Circuits and Systems, 2013, 4, 451-458
Published Online November 2013 (
Open Access CS
High Accurate Howland Current Source: Output
Constraints Analysis
Pedro Bertemes-Filho*, Alexandre Felipe, Volney C. Vincence
Department of Electrical Engineering, State University of Santa Catarina (UDESC), Joinville, Brazil
Email: *
Received July 24, 2013; revised August 24, 2013; accepted August 31, 2013
Copyright © 2013 Pedro Bertemes-Filho et al. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Howland circuits have been widely used as powerful source for exciting tissue over a wide frequency range. When a
Howland source is designed, the components are chosen so that the designed source has the desired characteristics.
However, the operational amplifier limitations and resistor tolerances cause undesired behaviors. This work proposes to
take into account the influence of the random distribution of the resistors in the modified Howland circuit over the fre-
quency range of 10 Hz to 10 MHz. Both output current and impedance of the circuit are deduced either considering or
the operational amplifiers parameters. The probability density function due to small changes in the resistors of the cir-
cuit was calculated by using the analytical modeling. Results showed that both output current and impedance are very
sensitive to the resistors variations. In order to get higher output impedances, high operational amplifier gains are re-
quired. The operational amplifier open-loop gain increases as increasing the sensitivity of the output impedance. The
analysis done in this work can be used as a powerful co-adjuvant tool when projecting this type of circuit in Spice
simulators. This might improve the implementations of practical current sources used in electrical bioimpedance.
Keywords: Howland Current Source; Electrical Bioimpedance; Probability Density Function; Resistors Mismatching
1. Introduction
Because of its simplicity, stability and other advantages,
the voltage controlled current source (VCCS) has been wi-
dely used in many applications, such as in neuron-stimu-
lation systems [1-3], single-electrode capacitive sensors
[4], electrical impedance tomography (EIT) systems both
for industrial and medical applications [5,6] and bioim-
pedance analysis (BIA) for tissue characterization [7,8].
It is also been used for exciting tissue for cancer charac-
terization in electrical impedance spectroscopy [9-11].
Most VCCS circuits in BIA use the Howland current
source (HCS) [12]. The first HCS circuit was proposed
by Howland in 1962 [13] for converting a voltage into a
current. However, it suffers from output voltage compli-
ance [7]. Therefore, the modified version of the circuit
has been proposed and widely used [14,15].
The most important requirement in EIS systems is to
assure that the injecting current, also called source cur-
rent, has constant amplitude over a wide frequency range,
which should be obtained by a high output impedance
circuit [7]. However, stray capacitances [4] and non-
idealities of the operational amplifiers used for the design
[15] reduce the current amplitude and introduce phase
shift errors at higher frequencies. However, some of
these requirements are essentially in conflict with each
other. For example, the frequency of the current source is
practically limited to 100 kHz when the output imped-
ance is required to be sufficiently larger, say larger than 1
M [1]. In most publications, the analysis of the circuit
is based on either simplified ideal opamp circuits or us-
ing simulation tools, e.g. PSpice or Multisim from NI
[12,16]. In many cases, however, the formulas they used
are not suitable because the calculation errors are too
large with a high-frequency current source [16]. Fur-
thermore, the formulas do not take into account the mis-
match between the resistors used to design such a current
source. The HCS circuit is sensitive upon this mismatch-
ing [7].
The objective of this work is to investigate the prob-
ability density function for the analysis of the Howland
circuit. It also investigates the sensibility of the Howland
current source over the frequency range of 10 Hz to 10
MHz due to value mismatching between resistors.
*Corresponding author.
Figure 1. Schematic diagram of the current source with
grounded load [17], where Vd is the differential voltage ac-
ross the input terminals.
2. Howland Current Source
Figure 1 shows the modified Howland current source
(HCS) circuit used for modeling the output characteris-
tics taking into account the effects of the resistors toler-
ances. If all resistors are perfectly matched and the op-
erational amplifier (opamp) has got a large gain, then the
output current is given by –Vin/r (assuming R2 = R3 = R5
= R, R4 = r and R1 = R + r). This approximation yields a
good result if the Opamp gain is sufficiently high and the
load impedance is small.
