs until the steady state is reached.
EET 307
POWER ELECTRONICS -1
CIRCUIT BASICS - PASSIVES
2
Prof R T Kennedy
2.2 Series R-L Circuit Delayed Switch -On at  from Voltage Zero
i (t )
v (t )
t

FIGURE 4
Fig.4 shows that the complete current response i(t) after a delayed turn-on  from voltage zero
exhibits the same properties as Fig. 2.
2.3 Series R-L Circuit Delayed Switch -On at    from Voltage Zero
v (t )
i (t )
0


0
 0
iss (t )
t
itr (t )
 
iss (t ) 
VM
sin(  t   )
Z
R
itr (t )  
VM
sin(    ) e  L e
Z
R t
L
FIGURE 5
At switch on the inductor current must start at zero yet the steady state curve iss(t) of Fig.5 shows
the current magnitude is non zero and negative at t   .
When    a positive decaying transient current transient itr(t) occurs at switch on  and
cancels the equal negative steady state value resulting in the current i(t) starting at zero at  . A
non sinusoidal current waveform occurs until the steady state is reached.
EET 307
POWER ELECTRONICS -1
CIRCUIT BASICS - PASSIVES
3
Prof R T Kennedy
2.4 Series R-L Circuit Delayed Switch -On at    from Voltage Zero
i (t )
v (t )
0


0
 0
 
t
itr (t )
iss (t )
V
i ss (t )  M sin( t   )
Z
R
itr (t )  
VM
sin(   ) e  L e
Z
R t
L
FIGURE 6
At switch on the inductor current must start at zero yet the steady state curve iss(t) of Fig.6 shows
the current magnitude is non zero and positive at t   .
When    a negative decaying transient current transient itr(t) occurs at switch on  and
cancels the equal positive steady state value resulting in the current i(t) starting at zero at  . A
non sinusoidal current waveform occurs until the steady state is reached.
2.5 Series R-L Circuit Delayed Switch -On at    from Voltage Zero
V
i (t )  i ss (t )  M sin( t   )
Z
v (t )
0
0
 0
 
itr (t )  0
t
iss (t )
FIGURE 7
At switch on when    there is no transient term and the current i(t) is the sinusoidal steady
state waveform as shown in Fig 7.
SUMMARY
EET 307
i (t )  steady state  transient
positive
zero
negative
 
 
 
POWER ELECTRONICS -1
CIRCUIT BASICS - PASSIVES
4
Prof R T Kennedy
3. Resistor Circuit Switch -On
i (t )

v(t )  VM sin t
R
  turn on delay from voltage zero
  2    50
V
i (t )  M sin  t
R
v(t )  VM sin  t
FIGURE 8
The particular case of a resistive load is shown in Fig. 8.
v (t )
0
i1 (t )
0
i2 (t )
0
i3 (t )
0
t
FIGURE 9
Resistive circuits do not have a transient term. The current rises (instantaneously) at switch on to
the value determined by the instantaneous voltage and then continues as the steady state value.
Switch on at voltage zero is represented by i1(t)
Switch on in the voltage positive half cycle is represented by i2(t)
Switch on in the voltage negative half cycle is represented by i3(t)
After switch on the current continues at the steady state value independent of where in the cycle
switch on occurred.
EET 307
POWER ELECTRONICS -1
CIRCUIT BASICS - PASSIVES
5
Prof R T Kennedy
4.1 ‘Pure’ Inductor Circuit Switch -On at Voltage Zero   0
i (t )

v(t )  VM sin t
L
  2    50
  turn on delay from voltage zero
v(t )  VM sin  t
V
V 

  V
i(t )  M sin( t   )  sin(   )  M sin( t  )  sin(  )  M (1  cos  t )
L
 L
2
2  L
v (t )
i (t )
FIGURE 10
V
dc shift )  M
L
0
t
FIGURE 11
Fig.11 shows that the inductor current for the circuit of Fig.10 is positive for the complete cycle
and exhibits a dc shift.
i (t )
v (t )
V
dc shift )  M
L
0
idc (t )
0
t
 0
iss (t )
V
V
i dc (t )  M cos   M
L
L
V
i ss (t )  M sin( t   )
L
FIGURE 12
Fig. 12 shows that the transient term is replaced by a positive dc shift that cancels the equal
negative steady state value at   0 resulting in the current i(t) starting at zero at voltage zero.
EET 307
POWER ELECTRONICS -1
CIRCUIT BASICS - PASSIVES
6
Prof R T Kennedy
4.2 ‘Pure’ Inductor Circuit Switch-On at    from Voltage Zero
i (t )
v (t )
dc shift ) 
0
VM cos 
L
 ve region
idc (t )
0
  
2
t
iss (t )

V
i ss (t )  M sin( t   )
L
V
idc (t )  M cos 
L
FIGURE 13
When    a positive dc shift idc(t) occurs at switch on  and cancels the equal negative
steady state value at  resulting in the current i(t) starting at zero at  . The current i(t) as
shown in Fig. 13 exhibits a positive shift that reduces to zero as    (90 o ) .
The current is the steady state value shifted upwards by a constant value dependent on  .
4.2 ‘Pure’ Inductor Circuit Switch -On at    from Voltage Zero
v (t )
i (t )
 ve region
0
dc shift ) 
0
VM cos 
L
idc (t )
  
iss (t )

2
i ss (t ) 
VM
sin(  t   )
L
t
i dc (t ) 
VM
cos 
L
FIGURE 14
When    a negative dc shift idc(t) occurs at switch on  and cancels the equal positive steady
state value at  resulting in the current i(t) starting at zero at  . The current i(t) exhibits a
positive shift as shown in Fig. 14 that reduces to zero as    (90 o ) .
The current is the steady state value shifted downwards by a constant value dependent on  .
EET 307
POWER ELECTRONICS -1
CIRCUIT BASICS - PASSIVES
7
Prof R T Kennedy
4.3 ‘Pure’ Inductor Circuit Switch -On at    from Voltage Zero
i(t )  iss (t )
v (t )
idc (t )  0
0
  
t

2
V
i ss (t )  M sin( t   )
L
FIGURE 15
When    there is no dc shift idc(t) at switch on  and the current i(t) is the steady state value
as shown in Fig.15.
4.4 DC Shift
V
DC shift  M cos 
L
1.25
 
1
negative shift
0.75
0.5
0.25
DC shift
0
 
0.25
0.5
0.75
zero shift
 
positive shift
1
1.25
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180

FIGURE 16
The DC shift is dependent on  as shown in Fig.16.
EET 307
POWER ELECTRONICS -1
CIRCUIT BASICS - PASSIVES
8
Prof R T Kennedy