In order to get good Common Mode Rejection Ratio
(CMRR) and high output impedance it is necessary to
trim the resistors and choose FET inputs amplifiers. On
the other hand, the non-ideal characteristics of the opera-
tional amplifiers (i.e., input impedance Zin, output resis-
tance Ro and open-loop gain A) reduce frequency band-
width of both output current and impedance.
2.1. Circuit Nodal Equations
The circuit can be characterized by the nodal Equations
(1)-(4), where V1 is the voltage at non-inverting input of
the Opamp, V2 is the inverting input of the Opamp and V3
is the voltage at the output of the Opamp. After deter-
mining the voltages the output current is calculated by
Equation (4). The transconductance gain is calculated by
Iout/Vin with Vout grounded. On the other hand, the output
impedance Zout is calculated by Iout/Vout with Vin groun-
1 out112
  (1)
2in 21
  (2)
3 23out
3out1 out
 (4)
AAj j
where Ao and ωco are the open-loop gain modulus at zero
frequency and the open-loop gain corner frequency, res-
in in
Zj RjC
where Rin and Cin are both input resistance and capaci-
tance of the opamp, respectively.
The Norton theorem can be applied to the circuit
shown in Figure 1 then the shunt impedance ZN of the
circuit can be calculated by Equation (7) and the trans-
conductance gain GN can be calculated by Equation (8).
out out0
 (7)
out in0
3. Probability Distribution Function
The probability distribution functions (PDF) of the
source parameters are calculated by classifying and
counting the elements falling in each class. It is expected
that any combination of resistors of ±1% tolerance (δ)
may produce either significant reduction in the output
impedance or a large variation in the transconductance
gain (Iout/Vin).
The box below shows the pseudo-code program de-
veloped in a very high level for the determination of some
statistics of the circuit, which was projected by using re-
sistors whose values distribution are known.
= empty lis
or the desired number of samples
- R1 = random resistor(R + r, tolerance of R1)
- R2 = random resistor(R, tolerance of R2)
- R3 = random resistor(R, tolerance of R3)
- R4 = random resistor(r, tolerance of R4)
- R5 = random resistor(R, tolerance of R5)
- H.append(Howland Circuit(R1, R2, R3, R4, R5, selected OPA))
stimate Zout distribution of H[]
stimate Iout distribution of H[]
Calculate other parameters of H[]
This program was implemented in R scripting lan-
guage, which is a language and environment for statisti-
cal computing and graphics. In this work the resistors
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were distributed uniformly by using the run if R function,
and the estimation of probability density functions was
made with the density R function. More detailed infor-
mation can be seen in the link
A Howland current source is designed by using R =
100 k, r = 1 k and it is assumed that each resistor has
a tolerance δ of ±1% with no correlation to each other.
It was investigated the effect of the tolerance of the re-
sistors in the transconductance gain when designing a
Howland current source, Equation (4) was used by as-
suming a tolerance of ±1% for each discrete frequency
over the range 10 Hz to 1 MHz. It was used the opera-
tional amplifier OPA655 from Texas Instruments, where
the technical specifications are shown in Table 1.
The result is shown in Figure 2, where the probability
density is represented by a gray scale. The white points
represent the values where the transconductance might
not be achieved at that frequency. It can also be seen that
the frequency dependency mater is much more relevant
than the tolerance of the resistors.
The transconductance gain GN can be approximately
calculated by the ratio –R3/(R2R4), thus if the tolerance of
the resistors are equal to δ, the tolerance of the transcon-
ductance gain will have a tolerance of approximately 3δ.
It implies that the use of a wideband operational ampli-
fier will produce a very stable transconductance gain. On
the other hand, the output impedance Zout will be signifi-
cantly changed by the tolerance of the resistors.
Figure 2. Probability distribution of the transconductance
gain, using the OPA655 and resistors of ±1% tolerance.
Table 1. The main technical specifications of the operational
amplifiers used in this work.
Rin (M) Cin (pF) R0 () A0 (dB) ωc0 (πHz)
OPA657 1000 0.7 0.02 75 600,000
OPA656 1000 0.7 0.01 66 200,000
OPA655 1000 1.2 0.04 58 400,000
LMH6654 4 1.8 0.08 80 50,000
OP07C 50 NF 60 115 4
TL081 1000 3.0 10 105 50
uA741 2 1.4 75 97 30
Note that A0 of the opamp has been extracted from the plots of the open-loop
gain versus frequency presented in the datasheet.
4. Output Impedance Modeling
4.1. Model 1: Infinite Open-Loop Gain
By solving Equation (4) and consider a suficiently high
open-loop gain A, the output impedance can be briefly
calculated by Equation (9)
245 124
2524 13
 (9)
It can be seen in Equation (9) that the relative error of
the numerator is approximately theree times larger than
the resistor tolerances. The denominator of the equation
would be zero if the resistors were perfectly matched.
Therefore, the output would be very dependent on their
values. By assuming equal tolerances for all resistors
with a fraction nominal value of each resitor, and also
using an operational amplifier with high open-loop gain
A, the minimum ouput impedance can be calculated by
Equation (10).
 
out,min 2
where R = R2 = R3 = R5, r = R4 and R + r = R1.
It can be seen in Equation (10) that if R is much
greater than r and assuming equal tolerances for all resis-
tors, then variations on the resistor R4 (=r) become much
less signicant for the output impedance compared to the
variations on other resistors.
4.2. Model 2: Finite Open-Loop Gain
By sampling randomly the resistors to fall next to theirs
nominal values and calculating Zout, it would imply that
we are assigning to Zout randomly values which are se-
lected near by a pole of a function. It means that high
values can be obtained as well as the small ones, which,
in turns, are much more difficult to happen. Furthermore,
as it can be seen in Figure 3, the output impedance has a
finite upper limit.
Both Vout and Iout can be written by two polynomial
functions, which are both related to A, Zin, Ro and the
resistors. Considering Zin sufficiently large and Ro suffi-
ciently small, Zout can be calculated according to Equa-
tion (11) as a function of the open-loop gain A.
211 23
213231 22
 (11)
where RE2 = R2(R4 + R5) and RE1 = R4(R1 + R5).
Perturbations in the coefficients of the gain A in both
numerator and denominator of Equation (11) may lead it
to negative values and then to instability. Therefore, they
are much more important than the independent terms.
The numerator can be approximately bounded by r(A + 2)
R(2R + r)(1 ± 3δ), by assuming R = R2 = R3 = R5, r = R4
and R + r = R1. On the other hand, the coefficient of the
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Figure 3. Probability distribution of the output impedance,
using the OPA655 and resistor tolerances of ±1%.
gain A in the denominator of Equation (11) may assume
both positive and negative values in the range of 2δR(R
+ r) to +2δR(R + r), and the independent term is 4R(R +
r)(1 + 2δ).
By considering small perturbations, Zout can be ap-
proximated by A/(a1A + a0), where the denominator of
this expression in the complex plane is a straight line
crossing a0 with an angle θ(A). Therefore, it can be cal-
culated that the modulus of Zout is inversely proportional
to the distance from a point in that line to the origin and
the phase is the difference between the arguments of such
a point and the gain A. As a result, the modulus of Zout
can be written as a function of its phase, as shown in
Equation (12).
2rR rR
aRrR Z
1254 1
aRRRRR (14)
 A
By doing a simple geometric analysis in Equation (12),
it can be observed that the distribution of Zout in the com-
plex plane results into a circle centered at the imaginary
axis with one of its vertices at the origin. It can be con-
cluded that small Zout has a phase near zero or 180 de-
grees, as the resistors are combined to give a higher Zout
and then the phase tends to 90 degrees, as shown in
Equation (16) by assuming that the Equation (17) is sat-
rR rR
25 4130
 
By considering negligible variations in the denomina-
tor of Equation (12) and also assuming only real values
for the coefficient a0 (see Equation 13), the Zout phase can
be given by “θ(a1 + a0/A)” and the modulus by “rR(2R
+ r)/|a1 + a0|”. The term “a1 + a0/A” represents a hori-
zontal line in the complex plane, so that the product be-
tween “|a1 + a0/A|” and “sin[θ(a1 + a0/A)]” is always con-
stant. Therefore, Zout can be approximated by a circum-
ference of radius |Zout,max|/2 according to Equation (18).
out out,maxout
sinZZ Z
Figure 4 shows both real and imaginary parts of Zout
calculated by using a set of random resistors (r = 1 k ±
0.1% and R = 100 k ± 0.1%) and the operational ampli-
fiers TL081 (graphics 1, 2 and 3), uA741 (graphics 4, 5
and 6) and OPA655 (graphics 7, 8 and 9), according to
the parameters shown in Table 1. Simulations were made
with and without the output resistance R0. The red points
were calculated values by using randomly values for the
resistors in the range +0.1% and 0.1%. The blue circle
represents the geometric space in the complex plan
which contains the calculated points according to Equa-
tion 12. The simulations show the results at three differ-
ent frequencies: 10 KHz (first column); 100 kHz (second
column); and 1 MHz (third column). It can be seen that
the MMHCS circuit designed by using the OPA655 has
the highest output impedance, which is approximately
1.02 M at 10 kHz. It can also be seen that the probabil-
ity of getting the maximum output impedance does de-
pend on the frequency and opamp used which, in turns,
depends on the open-loop gain corner frequency. The
graphic 6 shows that Zout is equal to zero at 1 MHz when
using the opamp uA741.
Figure 5 shows the output impedance of the MMHCS
designed by the operational amplifiers TL081 (first col-
umn) and OPA655 (second column) at 10 kHz, using
resistor tolerances of 0.1% (graphics “a” and “b”), 1%
(graphics “c” and “d”) and 10% (graphics “e” and “f”). It
can be calculated that the maximum Zout is approximately
213.6 k when using the opamp TL081 whereas 1.06
M for the OPA655 one, considering resistor tolerances
of 0.1%. However, the probability of getting such a value
is much smaller for the OPA655 case. It can also be seen
that the probability density on getting maximum Zout
increases significantly as increasing the resistor toler-
5. Mirrored Howland Current Source:
In order to improve the stability of the circuit, two sin-
gle-ended Howland circuits can be set together in order
to have a differential output rrent, which is also called cu
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Figure 4. The real and imaginary part of the output impedance by using resistor tolerance of 0.1%.
Figure 5. Changes in the output impedance at 10 kHz by using different resistor tolerances.
as Mirrored Modified Howland Current Source
(MMHCS) and it is shown in Figure 6 [17]. This type of
current source uses two equal input voltages with 180
degrees phase shift from each other but also has a unique
reference for both sides of the circuit, which, in turns,
reduce the feed-through capacitance between output and
input. This topology may significantly reduce the even
harmonics of the output current, and then improving its
linearity and reducing the amount of output voltage level
at each side of this current source.
The MMHCS can be represented by its Norton equiva-
lent circuit, as shown in Figure 7. The output voltage at
each side of the load can then be calculated according to
Equations (19) and (20), where IN1 and IN2 are the short-
circuit currents and ZN1 and ZN2 are the output impedance
of each side of the circuit.
Figure 6. Schematic diagram of the mirrored modified
Howland current source.
Figure 7. Equivalent Norton circuit for the mirrored modi-
fied Howland current source.
122 221
 (19)
 
 (20)
The output current in the load impedance ZL can then
be calculated according to Equation (21). As a result, the
differential output voltage Vdiff can be calculated by
Equation (22).
112 2
 (21)
12 1212
 (22)
where VN1 = ZN1IN1 and VN2 = ZN2IN2.
It is important to note that if the output impedances ZN1
and ZN2 have got similar high values, then it is expected
that the difference ZN1 ZN2 be very small whereas the
sum ZN1 + ZN2 be very high. It can be seen in (22) that the
differential voltage Vdiff reduces to the product between
(IN1 IN2) and ZN1,2, for example if the output impedances
of the current source are hundred times larger than the
load impedance. A difference of 1% in IN1 or IN2 can du-
plicate the voltage in one of the load terminals. The dif-
ference at the output which is seeing by each terminal
leads to a common mode voltage, which is dependent of
the load charge. This results from the fact that each side
of the current source tries to fix the output current to a
different value.
6. Discussions
6.1. Output Impedance
It is predicted by Equation (12) that if the open-loop gain
A is large, then it is very difficult to set the output im-
pedance to its optimal value. It was showed that as de-
creasing the gain A the size of the impedance circle is
also decreased and then the region away from the origin
become more statistically populated, as shown in Figure
Researches have been using the differential (mirrored)
output current sources for getting a more stable circuit
and higher output impedance. The output impedance of
this type of current source can be calculated by sum of
the output impedances from both single-ended circuits
(i.e. ZN1 + ZN2). It is thought having only advantage rea-
sons for doing that but it also suffers from high
open-loop gain of the operational amplifiers. The real
part of Zout can be either positive or negative and its
imaginary part very small, then leading to a differential
Zout even smaller than the single-ended output imped-
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6.2. Operational Amplifier Limitations
As discussed before, from (22) it can be seen that small
current differences at each side of the mirrored current
source can increase the common mode voltage at the load
terminals. In order to preserve those differences in the
output current, the operational amplifiers have to supply
the differential voltage as well as the common mode
The output impedance of the current source must be
high otherwise the load current varies as varying the load
impedance. By designing a high output impedance cur-
rent source, the transconductance gains have to be pre-
cisely matched for preventing the operational amplifiers
to work one against each other. This might be explained
by the fact that they are connected in series, and then
both circuits are designed to reject any variation in its
output current.
6.3. Tuning the Mirrored Current Source
Tuning the mirrored Howland current source is not a
simple task as it has to be done in both side of the circuit
at the same time.
It was shown that the transconductance gain can be
approximately given by the ratio between the resistor -R3
and the product “R2R4”. Consequently, it can be inter-
preted as a current gain, where instead of having an input
voltage it has an input current (=Vin/R2). As a result, both
branches of the circuit which contains the resistors R2
(see Figure 6) should be connected in series in order to
prevent input current differences at each side of the cur-
rent source. Therefore, it can be used only one resistor R2
for both source sides.
In order to have equal output characteristics at both
sides of the mirrored modified Howland current source,
both sides have to have an equal ratio of R3/R4. This
might be solved by adding a potentiometer in series with
one of R4 for tuning the differential output current. In
practice, this can be done by short-circuiting the output
and measuring the output voltage while tuning the poten-
tiometer to have a null output voltage. It might be neces-
sary to connect a resistor at each side of the short-circuit
in order to prevent saturation by the operational amplifi-
By the fact that the resistor R4 has a very small influ-
ence in the output impedance, this last can be tuned by
varying either the resistor R5 or R1. The resistor R1 of
both sides can also be connected in series in order to re-
duce imbalances from these circuit sides, thus one of the
resistors R5 becomes the natural tuning element for con-
trolling the output impedance of the mirrored modified
Howland current source.
6.4. Limitations
The analytical analyses and the equations related to the
output of the Howland current source have not consid-
ered any stray capacitances which might be found in
practical circuits. Also, the numerical solutions for both
output current and impedance of the circuit have not
considered the common-mode input impedance of the
operational amplifiers, as well as the common-mode re-
jection ratio (CMRR). It has to be pointed out that the
probability density function was used by assuming that
the resistor variations behave as a normal distribution.
Therefore, care should be taken when analyzing the re-
sults obtained in this work, especially when they are re-
lated to practical circuits.
7. Conclusions
It was fully described the modeling of both output cur-
rent and impedance of the Howland current source by
considering the operational amplifier parameters and the
mismatching between electrodes. It was shown that the
output resistance of the operational amplifier does not
play a role in the output characteristic of the Howland
circuit whereas the open-loop gain causes a great impact
on it.
It was also shown that the higher the gain, the most
sensitive is the output impedance in respect to the resis-
tors tolerances, as illustrated in Figures 4 and 5. In order
to operate at high frequency, the open-loop gain of the
opamp has to be very high. This explain why the resistors
have to be precisely matched in order to obtain a
MMHCS circuit which can deliver a constant differential
output current into the load over a wide frequency range.
It can be concluded from this work that both side of
the MMHCS should be very symmetrical if high output
impedance from this type of circuit is desired. This is a
very significant achievement found in this work for the
design of high output impedance Howland current sources
used in bioimpedance measurements. This may lead to
empirical distribution of important characteristics of the
MMCHS during the design and it might save time and
money when this type of circuit is put in production scale.
Furthermore, the error estimation provided here can
guide researchers, who have no previous knowledge in
Howland current source, to design a high quality circuit
in terms of both output current and impedance spectra
according to the application requirements.
8. Acknowledgements
This work was supported by the State University of Santa
Catarina and the National Council for Scientific and
Technological Development (grant 237931/2012-5).
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